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Theorem lcmfun 16622

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 14955	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝑥 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) ↔ (∅ ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))))
2		uneq2 4128	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝑌 ∪ 𝑥) = (𝑌 ∪ ∅))
3		un0 4360	. . . . . . . . 9 ⊢ (𝑌 ∪ ∅) = 𝑌
4	2, 3	eqtrdi 2781	. . . . . . . 8 ⊢ (𝑥 = ∅ → (𝑌 ∪ 𝑥) = 𝑌)
5	4	fveq2d 6865	. . . . . . 7 ⊢ (𝑥 = ∅ → (lcm‘(𝑌 ∪ 𝑥)) = (lcm‘𝑌))
6		fveq2 6861	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (lcm‘𝑥) = (lcm‘∅))
7		lcmf0 16611	. . . . . . . . 9 ⊢ (lcm‘∅) = 1
8	6, 7	eqtrdi 2781	. . . . . . . 8 ⊢ (𝑥 = ∅ → (lcm‘𝑥) = 1)
9	8	oveq2d 7406	. . . . . . 7 ⊢ (𝑥 = ∅ → ((lcm‘𝑌) lcm (lcm‘𝑥)) = ((lcm‘𝑌) lcm 1))
10	5, 9	eqeq12d 2746	. . . . . 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 14955	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑥 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) ↔ (𝑦 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))))
13		uneq2 4128	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑌 ∪ 𝑥) = (𝑌 ∪ 𝑦))
14	13	fveq2d 6865	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (lcm‘(𝑌 ∪ 𝑥)) = (lcm‘(𝑌 ∪ 𝑦)))
15		fveq2 6861	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (lcm‘𝑥) = (lcm‘𝑦))
16	15	oveq2d 7406	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((lcm‘𝑌) lcm (lcm‘𝑥)) = ((lcm‘𝑌) lcm (lcm‘𝑦)))
17	14, 16	eqeq12d 2746	. . . . . 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 14955	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝑥 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) ↔ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))))
20		uneq2 4128	. . . . . . . 8 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝑌 ∪ 𝑥) = (𝑌 ∪ (𝑦 ∪ {𝑧})))
21	20	fveq2d 6865	. . . . . . 7 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (lcm‘(𝑌 ∪ 𝑥)) = (lcm‘(𝑌 ∪ (𝑦 ∪ {𝑧}))))
22		fveq2 6861	. . . . . . . 8 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (lcm‘𝑥) = (lcm‘(𝑦 ∪ {𝑧})))
23	22	oveq2d 7406	. . . . . . 7 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((lcm‘𝑌) lcm (lcm‘𝑥)) = ((lcm‘𝑌) lcm (lcm‘(𝑦 ∪ {𝑧}))))
24	21, 23	eqeq12d 2746	. . . . . 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 14955	. . . . . 6 ⊢ (𝑥 = 𝑍 → ((𝑥 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) ↔ (𝑍 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))))
27		uneq2 4128	. . . . . . . 8 ⊢ (𝑥 = 𝑍 → (𝑌 ∪ 𝑥) = (𝑌 ∪ 𝑍))
28	27	fveq2d 6865	. . . . . . 7 ⊢ (𝑥 = 𝑍 → (lcm‘(𝑌 ∪ 𝑥)) = (lcm‘(𝑌 ∪ 𝑍)))
29		fveq2 6861	. . . . . . . 8 ⊢ (𝑥 = 𝑍 → (lcm‘𝑥) = (lcm‘𝑍))
30	29	oveq2d 7406	. . . . . . 7 ⊢ (𝑥 = 𝑍 → ((lcm‘𝑌) lcm (lcm‘𝑥)) = ((lcm‘𝑌) lcm (lcm‘𝑍)))
31	28, 30	eqeq12d 2746	. . . . . 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 16605	. . . . . . . . . 10 ⊢ ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → (lcm‘𝑌) ∈ ℕ₀)
34	33	nn0zd 12562	. . . . . . . . 9 ⊢ ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → (lcm‘𝑌) ∈ ℤ)
35		lcm1 16587	. . . . . . . . 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 12458	. . . . . . . . . . 11 ⊢ ((lcm‘𝑌) ∈ ℕ₀ → (lcm‘𝑌) ∈ ℝ)
38		nn0ge0 12474	. . . . . . . . . . 11 ⊢ ((lcm‘𝑌) ∈ ℕ₀ → 0 ≤ (lcm‘𝑌))
39	37, 38	jca 511	. . . . . . . . . 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 15269	. . . . . . . . 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 2765	. . . . . . 7 ⊢ ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → ((lcm‘𝑌) lcm 1) = (lcm‘𝑌))
44	43	adantl 481	. . . . . 6 ⊢ ((∅ ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → ((lcm‘𝑌) lcm 1) = (lcm‘𝑌))
45	44	eqcomd 2736	. . . . 5 ⊢ ((∅ ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm‘𝑌) = ((lcm‘𝑌) lcm 1))
46		unass 4138	. . . . . . . . . . . . . 14 ⊢ ((𝑌 ∪ 𝑦) ∪ {𝑧}) = (𝑌 ∪ (𝑦 ∪ {𝑧}))
47	46	eqcomi 2739	. . . . . . . . . . . . 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 6865	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (lcm‘(𝑌 ∪ (𝑦 ∪ {𝑧}))) = (lcm‘((𝑌 ∪ 𝑦) ∪ {𝑧})))
50		simpl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → 𝑌 ⊆ ℤ)
51	50	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → 𝑌 ⊆ ℤ)
52		unss 4156	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) ↔ (𝑦 ∪ {𝑧}) ⊆ ℤ)
53		simpl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) → 𝑦 ⊆ ℤ)
54	52, 53	sylbir 235	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∪ {𝑧}) ⊆ ℤ → 𝑦 ⊆ ℤ)
55	54	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → 𝑦 ⊆ ℤ)
56	51, 55	unssd 4158	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (𝑌 ∪ 𝑦) ⊆ ℤ)
57	56	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (𝑌 ∪ 𝑦) ⊆ ℤ)
58		unfi 9141	. . . . . . . . . . . . . . . 16 ⊢ ((𝑌 ∈ Fin ∧ 𝑦 ∈ Fin) → (𝑌 ∪ 𝑦) ∈ Fin)
59	58	ex 412	. . . . . . . . . . . . . . 15 ⊢ (𝑌 ∈ Fin → (𝑦 ∈ Fin → (𝑌 ∪ 𝑦) ∈ Fin))
60	59	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → (𝑦 ∈ Fin → (𝑌 ∪ 𝑦) ∈ Fin))
61	60	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (𝑦 ∈ Fin → (𝑌 ∪ 𝑦) ∈ Fin))
62	61	impcom 407	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (𝑌 ∪ 𝑦) ∈ Fin)
63		vex 3454	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑧 ∈ V
64	63	snss 4752	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ ℤ ↔ {𝑧} ⊆ ℤ)
65	64	biimpri 228	. . . . . . . . . . . . . . . 16 ⊢ ({𝑧} ⊆ ℤ → 𝑧 ∈ ℤ)
66	65	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) → 𝑧 ∈ ℤ)
67	52, 66	sylbir 235	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∪ {𝑧}) ⊆ ℤ → 𝑧 ∈ ℤ)
68	67	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → 𝑧 ∈ ℤ)
69	68	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → 𝑧 ∈ ℤ)
70		lcmfunsn 16621	. . . . . . . . . . . 12 ⊢ (((𝑌 ∪ 𝑦) ⊆ ℤ ∧ (𝑌 ∪ 𝑦) ∈ Fin ∧ 𝑧 ∈ ℤ) → (lcm‘((𝑌 ∪ 𝑦) ∪ {𝑧})) = ((lcm‘(𝑌 ∪ 𝑦)) lcm 𝑧))
71	57, 62, 69, 70	syl3anc 1373	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (lcm‘((𝑌 ∪ 𝑦) ∪ {𝑧})) = ((lcm‘(𝑌 ∪ 𝑦)) lcm 𝑧))
72	49, 71	eqtrd 2765	. . . . . . . . . 10 ⊢ ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (lcm‘(𝑌 ∪ (𝑦 ∪ {𝑧}))) = ((lcm‘(𝑌 ∪ 𝑦)) lcm 𝑧))
73	72	adantr 480	. . . . . . . . 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 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‘(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 506	. . . . . . . . . . 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 7405	. . . . . . . . . 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 481	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm‘𝑌) ∈ ℤ)
80	79	adantl 481	. . . . . . . . . . . 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 461	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin))
83		lcmfcl 16605	. . . . . . . . . . . . . 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 12562	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (lcm‘𝑦) ∈ ℤ)
86		lcmass 16591	. . . . . . . . . . . 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 1373	. . . . . . . . . . 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 480	. . . . . . . . . 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 2765	. . . . . . . . 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 2765	. . . . . . . 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 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) ∧ 𝑦 ∈ Fin) → 𝑦 ⊆ ℤ)
92		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) ∧ 𝑦 ∈ Fin) → 𝑦 ∈ Fin)
93	66	adantr 480	. . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) → (𝑦 ∈ Fin → (𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ∧ 𝑧 ∈ ℤ)))
96	52, 95	sylbir 235	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∪ {𝑧}) ⊆ ℤ → (𝑦 ∈ Fin → (𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ∧ 𝑧 ∈ ℤ)))
97	96	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (𝑦 ∈ Fin → (𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ∧ 𝑧 ∈ ℤ)))
98	97	impcom 407	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ∧ 𝑧 ∈ ℤ))
99		lcmfunsn 16621	. . . . . . . . . . . 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 7406	. . . . . . . . . 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 2741	. . . . . . . . 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 480	. . . . . . . 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 257	. . . . . . 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 419	. . . . . 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 9134	. . . 4 ⊢ (𝑍 ∈ Fin → ((𝑍 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm‘(𝑌 ∪ 𝑍)) = ((lcm‘𝑌) lcm (lcm‘𝑍))))
108	107	expd 415	. . 3 ⊢ (𝑍 ∈ Fin → (𝑍 ⊆ ℤ → ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → (lcm‘(𝑌 ∪ 𝑍)) = ((lcm‘𝑌) lcm (lcm‘𝑍)))))
109	108	impcom 407	. 2 ⊢ ((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) → ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → (lcm‘(𝑌 ∪ 𝑍)) = ((lcm‘𝑌) lcm (lcm‘𝑍))))
110	109	impcom 407	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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∪ cun 3915 ⊆ wss 3917 ∅c0 4299 {csn 4592 class class class wbr 5110 ‘cfv 6514 (class class class)co 7390 Fincfn 8921 ℝcr 11074 0cc0 11075 1c1 11076 ≤ cle 11216 ℕ₀cn0 12449 ℤcz 12536 abscabs 15207 lcm clcm 16565 lcmclcmf 16566

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-inf2 9601 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-se 5595 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-isom 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8674 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-sup 9400 df-inf 9401 df-oi 9470 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-n0 12450 df-z 12537 df-uz 12801 df-rp 12959 df-fz 13476 df-fzo 13623 df-fl 13761 df-mod 13839 df-seq 13974 df-exp 14034 df-hash 14303 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-clim 15461 df-prod 15877 df-dvds 16230 df-gcd 16472 df-lcm 16567 df-lcmf 16568

This theorem is referenced by: lcmfass 16623

W3C validator