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Theorem lcmfun 16521
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 14867 . . . . . 6 (𝑥 = ∅ → ((𝑥 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) ↔ (∅ ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))))
2 uneq2 4117 . . . . . . . . 9 (𝑥 = ∅ → (𝑌𝑥) = (𝑌 ∪ ∅))
3 un0 4350 . . . . . . . . 9 (𝑌 ∪ ∅) = 𝑌
42, 3eqtrdi 2792 . . . . . . . 8 (𝑥 = ∅ → (𝑌𝑥) = 𝑌)
54fveq2d 6846 . . . . . . 7 (𝑥 = ∅ → (lcm‘(𝑌𝑥)) = (lcm𝑌))
6 fveq2 6842 . . . . . . . . 9 (𝑥 = ∅ → (lcm𝑥) = (lcm‘∅))
7 lcmf0 16510 . . . . . . . . 9 (lcm‘∅) = 1
86, 7eqtrdi 2792 . . . . . . . 8 (𝑥 = ∅ → (lcm𝑥) = 1)
98oveq2d 7373 . . . . . . 7 (𝑥 = ∅ → ((lcm𝑌) lcm (lcm𝑥)) = ((lcm𝑌) lcm 1))
105, 9eqeq12d 2752 . . . . . 6 (𝑥 = ∅ → ((lcm‘(𝑌𝑥)) = ((lcm𝑌) lcm (lcm𝑥)) ↔ (lcm𝑌) = ((lcm𝑌) lcm 1)))
111, 10imbi12d 344 . . . . 5 (𝑥 = ∅ → (((𝑥 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm‘(𝑌𝑥)) = ((lcm𝑌) lcm (lcm𝑥))) ↔ ((∅ ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm𝑌) = ((lcm𝑌) lcm 1))))
12 cleq1lem 14867 . . . . . 6 (𝑥 = 𝑦 → ((𝑥 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) ↔ (𝑦 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))))
13 uneq2 4117 . . . . . . . 8 (𝑥 = 𝑦 → (𝑌𝑥) = (𝑌𝑦))
1413fveq2d 6846 . . . . . . 7 (𝑥 = 𝑦 → (lcm‘(𝑌𝑥)) = (lcm‘(𝑌𝑦)))
15 fveq2 6842 . . . . . . . 8 (𝑥 = 𝑦 → (lcm𝑥) = (lcm𝑦))
1615oveq2d 7373 . . . . . . 7 (𝑥 = 𝑦 → ((lcm𝑌) lcm (lcm𝑥)) = ((lcm𝑌) lcm (lcm𝑦)))
1714, 16eqeq12d 2752 . . . . . 6 (𝑥 = 𝑦 → ((lcm‘(𝑌𝑥)) = ((lcm𝑌) lcm (lcm𝑥)) ↔ (lcm‘(𝑌𝑦)) = ((lcm𝑌) lcm (lcm𝑦))))
1812, 17imbi12d 344 . . . . 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 14867 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → ((𝑥 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) ↔ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))))
20 uneq2 4117 . . . . . . . 8 (𝑥 = (𝑦 ∪ {𝑧}) → (𝑌𝑥) = (𝑌 ∪ (𝑦 ∪ {𝑧})))
2120fveq2d 6846 . . . . . . 7 (𝑥 = (𝑦 ∪ {𝑧}) → (lcm‘(𝑌𝑥)) = (lcm‘(𝑌 ∪ (𝑦 ∪ {𝑧}))))
22 fveq2 6842 . . . . . . . 8 (𝑥 = (𝑦 ∪ {𝑧}) → (lcm𝑥) = (lcm‘(𝑦 ∪ {𝑧})))
2322oveq2d 7373 . . . . . . 7 (𝑥 = (𝑦 ∪ {𝑧}) → ((lcm𝑌) lcm (lcm𝑥)) = ((lcm𝑌) lcm (lcm‘(𝑦 ∪ {𝑧}))))
2421, 23eqeq12d 2752 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → ((lcm‘(𝑌𝑥)) = ((lcm𝑌) lcm (lcm𝑥)) ↔ (lcm‘(𝑌 ∪ (𝑦 ∪ {𝑧}))) = ((lcm𝑌) lcm (lcm‘(𝑦 ∪ {𝑧})))))
2519, 24imbi12d 344 . . . . 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 14867 . . . . . 6 (𝑥 = 𝑍 → ((𝑥 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) ↔ (𝑍 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))))
27 uneq2 4117 . . . . . . . 8 (𝑥 = 𝑍 → (𝑌𝑥) = (𝑌𝑍))
2827fveq2d 6846 . . . . . . 7 (𝑥 = 𝑍 → (lcm‘(𝑌𝑥)) = (lcm‘(𝑌𝑍)))
29 fveq2 6842 . . . . . . . 8 (𝑥 = 𝑍 → (lcm𝑥) = (lcm𝑍))
3029oveq2d 7373 . . . . . . 7 (𝑥 = 𝑍 → ((lcm𝑌) lcm (lcm𝑥)) = ((lcm𝑌) lcm (lcm𝑍)))
3128, 30eqeq12d 2752 . . . . . 6 (𝑥 = 𝑍 → ((lcm‘(𝑌𝑥)) = ((lcm𝑌) lcm (lcm𝑥)) ↔ (lcm‘(𝑌𝑍)) = ((lcm𝑌) lcm (lcm𝑍))))
3226, 31imbi12d 344 . . . . 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 16504 . . . . . . . . . 10 ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → (lcm𝑌) ∈ ℕ0)
3433nn0zd 12525 . . . . . . . . 9 ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → (lcm𝑌) ∈ ℤ)
35 lcm1 16486 . . . . . . . . 9 ((lcm𝑌) ∈ ℤ → ((lcm𝑌) lcm 1) = (abs‘(lcm𝑌)))
3634, 35syl 17 . . . . . . . 8 ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → ((lcm𝑌) lcm 1) = (abs‘(lcm𝑌)))
37 nn0re 12422 . . . . . . . . . . 11 ((lcm𝑌) ∈ ℕ0 → (lcm𝑌) ∈ ℝ)
38 nn0ge0 12438 . . . . . . . . . . 11 ((lcm𝑌) ∈ ℕ0 → 0 ≤ (lcm𝑌))
3937, 38jca 512 . . . . . . . . . 10 ((lcm𝑌) ∈ ℕ0 → ((lcm𝑌) ∈ ℝ ∧ 0 ≤ (lcm𝑌)))
4033, 39syl 17 . . . . . . . . 9 ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → ((lcm𝑌) ∈ ℝ ∧ 0 ≤ (lcm𝑌)))
41 absid 15181 . . . . . . . . 9 (((lcm𝑌) ∈ ℝ ∧ 0 ≤ (lcm𝑌)) → (abs‘(lcm𝑌)) = (lcm𝑌))
4240, 41syl 17 . . . . . . . 8 ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → (abs‘(lcm𝑌)) = (lcm𝑌))
4336, 42eqtrd 2776 . . . . . . 7 ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → ((lcm𝑌) lcm 1) = (lcm𝑌))
4443adantl 482 . . . . . 6 ((∅ ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → ((lcm𝑌) lcm 1) = (lcm𝑌))
4544eqcomd 2742 . . . . 5 ((∅ ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm𝑌) = ((lcm𝑌) lcm 1))
46 unass 4126 . . . . . . . . . . . . . 14 ((𝑌𝑦) ∪ {𝑧}) = (𝑌 ∪ (𝑦 ∪ {𝑧}))
4746eqcomi 2745 . . . . . . . . . . . . 13 (𝑌 ∪ (𝑦 ∪ {𝑧})) = ((𝑌𝑦) ∪ {𝑧})
4847a1i 11 . . . . . . . . . . . 12 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (𝑌 ∪ (𝑦 ∪ {𝑧})) = ((𝑌𝑦) ∪ {𝑧}))
4948fveq2d 6846 . . . . . . . . . . 11 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (lcm‘(𝑌 ∪ (𝑦 ∪ {𝑧}))) = (lcm‘((𝑌𝑦) ∪ {𝑧})))
50 simpl 483 . . . . . . . . . . . . . . 15 ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → 𝑌 ⊆ ℤ)
5150adantl 482 . . . . . . . . . . . . . 14 (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → 𝑌 ⊆ ℤ)
52 unss 4144 . . . . . . . . . . . . . . . 16 ((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) ↔ (𝑦 ∪ {𝑧}) ⊆ ℤ)
53 simpl 483 . . . . . . . . . . . . . . . 16 ((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) → 𝑦 ⊆ ℤ)
5452, 53sylbir 234 . . . . . . . . . . . . . . 15 ((𝑦 ∪ {𝑧}) ⊆ ℤ → 𝑦 ⊆ ℤ)
5554adantr 481 . . . . . . . . . . . . . 14 (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → 𝑦 ⊆ ℤ)
5651, 55unssd 4146 . . . . . . . . . . . . 13 (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (𝑌𝑦) ⊆ ℤ)
5756adantl 482 . . . . . . . . . . . 12 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (𝑌𝑦) ⊆ ℤ)
58 unfi 9116 . . . . . . . . . . . . . . . 16 ((𝑌 ∈ Fin ∧ 𝑦 ∈ Fin) → (𝑌𝑦) ∈ Fin)
5958ex 413 . . . . . . . . . . . . . . 15 (𝑌 ∈ Fin → (𝑦 ∈ Fin → (𝑌𝑦) ∈ Fin))
6059adantl 482 . . . . . . . . . . . . . 14 ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → (𝑦 ∈ Fin → (𝑌𝑦) ∈ Fin))
6160adantl 482 . . . . . . . . . . . . 13 (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (𝑦 ∈ Fin → (𝑌𝑦) ∈ Fin))
6261impcom 408 . . . . . . . . . . . 12 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (𝑌𝑦) ∈ Fin)
63 vex 3449 . . . . . . . . . . . . . . . . . 18 𝑧 ∈ V
6463snss 4746 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ ℤ ↔ {𝑧} ⊆ ℤ)
6564biimpri 227 . . . . . . . . . . . . . . . 16 ({𝑧} ⊆ ℤ → 𝑧 ∈ ℤ)
6665adantl 482 . . . . . . . . . . . . . . 15 ((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) → 𝑧 ∈ ℤ)
6752, 66sylbir 234 . . . . . . . . . . . . . 14 ((𝑦 ∪ {𝑧}) ⊆ ℤ → 𝑧 ∈ ℤ)
6867adantr 481 . . . . . . . . . . . . 13 (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → 𝑧 ∈ ℤ)
6968adantl 482 . . . . . . . . . . . 12 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → 𝑧 ∈ ℤ)
70 lcmfunsn 16520 . . . . . . . . . . . 12 (((𝑌𝑦) ⊆ ℤ ∧ (𝑌𝑦) ∈ Fin ∧ 𝑧 ∈ ℤ) → (lcm‘((𝑌𝑦) ∪ {𝑧})) = ((lcm‘(𝑌𝑦)) lcm 𝑧))
7157, 62, 69, 70syl3anc 1371 . . . . . . . . . . 11 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (lcm‘((𝑌𝑦) ∪ {𝑧})) = ((lcm‘(𝑌𝑦)) lcm 𝑧))
7249, 71eqtrd 2776 . . . . . . . . . 10 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (lcm‘(𝑌 ∪ (𝑦 ∪ {𝑧}))) = ((lcm‘(𝑌𝑦)) lcm 𝑧))
7372adantr 481 . . . . . . . . 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 615 . . . . . . . . . . . . 13 (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (𝑦 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)))
7574adantl 482 . . . . . . . . . . . 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 507 . . . . . . . . . . 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 7372 . . . . . . . . . 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 482 . . . . . . . . . . . . 13 (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm𝑌) ∈ ℤ)
8079adantl 482 . . . . . . . . . . . 12 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (lcm𝑌) ∈ ℤ)
8155anim2i 617 . . . . . . . . . . . . . . 15 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (𝑦 ∈ Fin ∧ 𝑦 ⊆ ℤ))
8281ancomd 462 . . . . . . . . . . . . . 14 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin))
83 lcmfcl 16504 . . . . . . . . . . . . . 14 ((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm𝑦) ∈ ℕ0)
8482, 83syl 17 . . . . . . . . . . . . 13 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (lcm𝑦) ∈ ℕ0)
8584nn0zd 12525 . . . . . . . . . . . 12 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (lcm𝑦) ∈ ℤ)
86 lcmass 16490 . . . . . . . . . . . 12 (((lcm𝑌) ∈ ℤ ∧ (lcm𝑦) ∈ ℤ ∧ 𝑧 ∈ ℤ) → (((lcm𝑌) lcm (lcm𝑦)) lcm 𝑧) = ((lcm𝑌) lcm ((lcm𝑦) lcm 𝑧)))
8780, 85, 69, 86syl3anc 1371 . . . . . . . . . . 11 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (((lcm𝑌) lcm (lcm𝑦)) lcm 𝑧) = ((lcm𝑌) lcm ((lcm𝑦) lcm 𝑧)))
8887adantr 481 . . . . . . . . . 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 2776 . . . . . . . . 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 2776 . . . . . . . 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 481 . . . . . . . . . . . . . . . . 17 (((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) ∧ 𝑦 ∈ Fin) → 𝑦 ⊆ ℤ)
92 simpr 485 . . . . . . . . . . . . . . . . 17 (((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) ∧ 𝑦 ∈ Fin) → 𝑦 ∈ Fin)
9366adantr 481 . . . . . . . . . . . . . . . . 17 (((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) ∧ 𝑦 ∈ Fin) → 𝑧 ∈ ℤ)
9491, 92, 933jca 1128 . . . . . . . . . . . . . . . 16 (((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) ∧ 𝑦 ∈ Fin) → (𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ∧ 𝑧 ∈ ℤ))
9594ex 413 . . . . . . . . . . . . . . 15 ((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) → (𝑦 ∈ Fin → (𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ∧ 𝑧 ∈ ℤ)))
9652, 95sylbir 234 . . . . . . . . . . . . . 14 ((𝑦 ∪ {𝑧}) ⊆ ℤ → (𝑦 ∈ Fin → (𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ∧ 𝑧 ∈ ℤ)))
9796adantr 481 . . . . . . . . . . . . 13 (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (𝑦 ∈ Fin → (𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ∧ 𝑧 ∈ ℤ)))
9897impcom 408 . . . . . . . . . . . 12 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ∧ 𝑧 ∈ ℤ))
99 lcmfunsn 16520 . . . . . . . . . . . 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 7373 . . . . . . . . . 10 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → ((lcm𝑌) lcm (lcm‘(𝑦 ∪ {𝑧}))) = ((lcm𝑌) lcm ((lcm𝑦) lcm 𝑧)))
102101eqeq2d 2747 . . . . . . . . 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 481 . . . . . . . 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 256 . . . . . . 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 420 . . . . . 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 9108 . . . 4 (𝑍 ∈ Fin → ((𝑍 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm‘(𝑌𝑍)) = ((lcm𝑌) lcm (lcm𝑍))))
108107expd 416 . . 3 (𝑍 ∈ Fin → (𝑍 ⊆ ℤ → ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → (lcm‘(𝑌𝑍)) = ((lcm𝑌) lcm (lcm𝑍)))))
109108impcom 408 . 2 ((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) → ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → (lcm‘(𝑌𝑍)) = ((lcm𝑌) lcm (lcm𝑍))))
110109impcom 408 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 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  cun 3908  wss 3910  c0 4282  {csn 4586   class class class wbr 5105  cfv 6496  (class class class)co 7357  Fincfn 8883  cr 11050  0cc0 11051  1c1 11052  cle 11190  0cn0 12413  cz 12499  abscabs 15119   lcm clcm 16464  lcmclcmf 16465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-inf 9379  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-rp 12916  df-fz 13425  df-fzo 13568  df-fl 13697  df-mod 13775  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-prod 15789  df-dvds 16137  df-gcd 16375  df-lcm 16466  df-lcmf 16467
This theorem is referenced by:  lcmfass  16522
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