J, 32 bytes

[:(5&e.*12&e.*17={:)[:+/\1+0=7|]

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Straightforward:

• Take list of digits
• 7|] mod 7
• 0= equals 0 (returns 1-0 list)
• 1+ add 1 (now list of 1-2)
• [:+/\ scan sum
• [:(5&e.*12&e.*17={:) is 5 an elm and is 12 an elm and is 17 the last?

J, ³² 30 bytes

4=5 12 17+/@e.~[:+/\0,~1+0=7|]

-2 thanks to xash

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original explanation

[:(5&e.*12&e.*17={:)[:+/\1+0=7|]

Try it online!

Straightforward:

• Take list of digits
• 7|] mod 7
• 0= equals 0 (returns 1-0 list)
• 1+ add 1 (now list of 1-2)
• [:+/\ scan sum
• [:(5&e.*12&e.*17={:) is 5 an elm and is 12 an elm and is 17 the last?

J, 36 bytes

3=[:(5 12&e.+/@,17={:)[:+/\1+e.&'07'

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still golfing

J, 32 bytes

[:(5&e.*12&e.*17={:)[:+/\1+0=7|]

Try it online!

Straightforward:

• Take list of digits
• 7|] mod 7
• 0= equals 0 (returns 1-0 list)
• 1+ add 1 (now list of 1-2)
• [:+/\ scan sum
• [:(5&e.*12&e.*17={:) is 5 an elm and is 12 an elm and is 17 the last?