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Theorem lcmfun 15764
Description: The lcm function for a union of sets of integers. (Contributed by AV, 27-Aug-2020.)
Assertion
Ref Expression
lcmfun (((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) ∧ (𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin)) → (lcm‘(𝑌𝑍)) = ((lcm𝑌) lcm (lcm𝑍)))

Proof of Theorem lcmfun
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cleq1lem 14130 . . . . . 6 (𝑥 = ∅ → ((𝑥 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) ↔ (∅ ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))))
2 uneq2 3983 . . . . . . . . 9 (𝑥 = ∅ → (𝑌𝑥) = (𝑌 ∪ ∅))
3 un0 4192 . . . . . . . . 9 (𝑌 ∪ ∅) = 𝑌
42, 3syl6eq 2829 . . . . . . . 8 (𝑥 = ∅ → (𝑌𝑥) = 𝑌)
54fveq2d 6450 . . . . . . 7 (𝑥 = ∅ → (lcm‘(𝑌𝑥)) = (lcm𝑌))
6 fveq2 6446 . . . . . . . . 9 (𝑥 = ∅ → (lcm𝑥) = (lcm‘∅))
7 lcmf0 15753 . . . . . . . . 9 (lcm‘∅) = 1
86, 7syl6eq 2829 . . . . . . . 8 (𝑥 = ∅ → (lcm𝑥) = 1)
98oveq2d 6938 . . . . . . 7 (𝑥 = ∅ → ((lcm𝑌) lcm (lcm𝑥)) = ((lcm𝑌) lcm 1))
105, 9eqeq12d 2792 . . . . . 6 (𝑥 = ∅ → ((lcm‘(𝑌𝑥)) = ((lcm𝑌) lcm (lcm𝑥)) ↔ (lcm𝑌) = ((lcm𝑌) lcm 1)))
111, 10imbi12d 336 . . . . 5 (𝑥 = ∅ → (((𝑥 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm‘(𝑌𝑥)) = ((lcm𝑌) lcm (lcm𝑥))) ↔ ((∅ ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm𝑌) = ((lcm𝑌) lcm 1))))
12 cleq1lem 14130 . . . . . 6 (𝑥 = 𝑦 → ((𝑥 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) ↔ (𝑦 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))))
13 uneq2 3983 . . . . . . . 8 (𝑥 = 𝑦 → (𝑌𝑥) = (𝑌𝑦))
1413fveq2d 6450 . . . . . . 7 (𝑥 = 𝑦 → (lcm‘(𝑌𝑥)) = (lcm‘(𝑌𝑦)))
15 fveq2 6446 . . . . . . . 8 (𝑥 = 𝑦 → (lcm𝑥) = (lcm𝑦))
1615oveq2d 6938 . . . . . . 7 (𝑥 = 𝑦 → ((lcm𝑌) lcm (lcm𝑥)) = ((lcm𝑌) lcm (lcm𝑦)))
1714, 16eqeq12d 2792 . . . . . 6 (𝑥 = 𝑦 → ((lcm‘(𝑌𝑥)) = ((lcm𝑌) lcm (lcm𝑥)) ↔ (lcm‘(𝑌𝑦)) = ((lcm𝑌) lcm (lcm𝑦))))
1812, 17imbi12d 336 . . . . 5 (𝑥 = 𝑦 → (((𝑥 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm‘(𝑌𝑥)) = ((lcm𝑌) lcm (lcm𝑥))) ↔ ((𝑦 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm‘(𝑌𝑦)) = ((lcm𝑌) lcm (lcm𝑦)))))
19 cleq1lem 14130 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → ((𝑥 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) ↔ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))))
20 uneq2 3983 . . . . . . . 8 (𝑥 = (𝑦 ∪ {𝑧}) → (𝑌𝑥) = (𝑌 ∪ (𝑦 ∪ {𝑧})))
2120fveq2d 6450 . . . . . . 7 (𝑥 = (𝑦 ∪ {𝑧}) → (lcm‘(𝑌𝑥)) = (lcm‘(𝑌 ∪ (𝑦 ∪ {𝑧}))))
22 fveq2 6446 . . . . . . . 8 (𝑥 = (𝑦 ∪ {𝑧}) → (lcm𝑥) = (lcm‘(𝑦 ∪ {𝑧})))
2322oveq2d 6938 . . . . . . 7 (𝑥 = (𝑦 ∪ {𝑧}) → ((lcm𝑌) lcm (lcm𝑥)) = ((lcm𝑌) lcm (lcm‘(𝑦 ∪ {𝑧}))))
2421, 23eqeq12d 2792 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → ((lcm‘(𝑌𝑥)) = ((lcm𝑌) lcm (lcm𝑥)) ↔ (lcm‘(𝑌 ∪ (𝑦 ∪ {𝑧}))) = ((lcm𝑌) lcm (lcm‘(𝑦 ∪ {𝑧})))))
2519, 24imbi12d 336 . . . . 5 (𝑥 = (𝑦 ∪ {𝑧}) → (((𝑥 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm‘(𝑌𝑥)) = ((lcm𝑌) lcm (lcm𝑥))) ↔ (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm‘(𝑌 ∪ (𝑦 ∪ {𝑧}))) = ((lcm𝑌) lcm (lcm‘(𝑦 ∪ {𝑧}))))))
26 cleq1lem 14130 . . . . . 6 (𝑥 = 𝑍 → ((𝑥 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) ↔ (𝑍 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))))
27 uneq2 3983 . . . . . . . 8 (𝑥 = 𝑍 → (𝑌𝑥) = (𝑌𝑍))
2827fveq2d 6450 . . . . . . 7 (𝑥 = 𝑍 → (lcm‘(𝑌𝑥)) = (lcm‘(𝑌𝑍)))
29 fveq2 6446 . . . . . . . 8 (𝑥 = 𝑍 → (lcm𝑥) = (lcm𝑍))
3029oveq2d 6938 . . . . . . 7 (𝑥 = 𝑍 → ((lcm𝑌) lcm (lcm𝑥)) = ((lcm𝑌) lcm (lcm𝑍)))
3128, 30eqeq12d 2792 . . . . . 6 (𝑥 = 𝑍 → ((lcm‘(𝑌𝑥)) = ((lcm𝑌) lcm (lcm𝑥)) ↔ (lcm‘(𝑌𝑍)) = ((lcm𝑌) lcm (lcm𝑍))))
3226, 31imbi12d 336 . . . . 5 (𝑥 = 𝑍 → (((𝑥 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm‘(𝑌𝑥)) = ((lcm𝑌) lcm (lcm𝑥))) ↔ ((𝑍 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm‘(𝑌𝑍)) = ((lcm𝑌) lcm (lcm𝑍)))))
33 lcmfcl 15747 . . . . . . . . . 10 ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → (lcm𝑌) ∈ ℕ0)
3433nn0zd 11832 . . . . . . . . 9 ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → (lcm𝑌) ∈ ℤ)
35 lcm1 15729 . . . . . . . . 9 ((lcm𝑌) ∈ ℤ → ((lcm𝑌) lcm 1) = (abs‘(lcm𝑌)))
3634, 35syl 17 . . . . . . . 8 ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → ((lcm𝑌) lcm 1) = (abs‘(lcm𝑌)))
37 nn0re 11652 . . . . . . . . . . 11 ((lcm𝑌) ∈ ℕ0 → (lcm𝑌) ∈ ℝ)
38 nn0ge0 11669 . . . . . . . . . . 11 ((lcm𝑌) ∈ ℕ0 → 0 ≤ (lcm𝑌))
3937, 38jca 507 . . . . . . . . . 10 ((lcm𝑌) ∈ ℕ0 → ((lcm𝑌) ∈ ℝ ∧ 0 ≤ (lcm𝑌)))
4033, 39syl 17 . . . . . . . . 9 ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → ((lcm𝑌) ∈ ℝ ∧ 0 ≤ (lcm𝑌)))
41 absid 14443 . . . . . . . . 9 (((lcm𝑌) ∈ ℝ ∧ 0 ≤ (lcm𝑌)) → (abs‘(lcm𝑌)) = (lcm𝑌))
4240, 41syl 17 . . . . . . . 8 ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → (abs‘(lcm𝑌)) = (lcm𝑌))
4336, 42eqtrd 2813 . . . . . . 7 ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → ((lcm𝑌) lcm 1) = (lcm𝑌))
4443adantl 475 . . . . . 6 ((∅ ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → ((lcm𝑌) lcm 1) = (lcm𝑌))
4544eqcomd 2783 . . . . 5 ((∅ ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm𝑌) = ((lcm𝑌) lcm 1))
46 unass 3992 . . . . . . . . . . . . . 14 ((𝑌𝑦) ∪ {𝑧}) = (𝑌 ∪ (𝑦 ∪ {𝑧}))
4746eqcomi 2786 . . . . . . . . . . . . 13 (𝑌 ∪ (𝑦 ∪ {𝑧})) = ((𝑌𝑦) ∪ {𝑧})
4847a1i 11 . . . . . . . . . . . 12 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (𝑌 ∪ (𝑦 ∪ {𝑧})) = ((𝑌𝑦) ∪ {𝑧}))
4948fveq2d 6450 . . . . . . . . . . 11 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (lcm‘(𝑌 ∪ (𝑦 ∪ {𝑧}))) = (lcm‘((𝑌𝑦) ∪ {𝑧})))
50 simpl 476 . . . . . . . . . . . . . . 15 ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → 𝑌 ⊆ ℤ)
5150adantl 475 . . . . . . . . . . . . . 14 (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → 𝑌 ⊆ ℤ)
52 unss 4009 . . . . . . . . . . . . . . . 16 ((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) ↔ (𝑦 ∪ {𝑧}) ⊆ ℤ)
53 simpl 476 . . . . . . . . . . . . . . . 16 ((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) → 𝑦 ⊆ ℤ)
5452, 53sylbir 227 . . . . . . . . . . . . . . 15 ((𝑦 ∪ {𝑧}) ⊆ ℤ → 𝑦 ⊆ ℤ)
5554adantr 474 . . . . . . . . . . . . . 14 (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → 𝑦 ⊆ ℤ)
5651, 55unssd 4011 . . . . . . . . . . . . 13 (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (𝑌𝑦) ⊆ ℤ)
5756adantl 475 . . . . . . . . . . . 12 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (𝑌𝑦) ⊆ ℤ)
58 unfi 8515 . . . . . . . . . . . . . . . 16 ((𝑌 ∈ Fin ∧ 𝑦 ∈ Fin) → (𝑌𝑦) ∈ Fin)
5958ex 403 . . . . . . . . . . . . . . 15 (𝑌 ∈ Fin → (𝑦 ∈ Fin → (𝑌𝑦) ∈ Fin))
6059adantl 475 . . . . . . . . . . . . . 14 ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → (𝑦 ∈ Fin → (𝑌𝑦) ∈ Fin))
6160adantl 475 . . . . . . . . . . . . 13 (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (𝑦 ∈ Fin → (𝑌𝑦) ∈ Fin))
6261impcom 398 . . . . . . . . . . . 12 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (𝑌𝑦) ∈ Fin)
63 vex 3400 . . . . . . . . . . . . . . . . . 18 𝑧 ∈ V
6463snss 4548 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ ℤ ↔ {𝑧} ⊆ ℤ)
6564biimpri 220 . . . . . . . . . . . . . . . 16 ({𝑧} ⊆ ℤ → 𝑧 ∈ ℤ)
6665adantl 475 . . . . . . . . . . . . . . 15 ((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) → 𝑧 ∈ ℤ)
6752, 66sylbir 227 . . . . . . . . . . . . . 14 ((𝑦 ∪ {𝑧}) ⊆ ℤ → 𝑧 ∈ ℤ)
6867adantr 474 . . . . . . . . . . . . 13 (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → 𝑧 ∈ ℤ)
6968adantl 475 . . . . . . . . . . . 12 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → 𝑧 ∈ ℤ)
70 lcmfunsn 15763 . . . . . . . . . . . 12 (((𝑌𝑦) ⊆ ℤ ∧ (𝑌𝑦) ∈ Fin ∧ 𝑧 ∈ ℤ) → (lcm‘((𝑌𝑦) ∪ {𝑧})) = ((lcm‘(𝑌𝑦)) lcm 𝑧))
7157, 62, 69, 70syl3anc 1439 . . . . . . . . . . 11 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (lcm‘((𝑌𝑦) ∪ {𝑧})) = ((lcm‘(𝑌𝑦)) lcm 𝑧))
7249, 71eqtrd 2813 . . . . . . . . . 10 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (lcm‘(𝑌 ∪ (𝑦 ∪ {𝑧}))) = ((lcm‘(𝑌𝑦)) lcm 𝑧))
7372adantr 474 . . . . . . . . 9 (((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) ∧ ((𝑦 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm‘(𝑌𝑦)) = ((lcm𝑌) lcm (lcm𝑦)))) → (lcm‘(𝑌 ∪ (𝑦 ∪ {𝑧}))) = ((lcm‘(𝑌𝑦)) lcm 𝑧))
7454anim1i 608 . . . . . . . . . . . . 13 (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (𝑦 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)))
7574adantl 475 . . . . . . . . . . . 12 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (𝑦 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)))
76 id 22 . . . . . . . . . . . 12 (((𝑦 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm‘(𝑌𝑦)) = ((lcm𝑌) lcm (lcm𝑦))) → ((𝑦 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm‘(𝑌𝑦)) = ((lcm𝑌) lcm (lcm𝑦))))
7775, 76mpan9 502 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) ∧ ((𝑦 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm‘(𝑌𝑦)) = ((lcm𝑌) lcm (lcm𝑦)))) → (lcm‘(𝑌𝑦)) = ((lcm𝑌) lcm (lcm𝑦)))
7877oveq1d 6937 . . . . . . . . . 10 (((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) ∧ ((𝑦 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm‘(𝑌𝑦)) = ((lcm𝑌) lcm (lcm𝑦)))) → ((lcm‘(𝑌𝑦)) lcm 𝑧) = (((lcm𝑌) lcm (lcm𝑦)) lcm 𝑧))
7934adantl 475 . . . . . . . . . . . . 13 (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm𝑌) ∈ ℤ)
8079adantl 475 . . . . . . . . . . . 12 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (lcm𝑌) ∈ ℤ)
8155anim2i 610 . . . . . . . . . . . . . . 15 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (𝑦 ∈ Fin ∧ 𝑦 ⊆ ℤ))
8281ancomd 455 . . . . . . . . . . . . . 14 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin))
83 lcmfcl 15747 . . . . . . . . . . . . . 14 ((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm𝑦) ∈ ℕ0)
8482, 83syl 17 . . . . . . . . . . . . 13 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (lcm𝑦) ∈ ℕ0)
8584nn0zd 11832 . . . . . . . . . . . 12 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (lcm𝑦) ∈ ℤ)
86 lcmass 15733 . . . . . . . . . . . 12 (((lcm𝑌) ∈ ℤ ∧ (lcm𝑦) ∈ ℤ ∧ 𝑧 ∈ ℤ) → (((lcm𝑌) lcm (lcm𝑦)) lcm 𝑧) = ((lcm𝑌) lcm ((lcm𝑦) lcm 𝑧)))
8780, 85, 69, 86syl3anc 1439 . . . . . . . . . . 11 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (((lcm𝑌) lcm (lcm𝑦)) lcm 𝑧) = ((lcm𝑌) lcm ((lcm𝑦) lcm 𝑧)))
8887adantr 474 . . . . . . . . . 10 (((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) ∧ ((𝑦 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm‘(𝑌𝑦)) = ((lcm𝑌) lcm (lcm𝑦)))) → (((lcm𝑌) lcm (lcm𝑦)) lcm 𝑧) = ((lcm𝑌) lcm ((lcm𝑦) lcm 𝑧)))
8978, 88eqtrd 2813 . . . . . . . . 9 (((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) ∧ ((𝑦 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm‘(𝑌𝑦)) = ((lcm𝑌) lcm (lcm𝑦)))) → ((lcm‘(𝑌𝑦)) lcm 𝑧) = ((lcm𝑌) lcm ((lcm𝑦) lcm 𝑧)))
9073, 89eqtrd 2813 . . . . . . . 8 (((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) ∧ ((𝑦 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm‘(𝑌𝑦)) = ((lcm𝑌) lcm (lcm𝑦)))) → (lcm‘(𝑌 ∪ (𝑦 ∪ {𝑧}))) = ((lcm𝑌) lcm ((lcm𝑦) lcm 𝑧)))
9153adantr 474 . . . . . . . . . . . . . . . . 17 (((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) ∧ 𝑦 ∈ Fin) → 𝑦 ⊆ ℤ)
92 simpr 479 . . . . . . . . . . . . . . . . 17 (((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) ∧ 𝑦 ∈ Fin) → 𝑦 ∈ Fin)
9366adantr 474 . . . . . . . . . . . . . . . . 17 (((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) ∧ 𝑦 ∈ Fin) → 𝑧 ∈ ℤ)
9491, 92, 933jca 1119 . . . . . . . . . . . . . . . 16 (((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) ∧ 𝑦 ∈ Fin) → (𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ∧ 𝑧 ∈ ℤ))
9594ex 403 . . . . . . . . . . . . . . 15 ((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) → (𝑦 ∈ Fin → (𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ∧ 𝑧 ∈ ℤ)))
9652, 95sylbir 227 . . . . . . . . . . . . . 14 ((𝑦 ∪ {𝑧}) ⊆ ℤ → (𝑦 ∈ Fin → (𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ∧ 𝑧 ∈ ℤ)))
9796adantr 474 . . . . . . . . . . . . 13 (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (𝑦 ∈ Fin → (𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ∧ 𝑧 ∈ ℤ)))
9897impcom 398 . . . . . . . . . . . 12 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ∧ 𝑧 ∈ ℤ))
99 lcmfunsn 15763 . . . . . . . . . . . 12 ((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ∧ 𝑧 ∈ ℤ) → (lcm‘(𝑦 ∪ {𝑧})) = ((lcm𝑦) lcm 𝑧))
10098, 99syl 17 . . . . . . . . . . 11 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (lcm‘(𝑦 ∪ {𝑧})) = ((lcm𝑦) lcm 𝑧))
101100oveq2d 6938 . . . . . . . . . 10 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → ((lcm𝑌) lcm (lcm‘(𝑦 ∪ {𝑧}))) = ((lcm𝑌) lcm ((lcm𝑦) lcm 𝑧)))
102101eqeq2d 2787 . . . . . . . . 9 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → ((lcm‘(𝑌 ∪ (𝑦 ∪ {𝑧}))) = ((lcm𝑌) lcm (lcm‘(𝑦 ∪ {𝑧}))) ↔ (lcm‘(𝑌 ∪ (𝑦 ∪ {𝑧}))) = ((lcm𝑌) lcm ((lcm𝑦) lcm 𝑧))))
103102adantr 474 . . . . . . . 8 (((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) ∧ ((𝑦 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm‘(𝑌𝑦)) = ((lcm𝑌) lcm (lcm𝑦)))) → ((lcm‘(𝑌 ∪ (𝑦 ∪ {𝑧}))) = ((lcm𝑌) lcm (lcm‘(𝑦 ∪ {𝑧}))) ↔ (lcm‘(𝑌 ∪ (𝑦 ∪ {𝑧}))) = ((lcm𝑌) lcm ((lcm𝑦) lcm 𝑧))))
10490, 103mpbird 249 . . . . . . 7 (((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) ∧ ((𝑦 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm‘(𝑌𝑦)) = ((lcm𝑌) lcm (lcm𝑦)))) → (lcm‘(𝑌 ∪ (𝑦 ∪ {𝑧}))) = ((lcm𝑌) lcm (lcm‘(𝑦 ∪ {𝑧}))))
105104exp31 412 . . . . . 6 (𝑦 ∈ Fin → (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (((𝑦 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm‘(𝑌𝑦)) = ((lcm𝑌) lcm (lcm𝑦))) → (lcm‘(𝑌 ∪ (𝑦 ∪ {𝑧}))) = ((lcm𝑌) lcm (lcm‘(𝑦 ∪ {𝑧}))))))
106105com23 86 . . . . 5 (𝑦 ∈ Fin → (((𝑦 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm‘(𝑌𝑦)) = ((lcm𝑌) lcm (lcm𝑦))) → (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm‘(𝑌 ∪ (𝑦 ∪ {𝑧}))) = ((lcm𝑌) lcm (lcm‘(𝑦 ∪ {𝑧}))))))
10711, 18, 25, 32, 45, 106findcard2 8488 . . . 4 (𝑍 ∈ Fin → ((𝑍 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm‘(𝑌𝑍)) = ((lcm𝑌) lcm (lcm𝑍))))
108107expd 406 . . 3 (𝑍 ∈ Fin → (𝑍 ⊆ ℤ → ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → (lcm‘(𝑌𝑍)) = ((lcm𝑌) lcm (lcm𝑍)))))
109108impcom 398 . 2 ((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) → ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → (lcm‘(𝑌𝑍)) = ((lcm𝑌) lcm (lcm𝑍))))
110109impcom 398 1 (((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) ∧ (𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin)) → (lcm‘(𝑌𝑍)) = ((lcm𝑌) lcm (lcm𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1071   = wceq 1601  wcel 2106  cun 3789  wss 3791  c0 4140  {csn 4397   class class class wbr 4886  cfv 6135  (class class class)co 6922  Fincfn 8241  cr 10271  0cc0 10272  1c1 10273  cle 10412  0cn0 11642  cz 11728  abscabs 14381   lcm clcm 15707  lcmclcmf 15708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-inf2 8835  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349  ax-pre-sup 10350
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-pss 3807  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-se 5315  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-isom 6144  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-oadd 7847  df-er 8026  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-sup 8636  df-inf 8637  df-oi 8704  df-card 9098  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-div 11033  df-nn 11375  df-2 11438  df-3 11439  df-n0 11643  df-z 11729  df-uz 11993  df-rp 12138  df-fz 12644  df-fzo 12785  df-fl 12912  df-mod 12988  df-seq 13120  df-exp 13179  df-hash 13436  df-cj 14246  df-re 14247  df-im 14248  df-sqrt 14382  df-abs 14383  df-clim 14627  df-prod 15039  df-dvds 15388  df-gcd 15623  df-lcm 15709  df-lcmf 15710
This theorem is referenced by:  lcmfass  15765
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