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Theorem lcmfun 15977
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 14330 . . . . . 6 (𝑥 = ∅ → ((𝑥 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) ↔ (∅ ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))))
2 uneq2 4130 . . . . . . . . 9 (𝑥 = ∅ → (𝑌𝑥) = (𝑌 ∪ ∅))
3 un0 4341 . . . . . . . . 9 (𝑌 ∪ ∅) = 𝑌
42, 3syl6eq 2869 . . . . . . . 8 (𝑥 = ∅ → (𝑌𝑥) = 𝑌)
54fveq2d 6667 . . . . . . 7 (𝑥 = ∅ → (lcm‘(𝑌𝑥)) = (lcm𝑌))
6 fveq2 6663 . . . . . . . . 9 (𝑥 = ∅ → (lcm𝑥) = (lcm‘∅))
7 lcmf0 15966 . . . . . . . . 9 (lcm‘∅) = 1
86, 7syl6eq 2869 . . . . . . . 8 (𝑥 = ∅ → (lcm𝑥) = 1)
98oveq2d 7161 . . . . . . 7 (𝑥 = ∅ → ((lcm𝑌) lcm (lcm𝑥)) = ((lcm𝑌) lcm 1))
105, 9eqeq12d 2834 . . . . . 6 (𝑥 = ∅ → ((lcm‘(𝑌𝑥)) = ((lcm𝑌) lcm (lcm𝑥)) ↔ (lcm𝑌) = ((lcm𝑌) lcm 1)))
111, 10imbi12d 346 . . . . 5 (𝑥 = ∅ → (((𝑥 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm‘(𝑌𝑥)) = ((lcm𝑌) lcm (lcm𝑥))) ↔ ((∅ ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm𝑌) = ((lcm𝑌) lcm 1))))
12 cleq1lem 14330 . . . . . 6 (𝑥 = 𝑦 → ((𝑥 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) ↔ (𝑦 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))))
13 uneq2 4130 . . . . . . . 8 (𝑥 = 𝑦 → (𝑌𝑥) = (𝑌𝑦))
1413fveq2d 6667 . . . . . . 7 (𝑥 = 𝑦 → (lcm‘(𝑌𝑥)) = (lcm‘(𝑌𝑦)))
15 fveq2 6663 . . . . . . . 8 (𝑥 = 𝑦 → (lcm𝑥) = (lcm𝑦))
1615oveq2d 7161 . . . . . . 7 (𝑥 = 𝑦 → ((lcm𝑌) lcm (lcm𝑥)) = ((lcm𝑌) lcm (lcm𝑦)))
1714, 16eqeq12d 2834 . . . . . 6 (𝑥 = 𝑦 → ((lcm‘(𝑌𝑥)) = ((lcm𝑌) lcm (lcm𝑥)) ↔ (lcm‘(𝑌𝑦)) = ((lcm𝑌) lcm (lcm𝑦))))
1812, 17imbi12d 346 . . . . 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 14330 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → ((𝑥 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) ↔ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))))
20 uneq2 4130 . . . . . . . 8 (𝑥 = (𝑦 ∪ {𝑧}) → (𝑌𝑥) = (𝑌 ∪ (𝑦 ∪ {𝑧})))
2120fveq2d 6667 . . . . . . 7 (𝑥 = (𝑦 ∪ {𝑧}) → (lcm‘(𝑌𝑥)) = (lcm‘(𝑌 ∪ (𝑦 ∪ {𝑧}))))
22 fveq2 6663 . . . . . . . 8 (𝑥 = (𝑦 ∪ {𝑧}) → (lcm𝑥) = (lcm‘(𝑦 ∪ {𝑧})))
2322oveq2d 7161 . . . . . . 7 (𝑥 = (𝑦 ∪ {𝑧}) → ((lcm𝑌) lcm (lcm𝑥)) = ((lcm𝑌) lcm (lcm‘(𝑦 ∪ {𝑧}))))
2421, 23eqeq12d 2834 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → ((lcm‘(𝑌𝑥)) = ((lcm𝑌) lcm (lcm𝑥)) ↔ (lcm‘(𝑌 ∪ (𝑦 ∪ {𝑧}))) = ((lcm𝑌) lcm (lcm‘(𝑦 ∪ {𝑧})))))
2519, 24imbi12d 346 . . . . 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 14330 . . . . . 6 (𝑥 = 𝑍 → ((𝑥 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) ↔ (𝑍 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))))
27 uneq2 4130 . . . . . . . 8 (𝑥 = 𝑍 → (𝑌𝑥) = (𝑌𝑍))
2827fveq2d 6667 . . . . . . 7 (𝑥 = 𝑍 → (lcm‘(𝑌𝑥)) = (lcm‘(𝑌𝑍)))
29 fveq2 6663 . . . . . . . 8 (𝑥 = 𝑍 → (lcm𝑥) = (lcm𝑍))
3029oveq2d 7161 . . . . . . 7 (𝑥 = 𝑍 → ((lcm𝑌) lcm (lcm𝑥)) = ((lcm𝑌) lcm (lcm𝑍)))
3128, 30eqeq12d 2834 . . . . . 6 (𝑥 = 𝑍 → ((lcm‘(𝑌𝑥)) = ((lcm𝑌) lcm (lcm𝑥)) ↔ (lcm‘(𝑌𝑍)) = ((lcm𝑌) lcm (lcm𝑍))))
3226, 31imbi12d 346 . . . . 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 15960 . . . . . . . . . 10 ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → (lcm𝑌) ∈ ℕ0)
3433nn0zd 12073 . . . . . . . . 9 ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → (lcm𝑌) ∈ ℤ)
35 lcm1 15942 . . . . . . . . 9 ((lcm𝑌) ∈ ℤ → ((lcm𝑌) lcm 1) = (abs‘(lcm𝑌)))
3634, 35syl 17 . . . . . . . 8 ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → ((lcm𝑌) lcm 1) = (abs‘(lcm𝑌)))
37 nn0re 11894 . . . . . . . . . . 11 ((lcm𝑌) ∈ ℕ0 → (lcm𝑌) ∈ ℝ)
38 nn0ge0 11910 . . . . . . . . . . 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 14644 . . . . . . . . 9 (((lcm𝑌) ∈ ℝ ∧ 0 ≤ (lcm𝑌)) → (abs‘(lcm𝑌)) = (lcm𝑌))
4240, 41syl 17 . . . . . . . 8 ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → (abs‘(lcm𝑌)) = (lcm𝑌))
4336, 42eqtrd 2853 . . . . . . 7 ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → ((lcm𝑌) lcm 1) = (lcm𝑌))
4443adantl 482 . . . . . 6 ((∅ ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → ((lcm𝑌) lcm 1) = (lcm𝑌))
4544eqcomd 2824 . . . . 5 ((∅ ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm𝑌) = ((lcm𝑌) lcm 1))
46 unass 4139 . . . . . . . . . . . . . 14 ((𝑌𝑦) ∪ {𝑧}) = (𝑌 ∪ (𝑦 ∪ {𝑧}))
4746eqcomi 2827 . . . . . . . . . . . . 13 (𝑌 ∪ (𝑦 ∪ {𝑧})) = ((𝑌𝑦) ∪ {𝑧})
4847a1i 11 . . . . . . . . . . . 12 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (𝑌 ∪ (𝑦 ∪ {𝑧})) = ((𝑌𝑦) ∪ {𝑧}))
4948fveq2d 6667 . . . . . . . . . . 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 4157 . . . . . . . . . . . . . . . 16 ((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) ↔ (𝑦 ∪ {𝑧}) ⊆ ℤ)
53 simpl 483 . . . . . . . . . . . . . . . 16 ((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) → 𝑦 ⊆ ℤ)
5452, 53sylbir 236 . . . . . . . . . . . . . . 15 ((𝑦 ∪ {𝑧}) ⊆ ℤ → 𝑦 ⊆ ℤ)
5554adantr 481 . . . . . . . . . . . . . 14 (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → 𝑦 ⊆ ℤ)
5651, 55unssd 4159 . . . . . . . . . . . . 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 8773 . . . . . . . . . . . . . . . 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 3495 . . . . . . . . . . . . . . . . . 18 𝑧 ∈ V
6463snss 4710 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ ℤ ↔ {𝑧} ⊆ ℤ)
6564biimpri 229 . . . . . . . . . . . . . . . 16 ({𝑧} ⊆ ℤ → 𝑧 ∈ ℤ)
6665adantl 482 . . . . . . . . . . . . . . 15 ((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) → 𝑧 ∈ ℤ)
6752, 66sylbir 236 . . . . . . . . . . . . . 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 15976 . . . . . . . . . . . 12 (((𝑌𝑦) ⊆ ℤ ∧ (𝑌𝑦) ∈ Fin ∧ 𝑧 ∈ ℤ) → (lcm‘((𝑌𝑦) ∪ {𝑧})) = ((lcm‘(𝑌𝑦)) lcm 𝑧))
7157, 62, 69, 70syl3anc 1363 . . . . . . . . . . 11 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (lcm‘((𝑌𝑦) ∪ {𝑧})) = ((lcm‘(𝑌𝑦)) lcm 𝑧))
7249, 71eqtrd 2853 . . . . . . . . . 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 614 . . . . . . . . . . . . 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 7160 . . . . . . . . . 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 616 . . . . . . . . . . . . . . 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 15960 . . . . . . . . . . . . . 14 ((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm𝑦) ∈ ℕ0)
8482, 83syl 17 . . . . . . . . . . . . 13 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (lcm𝑦) ∈ ℕ0)
8584nn0zd 12073 . . . . . . . . . . . 12 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (lcm𝑦) ∈ ℤ)
86 lcmass 15946 . . . . . . . . . . . 12 (((lcm𝑌) ∈ ℤ ∧ (lcm𝑦) ∈ ℤ ∧ 𝑧 ∈ ℤ) → (((lcm𝑌) lcm (lcm𝑦)) lcm 𝑧) = ((lcm𝑌) lcm ((lcm𝑦) lcm 𝑧)))
8780, 85, 69, 86syl3anc 1363 . . . . . . . . . . 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 2853 . . . . . . . . 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 2853 . . . . . . . 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 1120 . . . . . . . . . . . . . . . 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 236 . . . . . . . . . . . . . 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 15976 . . . . . . . . . . . 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 7161 . . . . . . . . . 10 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → ((lcm𝑌) lcm (lcm‘(𝑦 ∪ {𝑧}))) = ((lcm𝑌) lcm ((lcm𝑦) lcm 𝑧)))
102101eqeq2d 2829 . . . . . . . . 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 258 . . . . . . 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 8746 . . . 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 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  cun 3931  wss 3933  c0 4288  {csn 4557   class class class wbr 5057  cfv 6348  (class class class)co 7145  Fincfn 8497  cr 10524  0cc0 10525  1c1 10526  cle 10664  0cn0 11885  cz 11969  abscabs 14581   lcm clcm 15920  lcmclcmf 15921
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-sup 8894  df-inf 8895  df-oi 8962  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-fz 12881  df-fzo 13022  df-fl 13150  df-mod 13226  df-seq 13358  df-exp 13418  df-hash 13679  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-clim 14833  df-prod 15248  df-dvds 15596  df-gcd 15832  df-lcm 15922  df-lcmf 15923
This theorem is referenced by:  lcmfass  15978
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