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Theorem lcmfun 16578

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‘(𝑌 ∪ 𝑍)) = ((lcm‘𝑌) lcm (lcm‘𝑍)))

Proof of Theorem lcmfun Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cleq1lem 14925	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝑥 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) ↔ (∅ ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))))
2		uneq2 4156	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝑌 ∪ 𝑥) = (𝑌 ∪ ∅))
3		un0 4389	. . . . . . . . 9 ⊢ (𝑌 ∪ ∅) = 𝑌
4	2, 3	eqtrdi 2788	. . . . . . . 8 ⊢ (𝑥 = ∅ → (𝑌 ∪ 𝑥) = 𝑌)
5	4	fveq2d 6892	. . . . . . 7 ⊢ (𝑥 = ∅ → (lcm‘(𝑌 ∪ 𝑥)) = (lcm‘𝑌))
6		fveq2 6888	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (lcm‘𝑥) = (lcm‘∅))
7		lcmf0 16567	. . . . . . . . 9 ⊢ (lcm‘∅) = 1
8	6, 7	eqtrdi 2788	. . . . . . . 8 ⊢ (𝑥 = ∅ → (lcm‘𝑥) = 1)
9	8	oveq2d 7421	. . . . . . 7 ⊢ (𝑥 = ∅ → ((lcm‘𝑌) lcm (lcm‘𝑥)) = ((lcm‘𝑌) lcm 1))
10	5, 9	eqeq12d 2748	. . . . . 6 ⊢ (𝑥 = ∅ → ((lcm‘(𝑌 ∪ 𝑥)) = ((lcm‘𝑌) lcm (lcm‘𝑥)) ↔ (lcm‘𝑌) = ((lcm‘𝑌) lcm 1)))
11	1, 10	imbi12d 344	. . . . 5 ⊢ (𝑥 = ∅ → (((𝑥 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm‘(𝑌 ∪ 𝑥)) = ((lcm‘𝑌) lcm (lcm‘𝑥))) ↔ ((∅ ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm‘𝑌) = ((lcm‘𝑌) lcm 1))))
12		cleq1lem 14925	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑥 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) ↔ (𝑦 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))))
13		uneq2 4156	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑌 ∪ 𝑥) = (𝑌 ∪ 𝑦))
14	13	fveq2d 6892	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (lcm‘(𝑌 ∪ 𝑥)) = (lcm‘(𝑌 ∪ 𝑦)))
15		fveq2 6888	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (lcm‘𝑥) = (lcm‘𝑦))
16	15	oveq2d 7421	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((lcm‘𝑌) lcm (lcm‘𝑥)) = ((lcm‘𝑌) lcm (lcm‘𝑦)))
17	14, 16	eqeq12d 2748	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((lcm‘(𝑌 ∪ 𝑥)) = ((lcm‘𝑌) lcm (lcm‘𝑥)) ↔ (lcm‘(𝑌 ∪ 𝑦)) = ((lcm‘𝑌) lcm (lcm‘𝑦))))
18	12, 17	imbi12d 344	. . . . 5 ⊢ (𝑥 = 𝑦 → (((𝑥 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm‘(𝑌 ∪ 𝑥)) = ((lcm‘𝑌) lcm (lcm‘𝑥))) ↔ ((𝑦 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm‘(𝑌 ∪ 𝑦)) = ((lcm‘𝑌) lcm (lcm‘𝑦)))))
19		cleq1lem 14925	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝑥 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) ↔ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))))
20		uneq2 4156	. . . . . . . 8 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝑌 ∪ 𝑥) = (𝑌 ∪ (𝑦 ∪ {𝑧})))
21	20	fveq2d 6892	. . . . . . 7 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (lcm‘(𝑌 ∪ 𝑥)) = (lcm‘(𝑌 ∪ (𝑦 ∪ {𝑧}))))
22		fveq2 6888	. . . . . . . 8 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (lcm‘𝑥) = (lcm‘(𝑦 ∪ {𝑧})))
23	22	oveq2d 7421	. . . . . . 7 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((lcm‘𝑌) lcm (lcm‘𝑥)) = ((lcm‘𝑌) lcm (lcm‘(𝑦 ∪ {𝑧}))))
24	21, 23	eqeq12d 2748	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((lcm‘(𝑌 ∪ 𝑥)) = ((lcm‘𝑌) lcm (lcm‘𝑥)) ↔ (lcm‘(𝑌 ∪ (𝑦 ∪ {𝑧}))) = ((lcm‘𝑌) lcm (lcm‘(𝑦 ∪ {𝑧})))))
25	19, 24	imbi12d 344	. . . . 5 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (((𝑥 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm‘(𝑌 ∪ 𝑥)) = ((lcm‘𝑌) lcm (lcm‘𝑥))) ↔ (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm‘(𝑌 ∪ (𝑦 ∪ {𝑧}))) = ((lcm‘𝑌) lcm (lcm‘(𝑦 ∪ {𝑧}))))))
26		cleq1lem 14925	. . . . . 6 ⊢ (𝑥 = 𝑍 → ((𝑥 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) ↔ (𝑍 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))))
27		uneq2 4156	. . . . . . . 8 ⊢ (𝑥 = 𝑍 → (𝑌 ∪ 𝑥) = (𝑌 ∪ 𝑍))
28	27	fveq2d 6892	. . . . . . 7 ⊢ (𝑥 = 𝑍 → (lcm‘(𝑌 ∪ 𝑥)) = (lcm‘(𝑌 ∪ 𝑍)))
29		fveq2 6888	. . . . . . . 8 ⊢ (𝑥 = 𝑍 → (lcm‘𝑥) = (lcm‘𝑍))
30	29	oveq2d 7421	. . . . . . 7 ⊢ (𝑥 = 𝑍 → ((lcm‘𝑌) lcm (lcm‘𝑥)) = ((lcm‘𝑌) lcm (lcm‘𝑍)))
31	28, 30	eqeq12d 2748	. . . . . 6 ⊢ (𝑥 = 𝑍 → ((lcm‘(𝑌 ∪ 𝑥)) = ((lcm‘𝑌) lcm (lcm‘𝑥)) ↔ (lcm‘(𝑌 ∪ 𝑍)) = ((lcm‘𝑌) lcm (lcm‘𝑍))))
32	26, 31	imbi12d 344	. . . . 5 ⊢ (𝑥 = 𝑍 → (((𝑥 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm‘(𝑌 ∪ 𝑥)) = ((lcm‘𝑌) lcm (lcm‘𝑥))) ↔ ((𝑍 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm‘(𝑌 ∪ 𝑍)) = ((lcm‘𝑌) lcm (lcm‘𝑍)))))
33		lcmfcl 16561	. . . . . . . . . 10 ⊢ ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → (lcm‘𝑌) ∈ ℕ₀)
34	33	nn0zd 12580	. . . . . . . . 9 ⊢ ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → (lcm‘𝑌) ∈ ℤ)
35		lcm1 16543	. . . . . . . . 9 ⊢ ((lcm‘𝑌) ∈ ℤ → ((lcm‘𝑌) lcm 1) = (abs‘(lcm‘𝑌)))
36	34, 35	syl 17	. . . . . . . 8 ⊢ ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → ((lcm‘𝑌) lcm 1) = (abs‘(lcm‘𝑌)))
37		nn0re 12477	. . . . . . . . . . 11 ⊢ ((lcm‘𝑌) ∈ ℕ₀ → (lcm‘𝑌) ∈ ℝ)
38		nn0ge0 12493	. . . . . . . . . . 11 ⊢ ((lcm‘𝑌) ∈ ℕ₀ → 0 ≤ (lcm‘𝑌))
39	37, 38	jca 512	. . . . . . . . . 10 ⊢ ((lcm‘𝑌) ∈ ℕ₀ → ((lcm‘𝑌) ∈ ℝ ∧ 0 ≤ (lcm‘𝑌)))
40	33, 39	syl 17	. . . . . . . . 9 ⊢ ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → ((lcm‘𝑌) ∈ ℝ ∧ 0 ≤ (lcm‘𝑌)))
41		absid 15239	. . . . . . . . 9 ⊢ (((lcm‘𝑌) ∈ ℝ ∧ 0 ≤ (lcm‘𝑌)) → (abs‘(lcm‘𝑌)) = (lcm‘𝑌))
42	40, 41	syl 17	. . . . . . . 8 ⊢ ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → (abs‘(lcm‘𝑌)) = (lcm‘𝑌))
43	36, 42	eqtrd 2772	. . . . . . 7 ⊢ ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → ((lcm‘𝑌) lcm 1) = (lcm‘𝑌))
44	43	adantl 482	. . . . . 6 ⊢ ((∅ ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → ((lcm‘𝑌) lcm 1) = (lcm‘𝑌))
45	44	eqcomd 2738	. . . . 5 ⊢ ((∅ ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm‘𝑌) = ((lcm‘𝑌) lcm 1))
46		unass 4165	. . . . . . . . . . . . . 14 ⊢ ((𝑌 ∪ 𝑦) ∪ {𝑧}) = (𝑌 ∪ (𝑦 ∪ {𝑧}))
47	46	eqcomi 2741	. . . . . . . . . . . . 13 ⊢ (𝑌 ∪ (𝑦 ∪ {𝑧})) = ((𝑌 ∪ 𝑦) ∪ {𝑧})
48	47	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (𝑌 ∪ (𝑦 ∪ {𝑧})) = ((𝑌 ∪ 𝑦) ∪ {𝑧}))
49	48	fveq2d 6892	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (lcm‘(𝑌 ∪ (𝑦 ∪ {𝑧}))) = (lcm‘((𝑌 ∪ 𝑦) ∪ {𝑧})))
50		simpl 483	. . . . . . . . . . . . . . 15 ⊢ ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → 𝑌 ⊆ ℤ)
51	50	adantl 482	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → 𝑌 ⊆ ℤ)
52		unss 4183	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) ↔ (𝑦 ∪ {𝑧}) ⊆ ℤ)
53		simpl 483	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) → 𝑦 ⊆ ℤ)
54	52, 53	sylbir 234	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∪ {𝑧}) ⊆ ℤ → 𝑦 ⊆ ℤ)
55	54	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → 𝑦 ⊆ ℤ)
56	51, 55	unssd 4185	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (𝑌 ∪ 𝑦) ⊆ ℤ)
57	56	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (𝑌 ∪ 𝑦) ⊆ ℤ)
58		unfi 9168	. . . . . . . . . . . . . . . 16 ⊢ ((𝑌 ∈ Fin ∧ 𝑦 ∈ Fin) → (𝑌 ∪ 𝑦) ∈ Fin)
59	58	ex 413	. . . . . . . . . . . . . . 15 ⊢ (𝑌 ∈ Fin → (𝑦 ∈ Fin → (𝑌 ∪ 𝑦) ∈ Fin))
60	59	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → (𝑦 ∈ Fin → (𝑌 ∪ 𝑦) ∈ Fin))
61	60	adantl 482	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (𝑦 ∈ Fin → (𝑌 ∪ 𝑦) ∈ Fin))
62	61	impcom 408	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (𝑌 ∪ 𝑦) ∈ Fin)
63		vex 3478	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑧 ∈ V
64	63	snss 4788	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ ℤ ↔ {𝑧} ⊆ ℤ)
65	64	biimpri 227	. . . . . . . . . . . . . . . 16 ⊢ ({𝑧} ⊆ ℤ → 𝑧 ∈ ℤ)
66	65	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) → 𝑧 ∈ ℤ)
67	52, 66	sylbir 234	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∪ {𝑧}) ⊆ ℤ → 𝑧 ∈ ℤ)
68	67	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → 𝑧 ∈ ℤ)
69	68	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → 𝑧 ∈ ℤ)
70		lcmfunsn 16577	. . . . . . . . . . . 12 ⊢ (((𝑌 ∪ 𝑦) ⊆ ℤ ∧ (𝑌 ∪ 𝑦) ∈ Fin ∧ 𝑧 ∈ ℤ) → (lcm‘((𝑌 ∪ 𝑦) ∪ {𝑧})) = ((lcm‘(𝑌 ∪ 𝑦)) lcm 𝑧))
71	57, 62, 69, 70	syl3anc 1371	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (lcm‘((𝑌 ∪ 𝑦) ∪ {𝑧})) = ((lcm‘(𝑌 ∪ 𝑦)) lcm 𝑧))
72	49, 71	eqtrd 2772	. . . . . . . . . 10 ⊢ ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (lcm‘(𝑌 ∪ (𝑦 ∪ {𝑧}))) = ((lcm‘(𝑌 ∪ 𝑦)) lcm 𝑧))
73	72	adantr 481	. . . . . . . . 9 ⊢ (((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) ∧ ((𝑦 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm‘(𝑌 ∪ 𝑦)) = ((lcm‘𝑌) lcm (lcm‘𝑦)))) → (lcm‘(𝑌 ∪ (𝑦 ∪ {𝑧}))) = ((lcm‘(𝑌 ∪ 𝑦)) lcm 𝑧))
74	54	anim1i 615	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (𝑦 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)))
75	74	adantl 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‘(𝑌 ∪ 𝑦)) = ((lcm‘𝑌) lcm (lcm‘𝑦))) → ((𝑦 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm‘(𝑌 ∪ 𝑦)) = ((lcm‘𝑌) lcm (lcm‘𝑦))))
77	75, 76	mpan9 507	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) ∧ ((𝑦 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm‘(𝑌 ∪ 𝑦)) = ((lcm‘𝑌) lcm (lcm‘𝑦)))) → (lcm‘(𝑌 ∪ 𝑦)) = ((lcm‘𝑌) lcm (lcm‘𝑦)))
78	77	oveq1d 7420	. . . . . . . . . 10 ⊢ (((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) ∧ ((𝑦 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm‘(𝑌 ∪ 𝑦)) = ((lcm‘𝑌) lcm (lcm‘𝑦)))) → ((lcm‘(𝑌 ∪ 𝑦)) lcm 𝑧) = (((lcm‘𝑌) lcm (lcm‘𝑦)) lcm 𝑧))
79	34	adantl 482	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm‘𝑌) ∈ ℤ)
80	79	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (lcm‘𝑌) ∈ ℤ)
81	55	anim2i 617	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (𝑦 ∈ Fin ∧ 𝑦 ⊆ ℤ))
82	81	ancomd 462	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin))
83		lcmfcl 16561	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘𝑦) ∈ ℕ₀)
84	82, 83	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (lcm‘𝑦) ∈ ℕ₀)
85	84	nn0zd 12580	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (lcm‘𝑦) ∈ ℤ)
86		lcmass 16547	. . . . . . . . . . . 12 ⊢ (((lcm‘𝑌) ∈ ℤ ∧ (lcm‘𝑦) ∈ ℤ ∧ 𝑧 ∈ ℤ) → (((lcm‘𝑌) lcm (lcm‘𝑦)) lcm 𝑧) = ((lcm‘𝑌) lcm ((lcm‘𝑦) lcm 𝑧)))
87	80, 85, 69, 86	syl3anc 1371	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (((lcm‘𝑌) lcm (lcm‘𝑦)) lcm 𝑧) = ((lcm‘𝑌) lcm ((lcm‘𝑦) lcm 𝑧)))
88	87	adantr 481	. . . . . . . . . 10 ⊢ (((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) ∧ ((𝑦 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm‘(𝑌 ∪ 𝑦)) = ((lcm‘𝑌) lcm (lcm‘𝑦)))) → (((lcm‘𝑌) lcm (lcm‘𝑦)) lcm 𝑧) = ((lcm‘𝑌) lcm ((lcm‘𝑦) lcm 𝑧)))
89	78, 88	eqtrd 2772	. . . . . . . . 9 ⊢ (((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) ∧ ((𝑦 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm‘(𝑌 ∪ 𝑦)) = ((lcm‘𝑌) lcm (lcm‘𝑦)))) → ((lcm‘(𝑌 ∪ 𝑦)) lcm 𝑧) = ((lcm‘𝑌) lcm ((lcm‘𝑦) lcm 𝑧)))
90	73, 89	eqtrd 2772	. . . . . . . 8 ⊢ (((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) ∧ ((𝑦 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm‘(𝑌 ∪ 𝑦)) = ((lcm‘𝑌) lcm (lcm‘𝑦)))) → (lcm‘(𝑌 ∪ (𝑦 ∪ {𝑧}))) = ((lcm‘𝑌) lcm ((lcm‘𝑦) lcm 𝑧)))
91	53	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) ∧ 𝑦 ∈ Fin) → 𝑦 ⊆ ℤ)
92		simpr 485	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) ∧ 𝑦 ∈ Fin) → 𝑦 ∈ Fin)
93	66	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) ∧ 𝑦 ∈ Fin) → 𝑧 ∈ ℤ)
94	91, 92, 93	3jca 1128	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) ∧ 𝑦 ∈ Fin) → (𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ∧ 𝑧 ∈ ℤ))
95	94	ex 413	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) → (𝑦 ∈ Fin → (𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ∧ 𝑧 ∈ ℤ)))
96	52, 95	sylbir 234	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∪ {𝑧}) ⊆ ℤ → (𝑦 ∈ Fin → (𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ∧ 𝑧 ∈ ℤ)))
97	96	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (𝑦 ∈ Fin → (𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ∧ 𝑧 ∈ ℤ)))
98	97	impcom 408	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ∧ 𝑧 ∈ ℤ))
99		lcmfunsn 16577	. . . . . . . . . . . 12 ⊢ ((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ∧ 𝑧 ∈ ℤ) → (lcm‘(𝑦 ∪ {𝑧})) = ((lcm‘𝑦) lcm 𝑧))
100	98, 99	syl 17	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (lcm‘(𝑦 ∪ {𝑧})) = ((lcm‘𝑦) lcm 𝑧))
101	100	oveq2d 7421	. . . . . . . . . 10 ⊢ ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → ((lcm‘𝑌) lcm (lcm‘(𝑦 ∪ {𝑧}))) = ((lcm‘𝑌) lcm ((lcm‘𝑦) lcm 𝑧)))
102	101	eqeq2d 2743	. . . . . . . . 9 ⊢ ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → ((lcm‘(𝑌 ∪ (𝑦 ∪ {𝑧}))) = ((lcm‘𝑌) lcm (lcm‘(𝑦 ∪ {𝑧}))) ↔ (lcm‘(𝑌 ∪ (𝑦 ∪ {𝑧}))) = ((lcm‘𝑌) lcm ((lcm‘𝑦) lcm 𝑧))))
103	102	adantr 481	. . . . . . . 8 ⊢ (((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) ∧ ((𝑦 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm‘(𝑌 ∪ 𝑦)) = ((lcm‘𝑌) lcm (lcm‘𝑦)))) → ((lcm‘(𝑌 ∪ (𝑦 ∪ {𝑧}))) = ((lcm‘𝑌) lcm (lcm‘(𝑦 ∪ {𝑧}))) ↔ (lcm‘(𝑌 ∪ (𝑦 ∪ {𝑧}))) = ((lcm‘𝑌) lcm ((lcm‘𝑦) lcm 𝑧))))
104	90, 103	mpbird 256	. . . . . . 7 ⊢ (((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) ∧ ((𝑦 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm‘(𝑌 ∪ 𝑦)) = ((lcm‘𝑌) lcm (lcm‘𝑦)))) → (lcm‘(𝑌 ∪ (𝑦 ∪ {𝑧}))) = ((lcm‘𝑌) lcm (lcm‘(𝑦 ∪ {𝑧}))))
105	104	exp31 420	. . . . . 6 ⊢ (𝑦 ∈ Fin → (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (((𝑦 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm‘(𝑌 ∪ 𝑦)) = ((lcm‘𝑌) lcm (lcm‘𝑦))) → (lcm‘(𝑌 ∪ (𝑦 ∪ {𝑧}))) = ((lcm‘𝑌) lcm (lcm‘(𝑦 ∪ {𝑧}))))))
106	105	com23 86	. . . . 5 ⊢ (𝑦 ∈ Fin → (((𝑦 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm‘(𝑌 ∪ 𝑦)) = ((lcm‘𝑌) lcm (lcm‘𝑦))) → (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm‘(𝑌 ∪ (𝑦 ∪ {𝑧}))) = ((lcm‘𝑌) lcm (lcm‘(𝑦 ∪ {𝑧}))))))
107	11, 18, 25, 32, 45, 106	findcard2 9160	. . . 4 ⊢ (𝑍 ∈ Fin → ((𝑍 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm‘(𝑌 ∪ 𝑍)) = ((lcm‘𝑌) lcm (lcm‘𝑍))))
108	107	expd 416	. . 3 ⊢ (𝑍 ∈ Fin → (𝑍 ⊆ ℤ → ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → (lcm‘(𝑌 ∪ 𝑍)) = ((lcm‘𝑌) lcm (lcm‘𝑍)))))
109	108	impcom 408	. 2 ⊢ ((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) → ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → (lcm‘(𝑌 ∪ 𝑍)) = ((lcm‘𝑌) lcm (lcm‘𝑍))))
110	109	impcom 408	1 ⊢ (((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) ∧ (𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin)) → (lcm‘(𝑌 ∪ 𝑍)) = ((lcm‘𝑌) lcm (lcm‘𝑍)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∪ cun 3945 ⊆ wss 3947 ∅c0 4321 {csn 4627 class class class wbr 5147 ‘cfv 6540 (class class class)co 7405 Fincfn 8935 ℝcr 11105 0cc0 11106 1c1 11107 ≤ cle 11245 ℕ₀cn0 12468 ℤcz 12554 abscabs 15177 lcm clcm 16521 lcmclcmf 16522

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 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-z 12555 df-uz 12819 df-rp 12971 df-fz 13481 df-fzo 13624 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-prod 15846 df-dvds 16194 df-gcd 16432 df-lcm 16523 df-lcmf 16524

This theorem is referenced by: lcmfass 16579

W3C validator