Description: The divides relation is transitive, a deduction version of dvdstr . (Contributed by metakunt, 12-May-2024)
Ref | Expression | ||
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Hypotheses | dvdstrd.1 | |- ( ph -> K e. ZZ ) |
|
dvdstrd.2 | |- ( ph -> M e. ZZ ) |
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dvdstrd.3 | |- ( ph -> N e. ZZ ) |
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dvdstrd.4 | |- ( ph -> K || M ) |
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dvdstrd.5 | |- ( ph -> M || N ) |
||
Assertion | dvdstrd | |- ( ph -> K || N ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvdstrd.1 | |- ( ph -> K e. ZZ ) |
|
2 | dvdstrd.2 | |- ( ph -> M e. ZZ ) |
|
3 | dvdstrd.3 | |- ( ph -> N e. ZZ ) |
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4 | dvdstrd.4 | |- ( ph -> K || M ) |
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5 | dvdstrd.5 | |- ( ph -> M || N ) |
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6 | dvdstr | |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K || M /\ M || N ) -> K || N ) ) |
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7 | 1 2 3 6 | syl3anc | |- ( ph -> ( ( K || M /\ M || N ) -> K || N ) ) |
8 | 4 5 7 | mp2and | |- ( ph -> K || N ) |