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Theorem lcmfun 16691

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 15007	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝑥 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) ↔ (∅ ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))))
2		uneq2 4118	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝑌 ∪ 𝑥) = (𝑌 ∪ ∅))
3		un0 4351	. . . . . . . . 9 ⊢ (𝑌 ∪ ∅) = 𝑌
4	2, 3	eqtrdi 2816	. . . . . . . 8 ⊢ (𝑥 = ∅ → (𝑌 ∪ 𝑥) = 𝑌)
5	4	fveq2d 6875	. . . . . . 7 ⊢ (𝑥 = ∅ → (lcm‘(𝑌 ∪ 𝑥)) = (lcm‘𝑌))
6		fveq2 6871	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (lcm‘𝑥) = (lcm‘∅))
7		lcmf0 16680	. . . . . . . . 9 ⊢ (lcm‘∅) = 1
8	6, 7	eqtrdi 2816	. . . . . . . 8 ⊢ (𝑥 = ∅ → (lcm‘𝑥) = 1)
9	8	oveq2d 7416	. . . . . . 7 ⊢ (𝑥 = ∅ → ((lcm‘𝑌) lcm (lcm‘𝑥)) = ((lcm‘𝑌) lcm 1))
10	5, 9	eqeq12d 2781	. . . . . 6 ⊢ (𝑥 = ∅ → ((lcm‘(𝑌 ∪ 𝑥)) = ((lcm‘𝑌) lcm (lcm‘𝑥)) ↔ (lcm‘𝑌) = ((lcm‘𝑌) lcm 1)))
11	1, 10	imbi12d 347	. . . . 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 15007	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑥 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) ↔ (𝑦 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))))
13		uneq2 4118	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑌 ∪ 𝑥) = (𝑌 ∪ 𝑦))
14	13	fveq2d 6875	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (lcm‘(𝑌 ∪ 𝑥)) = (lcm‘(𝑌 ∪ 𝑦)))
15		fveq2 6871	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (lcm‘𝑥) = (lcm‘𝑦))
16	15	oveq2d 7416	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((lcm‘𝑌) lcm (lcm‘𝑥)) = ((lcm‘𝑌) lcm (lcm‘𝑦)))
17	14, 16	eqeq12d 2781	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((lcm‘(𝑌 ∪ 𝑥)) = ((lcm‘𝑌) lcm (lcm‘𝑥)) ↔ (lcm‘(𝑌 ∪ 𝑦)) = ((lcm‘𝑌) lcm (lcm‘𝑦))))
18	12, 17	imbi12d 347	. . . . 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 15007	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝑥 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) ↔ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))))
20		uneq2 4118	. . . . . . . 8 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝑌 ∪ 𝑥) = (𝑌 ∪ (𝑦 ∪ {𝑧})))
21	20	fveq2d 6875	. . . . . . 7 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (lcm‘(𝑌 ∪ 𝑥)) = (lcm‘(𝑌 ∪ (𝑦 ∪ {𝑧}))))
22		fveq2 6871	. . . . . . . 8 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (lcm‘𝑥) = (lcm‘(𝑦 ∪ {𝑧})))
23	22	oveq2d 7416	. . . . . . 7 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((lcm‘𝑌) lcm (lcm‘𝑥)) = ((lcm‘𝑌) lcm (lcm‘(𝑦 ∪ {𝑧}))))
24	21, 23	eqeq12d 2781	. . . . . 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 347	. . . . 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 15007	. . . . . 6 ⊢ (𝑥 = 𝑍 → ((𝑥 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) ↔ (𝑍 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))))
27		uneq2 4118	. . . . . . . 8 ⊢ (𝑥 = 𝑍 → (𝑌 ∪ 𝑥) = (𝑌 ∪ 𝑍))
28	27	fveq2d 6875	. . . . . . 7 ⊢ (𝑥 = 𝑍 → (lcm‘(𝑌 ∪ 𝑥)) = (lcm‘(𝑌 ∪ 𝑍)))
29		fveq2 6871	. . . . . . . 8 ⊢ (𝑥 = 𝑍 → (lcm‘𝑥) = (lcm‘𝑍))
30	29	oveq2d 7416	. . . . . . 7 ⊢ (𝑥 = 𝑍 → ((lcm‘𝑌) lcm (lcm‘𝑥)) = ((lcm‘𝑌) lcm (lcm‘𝑍)))
31	28, 30	eqeq12d 2781	. . . . . 6 ⊢ (𝑥 = 𝑍 → ((lcm‘(𝑌 ∪ 𝑥)) = ((lcm‘𝑌) lcm (lcm‘𝑥)) ↔ (lcm‘(𝑌 ∪ 𝑍)) = ((lcm‘𝑌) lcm (lcm‘𝑍))))
32	26, 31	imbi12d 347	. . . . 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 16674	. . . . . . . . . 10 ⊢ ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → (lcm‘𝑌) ∈ ℕ₀)
34	33	nn0zd 12604	. . . . . . . . 9 ⊢ ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → (lcm‘𝑌) ∈ ℤ)
35		lcm1 16656	. . . . . . . . 9 ⊢ ((lcm‘𝑌) ∈ ℤ → ((lcm‘𝑌) lcm 1) = (abs‘(lcm‘𝑌)))
36	34, 35	syl 18	. . . . . . . 8 ⊢ ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → ((lcm‘𝑌) lcm 1) = (abs‘(lcm‘𝑌)))
37		nn0re 12501	. . . . . . . . . . 11 ⊢ ((lcm‘𝑌) ∈ ℕ₀ → (lcm‘𝑌) ∈ ℝ)
38		nn0ge0 12517	. . . . . . . . . . 11 ⊢ ((lcm‘𝑌) ∈ ℕ₀ → 0 ≤ (lcm‘𝑌))
39	37, 38	jca 520	. . . . . . . . . 10 ⊢ ((lcm‘𝑌) ∈ ℕ₀ → ((lcm‘𝑌) ∈ ℝ ∧ 0 ≤ (lcm‘𝑌)))
40	33, 39	syl 18	. . . . . . . . 9 ⊢ ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → ((lcm‘𝑌) ∈ ℝ ∧ 0 ≤ (lcm‘𝑌)))
41		absid 15335	. . . . . . . . 9 ⊢ (((lcm‘𝑌) ∈ ℝ ∧ 0 ≤ (lcm‘𝑌)) → (abs‘(lcm‘𝑌)) = (lcm‘𝑌))
42	40, 41	syl 18	. . . . . . . 8 ⊢ ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → (abs‘(lcm‘𝑌)) = (lcm‘𝑌))
43	36, 42	eqtrd 2800	. . . . . . 7 ⊢ ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → ((lcm‘𝑌) lcm 1) = (lcm‘𝑌))
44	43	adantl 486	. . . . . 6 ⊢ ((∅ ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → ((lcm‘𝑌) lcm 1) = (lcm‘𝑌))
45	44	eqcomd 2771	. . . . 5 ⊢ ((∅ ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm‘𝑌) = ((lcm‘𝑌) lcm 1))
46		unass 4127	. . . . . . . . . . . . . 14 ⊢ ((𝑌 ∪ 𝑦) ∪ {𝑧}) = (𝑌 ∪ (𝑦 ∪ {𝑧}))
47	46	eqcomi 2774	. . . . . . . . . . . . 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 6875	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (lcm‘(𝑌 ∪ (𝑦 ∪ {𝑧}))) = (lcm‘((𝑌 ∪ 𝑦) ∪ {𝑧})))
50		simpl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → 𝑌 ⊆ ℤ)
51	50	adantl 486	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → 𝑌 ⊆ ℤ)
52		unss 4145	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) ↔ (𝑦 ∪ {𝑧}) ⊆ ℤ)
53		simpl 487	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) → 𝑦 ⊆ ℤ)
54	52, 53	sylbir 238	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∪ {𝑧}) ⊆ ℤ → 𝑦 ⊆ ℤ)
55	54	adantr 485	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → 𝑦 ⊆ ℤ)
56	51, 55	unssd 4147	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (𝑌 ∪ 𝑦) ⊆ ℤ)
57	56	adantl 486	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (𝑌 ∪ 𝑦) ⊆ ℤ)
58		unfi 9143	. . . . . . . . . . . . . . . 16 ⊢ ((𝑌 ∈ Fin ∧ 𝑦 ∈ Fin) → (𝑌 ∪ 𝑦) ∈ Fin)
59	58	ex 417	. . . . . . . . . . . . . . 15 ⊢ (𝑌 ∈ Fin → (𝑦 ∈ Fin → (𝑌 ∪ 𝑦) ∈ Fin))
60	59	adantl 486	. . . . . . . . . . . . . 14 ⊢ ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → (𝑦 ∈ Fin → (𝑌 ∪ 𝑦) ∈ Fin))
61	60	adantl 486	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (𝑦 ∈ Fin → (𝑌 ∪ 𝑦) ∈ Fin))
62	61	impcom 412	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (𝑌 ∪ 𝑦) ∈ Fin)
63		vex 3461	. . . . . . . . . . . . . . . . 17 ⊢ 𝑧 ∈ V
64	63	snss 4746	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ ℤ ↔ {𝑧} ⊆ ℤ)
65	64	bilanri 511	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) → 𝑧 ∈ ℤ)
66	52, 65	sylbir 238	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∪ {𝑧}) ⊆ ℤ → 𝑧 ∈ ℤ)
67	66	adantr 485	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → 𝑧 ∈ ℤ)
68	67	adantl 486	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → 𝑧 ∈ ℤ)
69		lcmfunsn 16690	. . . . . . . . . . . 12 ⊢ (((𝑌 ∪ 𝑦) ⊆ ℤ ∧ (𝑌 ∪ 𝑦) ∈ Fin ∧ 𝑧 ∈ ℤ) → (lcm‘((𝑌 ∪ 𝑦) ∪ {𝑧})) = ((lcm‘(𝑌 ∪ 𝑦)) lcm 𝑧))
70	57, 62, 68, 69	syl3anc 1394	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (lcm‘((𝑌 ∪ 𝑦) ∪ {𝑧})) = ((lcm‘(𝑌 ∪ 𝑦)) lcm 𝑧))
71	49, 70	eqtrd 2800	. . . . . . . . . 10 ⊢ ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (lcm‘(𝑌 ∪ (𝑦 ∪ {𝑧}))) = ((lcm‘(𝑌 ∪ 𝑦)) lcm 𝑧))
72	71	adantr 485	. . . . . . . . 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))
73	54	anim1i 626	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (𝑦 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)))
74	73	adantl 486	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (𝑦 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)))
75		id 23	. . . . . . . . . . . 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))))
76	74, 75	mpan9 515	. . . . . . . . . . 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)))
77	76	oveq1d 7415	. . . . . . . . . 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))
78	34	adantl 486	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm‘𝑌) ∈ ℤ)
79	78	adantl 486	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (lcm‘𝑌) ∈ ℤ)
80	55	anim2i 628	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (𝑦 ∈ Fin ∧ 𝑦 ⊆ ℤ))
81	80	ancomd 466	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin))
82		lcmfcl 16674	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘𝑦) ∈ ℕ₀)
83	81, 82	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (lcm‘𝑦) ∈ ℕ₀)
84	83	nn0zd 12604	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (lcm‘𝑦) ∈ ℤ)
85		lcmass 16660	. . . . . . . . . . . 12 ⊢ (((lcm‘𝑌) ∈ ℤ ∧ (lcm‘𝑦) ∈ ℤ ∧ 𝑧 ∈ ℤ) → (((lcm‘𝑌) lcm (lcm‘𝑦)) lcm 𝑧) = ((lcm‘𝑌) lcm ((lcm‘𝑦) lcm 𝑧)))
86	79, 84, 68, 85	syl3anc 1394	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (((lcm‘𝑌) lcm (lcm‘𝑦)) lcm 𝑧) = ((lcm‘𝑌) lcm ((lcm‘𝑦) lcm 𝑧)))
87	86	adantr 485	. . . . . . . . . 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)))
88	77, 87	eqtrd 2800	. . . . . . . . 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)))
89	72, 88	eqtrd 2800	. . . . . . . 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)))
90	53	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) ∧ 𝑦 ∈ Fin) → 𝑦 ⊆ ℤ)
91		simpr 489	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) ∧ 𝑦 ∈ Fin) → 𝑦 ∈ Fin)
92	65	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) ∧ 𝑦 ∈ Fin) → 𝑧 ∈ ℤ)
93	90, 91, 92	3jca 1144	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) ∧ 𝑦 ∈ Fin) → (𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ∧ 𝑧 ∈ ℤ))
94	93	ex 417	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) → (𝑦 ∈ Fin → (𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ∧ 𝑧 ∈ ℤ)))
95	52, 94	sylbir 238	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∪ {𝑧}) ⊆ ℤ → (𝑦 ∈ Fin → (𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ∧ 𝑧 ∈ ℤ)))
96	95	adantr 485	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (𝑦 ∈ Fin → (𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ∧ 𝑧 ∈ ℤ)))
97	96	impcom 412	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ∧ 𝑧 ∈ ℤ))
98		lcmfunsn 16690	. . . . . . . . . . . 12 ⊢ ((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin ∧ 𝑧 ∈ ℤ) → (lcm‘(𝑦 ∪ {𝑧})) = ((lcm‘𝑦) lcm 𝑧))
99	97, 98	syl 18	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → (lcm‘(𝑦 ∪ {𝑧})) = ((lcm‘𝑦) lcm 𝑧))
100	99	oveq2d 7416	. . . . . . . . . 10 ⊢ ((𝑦 ∈ Fin ∧ ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin))) → ((lcm‘𝑌) lcm (lcm‘(𝑦 ∪ {𝑧}))) = ((lcm‘𝑌) lcm ((lcm‘𝑦) lcm 𝑧)))
101	100	eqeq2d 2776	. . . . . . . . 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))))
102	101	adantr 485	. . . . . . . 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))))
103	89, 102	mpbird 260	. . . . . . 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}))))
104	103	exp31 424	. . . . . 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}))))))
105	104	com23 87	. . . . 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}))))))
106	11, 18, 25, 32, 45, 105	findcard2 9137	. . . 4 ⊢ (𝑍 ∈ Fin → ((𝑍 ⊆ ℤ ∧ (𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin)) → (lcm‘(𝑌 ∪ 𝑍)) = ((lcm‘𝑌) lcm (lcm‘𝑍))))
107	106	expd 420	. . 3 ⊢ (𝑍 ∈ Fin → (𝑍 ⊆ ℤ → ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → (lcm‘(𝑌 ∪ 𝑍)) = ((lcm‘𝑌) lcm (lcm‘𝑍)))))
108	107	impcom 412	. 2 ⊢ ((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) → ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → (lcm‘(𝑌 ∪ 𝑍)) = ((lcm‘𝑌) lcm (lcm‘𝑍))))
109	108	impcom 412	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 209 ∧ wa 400 ∧ w3a 1101 = wceq 1563 ∈ wcel 2145 ∪ cun 3905 ⊆ wss 3907 ∅c0 4288 {csn 4585 class class class wbr 5104 ‘cfv 6525 (class class class)co 7400 Fincfn 8931 ℝcr 11087 0cc0 11088 1c1 11089 ≤ cle 11232 ℕ₀cn0 12492 ℤcz 12579 abscabs 15273 lcm clcm 16634 lcmclcmf 16635

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5231 ax-sep 5250 ax-nul 5260 ax-pow 5326 ax-pr 5394 ax-un 7722 ax-inf2 9598 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165 ax-pre-sup 11166

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-int 4908 df-iun 4953 df-br 5105 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6291 df-ord 6352 df-on 6353 df-lim 6354 df-suc 6355 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-isom 6534 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-om 7851 df-1st 7974 df-2nd 7975 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-1o 8441 df-2o 8442 df-er 8682 df-en 8932 df-dom 8933 df-sdom 8934 df-fin 8935 df-sup 9390 df-inf 9391 df-oi 9460 df-card 9913 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-div 11860 df-nn 12222 df-2 12291 df-3 12292 df-n0 12493 df-z 12580 df-uz 12851 df-rp 13005 df-fz 13524 df-fzo 13671 df-fl 13813 df-mod 13891 df-seq 14026 df-exp 14086 df-hash 14355 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15527 df-prod 15946 df-dvds 16299 df-gcd 16541 df-lcm 16636 df-lcmf 16637

This theorem is referenced by: lcmfass 16692

W3C validator