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Theorem lcmfun 16678
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 15017 . . . . . 6 (𝑥 = ∅ → ((𝑥 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) ↔ (∅ ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))))
2 uneq2 4171 . . . . . . . . 9 (𝑥 = ∅ → (𝑌𝑥) = (𝑌 ∪ ∅))
3 un0 4399 . . . . . . . . 9 (𝑌 ∪ ∅) = 𝑌
42, 3eqtrdi 2790 . . . . . . . 8 (𝑥 = ∅ → (𝑌𝑥) = 𝑌)
54fveq2d 6910 . . . . . . 7 (𝑥 = ∅ → (lcm‘(𝑌𝑥)) = (lcm𝑌))
6 fveq2 6906 . . . . . . . . 9 (𝑥 = ∅ → (lcm𝑥) = (lcm‘∅))
7 lcmf0 16667 . . . . . . . . 9 (lcm‘∅) = 1
86, 7eqtrdi 2790 . . . . . . . 8 (𝑥 = ∅ → (lcm𝑥) = 1)
98oveq2d 7446 . . . . . . 7 (𝑥 = ∅ → ((lcm𝑌) lcm (lcm𝑥)) = ((lcm𝑌) lcm 1))
105, 9eqeq12d 2750 . . . . . 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 15017 . . . . . 6 (𝑥 = 𝑦 → ((𝑥 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) ↔ (𝑦 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))))
13 uneq2 4171 . . . . . . . 8 (𝑥 = 𝑦 → (𝑌𝑥) = (𝑌𝑦))
1413fveq2d 6910 . . . . . . 7 (𝑥 = 𝑦 → (lcm‘(𝑌𝑥)) = (lcm‘(𝑌𝑦)))
15 fveq2 6906 . . . . . . . 8 (𝑥 = 𝑦 → (lcm𝑥) = (lcm𝑦))
1615oveq2d 7446 . . . . . . 7 (𝑥 = 𝑦 → ((lcm𝑌) lcm (lcm𝑥)) = ((lcm𝑌) lcm (lcm𝑦)))
1714, 16eqeq12d 2750 . . . . . 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 15017 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → ((𝑥 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) ↔ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))))
20 uneq2 4171 . . . . . . . 8 (𝑥 = (𝑦 ∪ {𝑧}) → (𝑌𝑥) = (𝑌 ∪ (𝑦 ∪ {𝑧})))
2120fveq2d 6910 . . . . . . 7 (𝑥 = (𝑦 ∪ {𝑧}) → (lcm‘(𝑌𝑥)) = (lcm‘(𝑌 ∪ (𝑦 ∪ {𝑧}))))
22 fveq2 6906 . . . . . . . 8 (𝑥 = (𝑦 ∪ {𝑧}) → (lcm𝑥) = (lcm‘(𝑦 ∪ {𝑧})))
2322oveq2d 7446 . . . . . . 7 (𝑥 = (𝑦 ∪ {𝑧}) → ((lcm𝑌) lcm (lcm𝑥)) = ((lcm𝑌) lcm (lcm‘(𝑦 ∪ {𝑧}))))
2421, 23eqeq12d 2750 . . . . . 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 15017 . . . . . 6 (𝑥 = 𝑍 → ((𝑥 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) ↔ (𝑍 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))))
27 uneq2 4171 . . . . . . . 8 (𝑥 = 𝑍 → (𝑌𝑥) = (𝑌𝑍))
2827fveq2d 6910 . . . . . . 7 (𝑥 = 𝑍 → (lcm‘(𝑌𝑥)) = (lcm‘(𝑌𝑍)))
29 fveq2 6906 . . . . . . . 8 (𝑥 = 𝑍 → (lcm𝑥) = (lcm𝑍))
3029oveq2d 7446 . . . . . . 7 (𝑥 = 𝑍 → ((lcm𝑌) lcm (lcm𝑥)) = ((lcm𝑌) lcm (lcm𝑍)))
3128, 30eqeq12d 2750 . . . . . 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 16661 . . . . . . . . . 10 ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → (lcm𝑌) ∈ ℕ0)
3433nn0zd 12636 . . . . . . . . 9 ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → (lcm𝑌) ∈ ℤ)
35 lcm1 16643 . . . . . . . . 9 ((lcm𝑌) ∈ ℤ → ((lcm𝑌) lcm 1) = (abs‘(lcm𝑌)))
3634, 35syl 17 . . . . . . . 8 ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → ((lcm𝑌) lcm 1) = (abs‘(lcm𝑌)))
37 nn0re 12532 . . . . . . . . . . 11 ((lcm𝑌) ∈ ℕ0 → (lcm𝑌) ∈ ℝ)
38 nn0ge0 12548 . . . . . . . . . . 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 15331 . . . . . . . . 9 (((lcm𝑌) ∈ ℝ ∧ 0 ≤ (lcm𝑌)) → (abs‘(lcm𝑌)) = (lcm𝑌))
4240, 41syl 17 . . . . . . . 8 ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → (abs‘(lcm𝑌)) = (lcm𝑌))
4336, 42eqtrd 2774 . . . . . . 7 ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → ((lcm𝑌) lcm 1) = (lcm𝑌))
4443adantl 481 . . . . . 6 ((∅ ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → ((lcm𝑌) lcm 1) = (lcm𝑌))
4544eqcomd 2740 . . . . 5 ((∅ ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm𝑌) = ((lcm𝑌) lcm 1))
46 unass 4181 . . . . . . . . . . . . . 14 ((𝑌𝑦) ∪ {𝑧}) = (𝑌 ∪ (𝑦 ∪ {𝑧}))
4746eqcomi 2743 . . . . . . . . . . . . 13 (𝑌 ∪ (𝑦 ∪ {𝑧})) = ((𝑌𝑦) ∪ {𝑧})
4847a1i 11 . . . . . . . . . . . 12 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (𝑌 ∪ (𝑦 ∪ {𝑧})) = ((𝑌𝑦) ∪ {𝑧}))
4948fveq2d 6910 . . . . . . . . . . 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 4199 . . . . . . . . . . . . . . . 16 ((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) ↔ (𝑦 ∪ {𝑧}) ⊆ ℤ)
53 simpl 482 . . . . . . . . . . . . . . . 16 ((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) → 𝑦 ⊆ ℤ)
5452, 53sylbir 235 . . . . . . . . . . . . . . 15 ((𝑦 ∪ {𝑧}) ⊆ ℤ → 𝑦 ⊆ ℤ)
5554adantr 480 . . . . . . . . . . . . . 14 (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → 𝑦 ⊆ ℤ)
5651, 55unssd 4201 . . . . . . . . . . . . 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 9209 . . . . . . . . . . . . . . . 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 3481 . . . . . . . . . . . . . . . . . 18 𝑧 ∈ V
6463snss 4789 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ ℤ ↔ {𝑧} ⊆ ℤ)
6564biimpri 228 . . . . . . . . . . . . . . . 16 ({𝑧} ⊆ ℤ → 𝑧 ∈ ℤ)
6665adantl 481 . . . . . . . . . . . . . . 15 ((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) → 𝑧 ∈ ℤ)
6752, 66sylbir 235 . . . . . . . . . . . . . 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 16677 . . . . . . . . . . . 12 (((𝑌𝑦) ⊆ ℤ ∧ (𝑌𝑦) ∈ Fin ∧ 𝑧 ∈ ℤ) → (lcm‘((𝑌𝑦) ∪ {𝑧})) = ((lcm‘(𝑌𝑦)) lcm 𝑧))
7157, 62, 69, 70syl3anc 1370 . . . . . . . . . . 11 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (lcm‘((𝑌𝑦) ∪ {𝑧})) = ((lcm‘(𝑌𝑦)) lcm 𝑧))
7249, 71eqtrd 2774 . . . . . . . . . 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 615 . . . . . . . . . . . . 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 7445 . . . . . . . . . 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 617 . . . . . . . . . . . . . . 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 16661 . . . . . . . . . . . . . 14 ((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm𝑦) ∈ ℕ0)
8482, 83syl 17 . . . . . . . . . . . . 13 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (lcm𝑦) ∈ ℕ0)
8584nn0zd 12636 . . . . . . . . . . . 12 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (lcm𝑦) ∈ ℤ)
86 lcmass 16647 . . . . . . . . . . . 12 (((lcm𝑌) ∈ ℤ ∧ (lcm𝑦) ∈ ℤ ∧ 𝑧 ∈ ℤ) → (((lcm𝑌) lcm (lcm𝑦)) lcm 𝑧) = ((lcm𝑌) lcm ((lcm𝑦) lcm 𝑧)))
8780, 85, 69, 86syl3anc 1370 . . . . . . . . . . 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 2774 . . . . . . . . 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 2774 . . . . . . . 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 1127 . . . . . . . . . . . . . . . 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 235 . . . . . . . . . . . . . 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 16677 . . . . . . . . . . . 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 7446 . . . . . . . . . 10 ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → ((lcm𝑌) lcm (lcm‘(𝑦 ∪ {𝑧}))) = ((lcm𝑌) lcm ((lcm𝑦) lcm 𝑧)))
102101eqeq2d 2745 . . . . . . . . 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 9202 . . . 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 206  wa 395  w3a 1086   = wceq 1536  wcel 2105  cun 3960  wss 3962  c0 4338  {csn 4630   class class class wbr 5147  cfv 6562  (class class class)co 7430  Fincfn 8983  cr 11151  0cc0 11152  1c1 11153  cle 11293  0cn0 12523  cz 12610  abscabs 15269   lcm clcm 16621  lcmclcmf 16622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-sup 9479  df-inf 9480  df-oi 9547  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-n0 12524  df-z 12611  df-uz 12876  df-rp 13032  df-fz 13544  df-fzo 13691  df-fl 13828  df-mod 13906  df-seq 14039  df-exp 14099  df-hash 14366  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-clim 15520  df-prod 15936  df-dvds 16287  df-gcd 16528  df-lcm 16623  df-lcmf 16624
This theorem is referenced by:  lcmfass  16679
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