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Theorem lcmfun 16587
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 14933 . . . . . 6 (𝑥 = ∅ → ((𝑥 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) ↔ (∅ ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))))
2 uneq2 4152 . . . . . . . . 9 (𝑥 = ∅ → (𝑌𝑥) = (𝑌 ∪ ∅))
3 un0 4385 . . . . . . . . 9 (𝑌 ∪ ∅) = 𝑌
42, 3eqtrdi 2782 . . . . . . . 8 (𝑥 = ∅ → (𝑌𝑥) = 𝑌)
54fveq2d 6888 . . . . . . 7 (𝑥 = ∅ → (lcm‘(𝑌𝑥)) = (lcm𝑌))
6 fveq2 6884 . . . . . . . . 9 (𝑥 = ∅ → (lcm𝑥) = (lcm‘∅))
7 lcmf0 16576 . . . . . . . . 9 (lcm‘∅) = 1
86, 7eqtrdi 2782 . . . . . . . 8 (𝑥 = ∅ → (lcm𝑥) = 1)
98oveq2d 7420 . . . . . . 7 (𝑥 = ∅ → ((lcm𝑌) lcm (lcm𝑥)) = ((lcm𝑌) lcm 1))
105, 9eqeq12d 2742 . . . . . 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 14933 . . . . . 6 (𝑥 = 𝑦 → ((𝑥 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) ↔ (𝑦 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))))
13 uneq2 4152 . . . . . . . 8 (𝑥 = 𝑦 → (𝑌𝑥) = (𝑌𝑦))
1413fveq2d 6888 . . . . . . 7 (𝑥 = 𝑦 → (lcm‘(𝑌𝑥)) = (lcm‘(𝑌𝑦)))
15 fveq2 6884 . . . . . . . 8 (𝑥 = 𝑦 → (lcm𝑥) = (lcm𝑦))
1615oveq2d 7420 . . . . . . 7 (𝑥 = 𝑦 → ((lcm𝑌) lcm (lcm𝑥)) = ((lcm𝑌) lcm (lcm𝑦)))
1714, 16eqeq12d 2742 . . . . . 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 14933 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → ((𝑥 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) ↔ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))))
20 uneq2 4152 . . . . . . . 8 (𝑥 = (𝑦 ∪ {𝑧}) → (𝑌𝑥) = (𝑌 ∪ (𝑦 ∪ {𝑧})))
2120fveq2d 6888 . . . . . . 7 (𝑥 = (𝑦 ∪ {𝑧}) → (lcm‘(𝑌𝑥)) = (lcm‘(𝑌 ∪ (𝑦 ∪ {𝑧}))))
22 fveq2 6884 . . . . . . . 8 (𝑥 = (𝑦 ∪ {𝑧}) → (lcm𝑥) = (lcm‘(𝑦 ∪ {𝑧})))
2322oveq2d 7420 . . . . . . 7 (𝑥 = (𝑦 ∪ {𝑧}) → ((lcm𝑌) lcm (lcm𝑥)) = ((lcm𝑌) lcm (lcm‘(𝑦 ∪ {𝑧}))))
2421, 23eqeq12d 2742 . . . . . 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 14933 . . . . . 6 (𝑥 = 𝑍 → ((𝑥 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) ↔ (𝑍 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))))
27 uneq2 4152 . . . . . . . 8 (𝑥 = 𝑍 → (𝑌𝑥) = (𝑌𝑍))
2827fveq2d 6888 . . . . . . 7 (𝑥 = 𝑍 → (lcm‘(𝑌𝑥)) = (lcm‘(𝑌𝑍)))
29 fveq2 6884 . . . . . . . 8 (𝑥 = 𝑍 → (lcm𝑥) = (lcm𝑍))
3029oveq2d 7420 . . . . . . 7 (𝑥 = 𝑍 → ((lcm𝑌) lcm (lcm𝑥)) = ((lcm𝑌) lcm (lcm𝑍)))
3128, 30eqeq12d 2742 . . . . . 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 16570 . . . . . . . . . 10 ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → (lcm𝑌) ∈ ℕ0)
3433nn0zd 12585 . . . . . . . . 9 ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → (lcm𝑌) ∈ ℤ)
35 lcm1 16552 . . . . . . . . 9 ((lcm𝑌) ∈ ℤ → ((lcm𝑌) lcm 1) = (abs‘(lcm𝑌)))
3634, 35syl 17 . . . . . . . 8 ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → ((lcm𝑌) lcm 1) = (abs‘(lcm𝑌)))
37 nn0re 12482 . . . . . . . . . . 11 ((lcm𝑌) ∈ ℕ0 → (lcm𝑌) ∈ ℝ)
38 nn0ge0 12498 . . . . . . . . . . 11 ((lcm𝑌) ∈ ℕ0 → 0 ≤ (lcm𝑌))
3937, 38jca 511 . . . . . . . . . 10 ((lcm𝑌) ∈ ℕ0 → ((lcm𝑌) ∈ ℝ ∧ 0 ≤ (lcm𝑌)))
4033, 39syl 17 . . . . . . . . 9 ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → ((lcm𝑌) ∈ ℝ ∧ 0 ≤ (lcm𝑌)))
41 absid 15247 . . . . . . . . 9 (((lcm𝑌) ∈ ℝ ∧ 0 ≤ (lcm𝑌)) → (abs‘(lcm𝑌)) = (lcm𝑌))
4240, 41syl 17 . . . . . . . 8 ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → (abs‘(lcm𝑌)) = (lcm𝑌))
4336, 42eqtrd 2766 . . . . . . 7 ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → ((lcm𝑌) lcm 1) = (lcm𝑌))
4443adantl 481 . . . . . 6 ((∅ ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → ((lcm𝑌) lcm 1) = (lcm𝑌))
4544eqcomd 2732 . . . . 5 ((∅ ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm𝑌) = ((lcm𝑌) lcm 1))
46 unass 4161 . . . . . . . . . . . . . 14 ((𝑌𝑦) ∪ {𝑧}) = (𝑌 ∪ (𝑦 ∪ {𝑧}))
4746eqcomi 2735 . . . . . . . . . . . . 13 (𝑌 ∪ (𝑦 ∪ {𝑧})) = ((𝑌𝑦) ∪ {𝑧})
4847a1i 11 . . . . . . . . . . . 12 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (𝑌 ∪ (𝑦 ∪ {𝑧})) = ((𝑌𝑦) ∪ {𝑧}))
4948fveq2d 6888 . . . . . . . . . . 11 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (lcm‘(𝑌 ∪ (𝑦 ∪ {𝑧}))) = (lcm‘((𝑌𝑦) ∪ {𝑧})))
50 simpl 482 . . . . . . . . . . . . . . 15 ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → 𝑌 ⊆ ℤ)
5150adantl 481 . . . . . . . . . . . . . 14 (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → 𝑌 ⊆ ℤ)
52 unss 4179 . . . . . . . . . . . . . . . 16 ((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) ↔ (𝑦 ∪ {𝑧}) ⊆ ℤ)
53 simpl 482 . . . . . . . . . . . . . . . 16 ((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) → 𝑦 ⊆ ℤ)
5452, 53sylbir 234 . . . . . . . . . . . . . . 15 ((𝑦 ∪ {𝑧}) ⊆ ℤ → 𝑦 ⊆ ℤ)
5554adantr 480 . . . . . . . . . . . . . 14 (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → 𝑦 ⊆ ℤ)
5651, 55unssd 4181 . . . . . . . . . . . . 13 (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (𝑌𝑦) ⊆ ℤ)
5756adantl 481 . . . . . . . . . . . 12 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (𝑌𝑦) ⊆ ℤ)
58 unfi 9171 . . . . . . . . . . . . . . . 16 ((𝑌 ∈ Fin ∧ 𝑦 ∈ Fin) → (𝑌𝑦) ∈ Fin)
5958ex 412 . . . . . . . . . . . . . . 15 (𝑌 ∈ Fin → (𝑦 ∈ Fin → (𝑌𝑦) ∈ Fin))
6059adantl 481 . . . . . . . . . . . . . 14 ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → (𝑦 ∈ Fin → (𝑌𝑦) ∈ Fin))
6160adantl 481 . . . . . . . . . . . . 13 (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (𝑦 ∈ Fin → (𝑌𝑦) ∈ Fin))
6261impcom 407 . . . . . . . . . . . 12 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (𝑌𝑦) ∈ Fin)
63 vex 3472 . . . . . . . . . . . . . . . . . 18 𝑧 ∈ V
6463snss 4784 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ ℤ ↔ {𝑧} ⊆ ℤ)
6564biimpri 227 . . . . . . . . . . . . . . . 16 ({𝑧} ⊆ ℤ → 𝑧 ∈ ℤ)
6665adantl 481 . . . . . . . . . . . . . . 15 ((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) → 𝑧 ∈ ℤ)
6752, 66sylbir 234 . . . . . . . . . . . . . 14 ((𝑦 ∪ {𝑧}) ⊆ ℤ → 𝑧 ∈ ℤ)
6867adantr 480 . . . . . . . . . . . . 13 (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → 𝑧 ∈ ℤ)
6968adantl 481 . . . . . . . . . . . 12 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → 𝑧 ∈ ℤ)
70 lcmfunsn 16586 . . . . . . . . . . . 12 (((𝑌𝑦) ⊆ ℤ ∧ (𝑌𝑦) ∈ Fin ∧ 𝑧 ∈ ℤ) → (lcm‘((𝑌𝑦) ∪ {𝑧})) = ((lcm‘(𝑌𝑦)) lcm 𝑧))
7157, 62, 69, 70syl3anc 1368 . . . . . . . . . . 11 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (lcm‘((𝑌𝑦) ∪ {𝑧})) = ((lcm‘(𝑌𝑦)) lcm 𝑧))
7249, 71eqtrd 2766 . . . . . . . . . 10 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (lcm‘(𝑌 ∪ (𝑦 ∪ {𝑧}))) = ((lcm‘(𝑌𝑦)) lcm 𝑧))
7372adantr 480 . . . . . . . . 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 614 . . . . . . . . . . . . 13 (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (𝑦 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)))
7574adantl 481 . . . . . . . . . . . 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 506 . . . . . . . . . . 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 7419 . . . . . . . . . 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 481 . . . . . . . . . . . . 13 (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm𝑌) ∈ ℤ)
8079adantl 481 . . . . . . . . . . . 12 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (lcm𝑌) ∈ ℤ)
8155anim2i 616 . . . . . . . . . . . . . . 15 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (𝑦 ∈ Fin ∧ 𝑦 ⊆ ℤ))
8281ancomd 461 . . . . . . . . . . . . . 14 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin))
83 lcmfcl 16570 . . . . . . . . . . . . . 14 ((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm𝑦) ∈ ℕ0)
8482, 83syl 17 . . . . . . . . . . . . 13 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (lcm𝑦) ∈ ℕ0)
8584nn0zd 12585 . . . . . . . . . . . 12 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (lcm𝑦) ∈ ℤ)
86 lcmass 16556 . . . . . . . . . . . 12 (((lcm𝑌) ∈ ℤ ∧ (lcm𝑦) ∈ ℤ ∧ 𝑧 ∈ ℤ) → (((lcm𝑌) lcm (lcm𝑦)) lcm 𝑧) = ((lcm𝑌) lcm ((lcm𝑦) lcm 𝑧)))
8780, 85, 69, 86syl3anc 1368 . . . . . . . . . . 11 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (((lcm𝑌) lcm (lcm𝑦)) lcm 𝑧) = ((lcm𝑌) lcm ((lcm𝑦) lcm 𝑧)))
8887adantr 480 . . . . . . . . . 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 2766 . . . . . . . . 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 2766 . . . . . . . 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 480 . . . . . . . . . . . . . . . . 17 (((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) ∧ 𝑦 ∈ Fin) → 𝑦 ⊆ ℤ)
92 simpr 484 . . . . . . . . . . . . . . . . 17 (((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) ∧ 𝑦 ∈ Fin) → 𝑦 ∈ Fin)
9366adantr 480 . . . . . . . . . . . . . . . . 17 (((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) ∧ 𝑦 ∈ Fin) → 𝑧 ∈ ℤ)
9491, 92, 933jca 1125 . . . . . . . . . . . . . . . 16 (((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) ∧ 𝑦 ∈ Fin) → (𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ∧ 𝑧 ∈ ℤ))
9594ex 412 . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . 13 (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (𝑦 ∈ Fin → (𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ∧ 𝑧 ∈ ℤ)))
9897impcom 407 . . . . . . . . . . . 12 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ∧ 𝑧 ∈ ℤ))
99 lcmfunsn 16586 . . . . . . . . . . . 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 7420 . . . . . . . . . 10 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → ((lcm𝑌) lcm (lcm‘(𝑦 ∪ {𝑧}))) = ((lcm𝑌) lcm ((lcm𝑦) lcm 𝑧)))
102101eqeq2d 2737 . . . . . . . . 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 480 . . . . . . . 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 257 . . . . . . 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 419 . . . . . 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 9163 . . . 4 (𝑍 ∈ Fin → ((𝑍 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm‘(𝑌𝑍)) = ((lcm𝑌) lcm (lcm𝑍))))
108107expd 415 . . 3 (𝑍 ∈ Fin → (𝑍 ⊆ ℤ → ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → (lcm‘(𝑌𝑍)) = ((lcm𝑌) lcm (lcm𝑍)))))
109108impcom 407 . 2 ((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) → ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → (lcm‘(𝑌𝑍)) = ((lcm𝑌) lcm (lcm𝑍))))
110109impcom 407 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 395  w3a 1084   = wceq 1533  wcel 2098  cun 3941  wss 3943  c0 4317  {csn 4623   class class class wbr 5141  cfv 6536  (class class class)co 7404  Fincfn 8938  cr 11108  0cc0 11109  1c1 11110  cle 11250  0cn0 12473  cz 12559  abscabs 15185   lcm clcm 16530  lcmclcmf 16531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-inf2 9635  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186  ax-pre-sup 11187
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408  df-1o 8464  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-sup 9436  df-inf 9437  df-oi 9504  df-card 9933  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11447  df-neg 11448  df-div 11873  df-nn 12214  df-2 12276  df-3 12277  df-n0 12474  df-z 12560  df-uz 12824  df-rp 12978  df-fz 13488  df-fzo 13631  df-fl 13760  df-mod 13838  df-seq 13970  df-exp 14031  df-hash 14294  df-cj 15050  df-re 15051  df-im 15052  df-sqrt 15186  df-abs 15187  df-clim 15436  df-prod 15854  df-dvds 16203  df-gcd 16441  df-lcm 16532  df-lcmf 16533
This theorem is referenced by:  lcmfass  16588
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