Theorem prodssdc 12339

Description: Change the index set to a subset in an upper integer product. (Contributed by Scott Fenton, 11-Dec-2017.) (Revised by Jim Kingdon, 6-Aug-2024.)

Hypotheses

Ref	Expression
prodss.1	⊢ (𝜑 → 𝐴 ⊆ 𝐵)
prodss.2	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ)
prodssdc.3	⊢ (𝜑 → ∃𝑛 ∈ (ℤ≥‘𝑀)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐵, 𝐶, 1))) ⇝ 𝑦))
prodssdc.a	⊢ (𝜑 → ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴)
prodssdc.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
prodss.4	⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 = 1)
prodss.5	⊢ (𝜑 → 𝐵 ⊆ (ℤ≥‘𝑀))
prodssdc.b	⊢ (𝜑 → ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐵)

Assertion

Ref	Expression
prodssdc	⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶)

Distinct variable groups:   𝐴,𝑗,𝑘,𝑛,𝑦   𝐵,𝑗,𝑘,𝑛,𝑦   𝐶,𝑗,𝑛,𝑦   𝑗,𝑀,𝑘,𝑛,𝑦   𝜑,𝑗,𝑘,𝑛,𝑦

Allowed substitution hint:   𝐶(𝑘)

Proof of Theorem prodssdc

Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2238	. . . 4 ⊢ (ℤ≥‘𝑀) = (ℤ≥‘𝑀)
2		prodssdc.m	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ)
3		prodssdc.3	. . . 4 ⊢ (𝜑 → ∃𝑛 ∈ (ℤ≥‘𝑀)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐵, 𝐶, 1))) ⇝ 𝑦))
4		prodss.1	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
5		prodss.5	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ (ℤ≥‘𝑀))
6	4, 5	sstrd 3258	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ (ℤ≥‘𝑀))
7		prodssdc.a	. . . 4 ⊢ (𝜑 → ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴)
8		simpr 110	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑀)) → 𝑚 ∈ (ℤ≥‘𝑀))
9		eleq1w 2299	. . . . . . . . . 10 ⊢ (𝑗 = 𝑚 → (𝑗 ∈ 𝐵 ↔ 𝑚 ∈ 𝐵))
10	9	dcbid 850	. . . . . . . . 9 ⊢ (𝑗 = 𝑚 → (DECID 𝑗 ∈ 𝐵 ↔ DECID 𝑚 ∈ 𝐵))
11		prodssdc.b	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐵)
12	11	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑀)) → ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐵)
13	10, 12, 8	rspcdva 2934	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑀)) → DECID 𝑚 ∈ 𝐵)
14		exmiddc 848	. . . . . . . 8 ⊢ (DECID 𝑚 ∈ 𝐵 → (𝑚 ∈ 𝐵 ∨ ¬ 𝑚 ∈ 𝐵))
15	13, 14	syl 14	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑀)) → (𝑚 ∈ 𝐵 ∨ ¬ 𝑚 ∈ 𝐵))
16		iftrue 3645	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ 𝐵 → if(𝑚 ∈ 𝐵, ⦋𝑚 / 𝑘⦌𝐶, 1) = ⦋𝑚 / 𝑘⦌𝐶)
17	16	adantl 277	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐵) → if(𝑚 ∈ 𝐵, ⦋𝑚 / 𝑘⦌𝐶, 1) = ⦋𝑚 / 𝑘⦌𝐶)
18		prodss.2	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ)
19	18	ex 115	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝐶 ∈ ℂ))
20	19	adantr 276	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑘 ∈ 𝐴 → 𝐶 ∈ ℂ))
21		eldif 3229	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (𝐵 ∖ 𝐴) ↔ (𝑘 ∈ 𝐵 ∧ ¬ 𝑘 ∈ 𝐴))
22		prodss.4	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 = 1)
23		ax-1cn 8266	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℂ
24	22, 23	eqeltrdi 2329	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 ∈ ℂ)
25	21, 24	sylan2br 288	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐵 ∧ ¬ 𝑘 ∈ 𝐴)) → 𝐶 ∈ ℂ)
26	25	expr 375	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (¬ 𝑘 ∈ 𝐴 → 𝐶 ∈ ℂ))
27		eleq1w 2299	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑘 → (𝑗 ∈ 𝐴 ↔ 𝑘 ∈ 𝐴))
28	27	dcbid 850	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑘 → (DECID 𝑗 ∈ 𝐴 ↔ DECID 𝑘 ∈ 𝐴))
29	7	adantr 276	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴)
30	5	sselda 3248	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝑘 ∈ (ℤ≥‘𝑀))
31	28, 29, 30	rspcdva 2934	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → DECID 𝑘 ∈ 𝐴)
32		exmiddc 848	. . . . . . . . . . . . . . 15 ⊢ (DECID 𝑘 ∈ 𝐴 → (𝑘 ∈ 𝐴 ∨ ¬ 𝑘 ∈ 𝐴))
33	31, 32	syl 14	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑘 ∈ 𝐴 ∨ ¬ 𝑘 ∈ 𝐴))
34	20, 26, 33	mpjaod 730	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ)
35	34	ralrimiva 2623	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑘 ∈ 𝐵 𝐶 ∈ ℂ)
36		nfcsb1v 3180	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘⦋𝑚 / 𝑘⦌𝐶
37	36	nfel1 2403	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘⦋𝑚 / 𝑘⦌𝐶 ∈ ℂ
38		csbeq1a 3156	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑚 → 𝐶 = ⦋𝑚 / 𝑘⦌𝐶)
39	38	eleq1d 2307	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑚 → (𝐶 ∈ ℂ ↔ ⦋𝑚 / 𝑘⦌𝐶 ∈ ℂ))
40	37, 39	rspc 2923	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ 𝐵 → (∀𝑘 ∈ 𝐵 𝐶 ∈ ℂ → ⦋𝑚 / 𝑘⦌𝐶 ∈ ℂ))
41	35, 40	mpan9 281	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐵) → ⦋𝑚 / 𝑘⦌𝐶 ∈ ℂ)
42	17, 41	eqeltrd 2315	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐵) → if(𝑚 ∈ 𝐵, ⦋𝑚 / 𝑘⦌𝐶, 1) ∈ ℂ)
43	42	ex 115	. . . . . . . . 9 ⊢ (𝜑 → (𝑚 ∈ 𝐵 → if(𝑚 ∈ 𝐵, ⦋𝑚 / 𝑘⦌𝐶, 1) ∈ ℂ))
44		iffalse 3648	. . . . . . . . . . 11 ⊢ (¬ 𝑚 ∈ 𝐵 → if(𝑚 ∈ 𝐵, ⦋𝑚 / 𝑘⦌𝐶, 1) = 1)
45	44, 23	eqeltrdi 2329	. . . . . . . . . 10 ⊢ (¬ 𝑚 ∈ 𝐵 → if(𝑚 ∈ 𝐵, ⦋𝑚 / 𝑘⦌𝐶, 1) ∈ ℂ)
46	45	a1i 9	. . . . . . . . 9 ⊢ (𝜑 → (¬ 𝑚 ∈ 𝐵 → if(𝑚 ∈ 𝐵, ⦋𝑚 / 𝑘⦌𝐶, 1) ∈ ℂ))
47	43, 46	jaod 729	. . . . . . . 8 ⊢ (𝜑 → ((𝑚 ∈ 𝐵 ∨ ¬ 𝑚 ∈ 𝐵) → if(𝑚 ∈ 𝐵, ⦋𝑚 / 𝑘⦌𝐶, 1) ∈ ℂ))
48	47	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑀)) → ((𝑚 ∈ 𝐵 ∨ ¬ 𝑚 ∈ 𝐵) → if(𝑚 ∈ 𝐵, ⦋𝑚 / 𝑘⦌𝐶, 1) ∈ ℂ))
49	15, 48	mpd 13	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑀)) → if(𝑚 ∈ 𝐵, ⦋𝑚 / 𝑘⦌𝐶, 1) ∈ ℂ)
50		nfcv 2392	. . . . . . 7 ⊢ Ⅎ𝑘𝑚
51		nfv 1581	. . . . . . . 8 ⊢ Ⅎ𝑘 𝑚 ∈ 𝐵
52		nfcv 2392	. . . . . . . 8 ⊢ Ⅎ𝑘1
53	51, 36, 52	nfif 3669	. . . . . . 7 ⊢ Ⅎ𝑘if(𝑚 ∈ 𝐵, ⦋𝑚 / 𝑘⦌𝐶, 1)
54		eleq1w 2299	. . . . . . . 8 ⊢ (𝑘 = 𝑚 → (𝑘 ∈ 𝐵 ↔ 𝑚 ∈ 𝐵))
55	54, 38	ifbieq1d 3663	. . . . . . 7 ⊢ (𝑘 = 𝑚 → if(𝑘 ∈ 𝐵, 𝐶, 1) = if(𝑚 ∈ 𝐵, ⦋𝑚 / 𝑘⦌𝐶, 1))
56		eqid 2238	. . . . . . 7 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐵, 𝐶, 1)) = (𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐵, 𝐶, 1))
57	50, 53, 55, 56	fvmptf 5795	. . . . . 6 ⊢ ((𝑚 ∈ (ℤ≥‘𝑀) ∧ if(𝑚 ∈ 𝐵, ⦋𝑚 / 𝑘⦌𝐶, 1) ∈ ℂ) → ((𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐵, 𝐶, 1))‘𝑚) = if(𝑚 ∈ 𝐵, ⦋𝑚 / 𝑘⦌𝐶, 1))
58	8, 49, 57	syl2anc 415	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑀)) → ((𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐵, 𝐶, 1))‘𝑚) = if(𝑚 ∈ 𝐵, ⦋𝑚 / 𝑘⦌𝐶, 1))
59		iftrue 3645	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ 𝐴 → if(𝑚 ∈ 𝐴, ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚), 1) = ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚))
60	59	adantl 277	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → if(𝑚 ∈ 𝐴, ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚), 1) = ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚))
61		simpr 110	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → 𝑚 ∈ 𝐴)
62	4	sselda 3248	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → 𝑚 ∈ 𝐵)
63	62, 41	syldan 282	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → ⦋𝑚 / 𝑘⦌𝐶 ∈ ℂ)
64		eqid 2238	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐶) = (𝑘 ∈ 𝐴 ↦ 𝐶)
65	64	fvmpts 5780	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ 𝐴 ∧ ⦋𝑚 / 𝑘⦌𝐶 ∈ ℂ) → ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = ⦋𝑚 / 𝑘⦌𝐶)
66	61, 63, 65	syl2anc 415	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = ⦋𝑚 / 𝑘⦌𝐶)
67	60, 66	eqtrd 2271	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → if(𝑚 ∈ 𝐴, ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚), 1) = ⦋𝑚 / 𝑘⦌𝐶)
68	67	ex 115	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑚 ∈ 𝐴 → if(𝑚 ∈ 𝐴, ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚), 1) = ⦋𝑚 / 𝑘⦌𝐶))
69	68	adantr 276	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐵) → (𝑚 ∈ 𝐴 → if(𝑚 ∈ 𝐴, ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚), 1) = ⦋𝑚 / 𝑘⦌𝐶))
70		iffalse 3648	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑚 ∈ 𝐴 → if(𝑚 ∈ 𝐴, ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚), 1) = 1)
71	70	adantl 277	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ 𝐵 ∧ ¬ 𝑚 ∈ 𝐴) → if(𝑚 ∈ 𝐴, ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚), 1) = 1)
72	71	adantl 277	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝐵 ∧ ¬ 𝑚 ∈ 𝐴)) → if(𝑚 ∈ 𝐴, ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚), 1) = 1)
73		eldif 3229	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (𝐵 ∖ 𝐴) ↔ (𝑚 ∈ 𝐵 ∧ ¬ 𝑚 ∈ 𝐴))
74	22	ralrimiva 2623	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑘 ∈ (𝐵 ∖ 𝐴)𝐶 = 1)
75	36	nfeq1 2402	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘⦋𝑚 / 𝑘⦌𝐶 = 1
76	38	eqeq1d 2247	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑚 → (𝐶 = 1 ↔ ⦋𝑚 / 𝑘⦌𝐶 = 1))
77	75, 76	rspc 2923	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ (𝐵 ∖ 𝐴) → (∀𝑘 ∈ (𝐵 ∖ 𝐴)𝐶 = 1 → ⦋𝑚 / 𝑘⦌𝐶 = 1))
78	74, 77	mpan9 281	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ (𝐵 ∖ 𝐴)) → ⦋𝑚 / 𝑘⦌𝐶 = 1)
79	73, 78	sylan2br 288	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝐵 ∧ ¬ 𝑚 ∈ 𝐴)) → ⦋𝑚 / 𝑘⦌𝐶 = 1)
80	72, 79	eqtr4d 2274	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝐵 ∧ ¬ 𝑚 ∈ 𝐴)) → if(𝑚 ∈ 𝐴, ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚), 1) = ⦋𝑚 / 𝑘⦌𝐶)
81	80	expr 375	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐵) → (¬ 𝑚 ∈ 𝐴 → if(𝑚 ∈ 𝐴, ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚), 1) = ⦋𝑚 / 𝑘⦌𝐶))
82		eleq1w 2299	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑚 → (𝑗 ∈ 𝐴 ↔ 𝑚 ∈ 𝐴))
83	82	dcbid 850	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑚 → (DECID 𝑗 ∈ 𝐴 ↔ DECID 𝑚 ∈ 𝐴))
84	7	adantr 276	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐵) → ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴)
85	5	sselda 3248	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐵) → 𝑚 ∈ (ℤ≥‘𝑀))
86	83, 84, 85	rspcdva 2934	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐵) → DECID 𝑚 ∈ 𝐴)
87		exmiddc 848	. . . . . . . . . . . 12 ⊢ (DECID 𝑚 ∈ 𝐴 → (𝑚 ∈ 𝐴 ∨ ¬ 𝑚 ∈ 𝐴))
88	86, 87	syl 14	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐵) → (𝑚 ∈ 𝐴 ∨ ¬ 𝑚 ∈ 𝐴))
89	69, 81, 88	mpjaod 730	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐵) → if(𝑚 ∈ 𝐴, ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚), 1) = ⦋𝑚 / 𝑘⦌𝐶)
90	89, 17	eqtr4d 2274	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐵) → if(𝑚 ∈ 𝐴, ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚), 1) = if(𝑚 ∈ 𝐵, ⦋𝑚 / 𝑘⦌𝐶, 1))
91	90	ex 115	. . . . . . . 8 ⊢ (𝜑 → (𝑚 ∈ 𝐵 → if(𝑚 ∈ 𝐴, ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚), 1) = if(𝑚 ∈ 𝐵, ⦋𝑚 / 𝑘⦌𝐶, 1)))
92	4	ssneld 3250	. . . . . . . . . . . 12 ⊢ (𝜑 → (¬ 𝑚 ∈ 𝐵 → ¬ 𝑚 ∈ 𝐴))
93	92	imp 124	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝑚 ∈ 𝐵) → ¬ 𝑚 ∈ 𝐴)
94	93, 70	syl 14	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝑚 ∈ 𝐵) → if(𝑚 ∈ 𝐴, ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚), 1) = 1)
95	44	adantl 277	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝑚 ∈ 𝐵) → if(𝑚 ∈ 𝐵, ⦋𝑚 / 𝑘⦌𝐶, 1) = 1)
96	94, 95	eqtr4d 2274	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑚 ∈ 𝐵) → if(𝑚 ∈ 𝐴, ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚), 1) = if(𝑚 ∈ 𝐵, ⦋𝑚 / 𝑘⦌𝐶, 1))
97	96	ex 115	. . . . . . . 8 ⊢ (𝜑 → (¬ 𝑚 ∈ 𝐵 → if(𝑚 ∈ 𝐴, ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚), 1) = if(𝑚 ∈ 𝐵, ⦋𝑚 / 𝑘⦌𝐶, 1)))
98	91, 97	jaod 729	. . . . . . 7 ⊢ (𝜑 → ((𝑚 ∈ 𝐵 ∨ ¬ 𝑚 ∈ 𝐵) → if(𝑚 ∈ 𝐴, ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚), 1) = if(𝑚 ∈ 𝐵, ⦋𝑚 / 𝑘⦌𝐶, 1)))
99	98	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑀)) → ((𝑚 ∈ 𝐵 ∨ ¬ 𝑚 ∈ 𝐵) → if(𝑚 ∈ 𝐴, ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚), 1) = if(𝑚 ∈ 𝐵, ⦋𝑚 / 𝑘⦌𝐶, 1)))
100	15, 99	mpd 13	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑀)) → if(𝑚 ∈ 𝐴, ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚), 1) = if(𝑚 ∈ 𝐵, ⦋𝑚 / 𝑘⦌𝐶, 1))
101	58, 100	eqtr4d 2274	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑀)) → ((𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐵, 𝐶, 1))‘𝑚) = if(𝑚 ∈ 𝐴, ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚), 1))
102	18	fmpttd 5857	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐶):𝐴⟶ℂ)
103	102	ffvelcdmda 5837	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) ∈ ℂ)
104	1, 2, 3, 6, 7, 101, 103	zproddc 12329	. . 3 ⊢ (𝜑 → ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = ( ⇝ ‘seq𝑀( · , (𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐵, 𝐶, 1)))))
105		simpr 110	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐵) → 𝑚 ∈ 𝐵)
106		eqid 2238	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝐵 ↦ 𝐶) = (𝑘 ∈ 𝐵 ↦ 𝐶)
107	106	fvmpts 5780	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ 𝐵 ∧ ⦋𝑚 / 𝑘⦌𝐶 ∈ ℂ) → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚) = ⦋𝑚 / 𝑘⦌𝐶)
108	105, 41, 107	syl2anc 415	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐵) → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚) = ⦋𝑚 / 𝑘⦌𝐶)
109	108	ifeq1d 3658	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐵) → if(𝑚 ∈ 𝐵, ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚), 1) = if(𝑚 ∈ 𝐵, ⦋𝑚 / 𝑘⦌𝐶, 1))
110	109	ex 115	. . . . . . . 8 ⊢ (𝜑 → (𝑚 ∈ 𝐵 → if(𝑚 ∈ 𝐵, ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚), 1) = if(𝑚 ∈ 𝐵, ⦋𝑚 / 𝑘⦌𝐶, 1)))
111		iffalse 3648	. . . . . . . . . 10 ⊢ (¬ 𝑚 ∈ 𝐵 → if(𝑚 ∈ 𝐵, ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚), 1) = 1)
112	111, 44	eqtr4d 2274	. . . . . . . . 9 ⊢ (¬ 𝑚 ∈ 𝐵 → if(𝑚 ∈ 𝐵, ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚), 1) = if(𝑚 ∈ 𝐵, ⦋𝑚 / 𝑘⦌𝐶, 1))
113	112	a1i 9	. . . . . . . 8 ⊢ (𝜑 → (¬ 𝑚 ∈ 𝐵 → if(𝑚 ∈ 𝐵, ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚), 1) = if(𝑚 ∈ 𝐵, ⦋𝑚 / 𝑘⦌𝐶, 1)))
114	110, 113	jaod 729	. . . . . . 7 ⊢ (𝜑 → ((𝑚 ∈ 𝐵 ∨ ¬ 𝑚 ∈ 𝐵) → if(𝑚 ∈ 𝐵, ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚), 1) = if(𝑚 ∈ 𝐵, ⦋𝑚 / 𝑘⦌𝐶, 1)))
115	114	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑀)) → ((𝑚 ∈ 𝐵 ∨ ¬ 𝑚 ∈ 𝐵) → if(𝑚 ∈ 𝐵, ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚), 1) = if(𝑚 ∈ 𝐵, ⦋𝑚 / 𝑘⦌𝐶, 1)))
116	15, 115	mpd 13	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑀)) → if(𝑚 ∈ 𝐵, ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚), 1) = if(𝑚 ∈ 𝐵, ⦋𝑚 / 𝑘⦌𝐶, 1))
117	58, 116	eqtr4d 2274	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑀)) → ((𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐵, 𝐶, 1))‘𝑚) = if(𝑚 ∈ 𝐵, ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚), 1))
118	34	fmpttd 5857	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐵 ↦ 𝐶):𝐵⟶ℂ)
119	118	ffvelcdmda 5837	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐵) → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚) ∈ ℂ)
120	1, 2, 3, 5, 11, 117, 119	zproddc 12329	. . 3 ⊢ (𝜑 → ∏𝑚 ∈ 𝐵 ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚) = ( ⇝ ‘seq𝑀( · , (𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐵, 𝐶, 1)))))
121	104, 120	eqtr4d 2274	. 2 ⊢ (𝜑 → ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = ∏𝑚 ∈ 𝐵 ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚))
122	18	ralrimiva 2623	. . 3 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐶 ∈ ℂ)
123		prodfct 12337	. . 3 ⊢ (∀𝑘 ∈ 𝐴 𝐶 ∈ ℂ → ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = ∏𝑘 ∈ 𝐴 𝐶)
124	122, 123	syl 14	. 2 ⊢ (𝜑 → ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = ∏𝑘 ∈ 𝐴 𝐶)
125		prodfct 12337	. . 3 ⊢ (∀𝑘 ∈ 𝐵 𝐶 ∈ ℂ → ∏𝑚 ∈ 𝐵 ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚) = ∏𝑘 ∈ 𝐵 𝐶)
126	35, 125	syl 14	. 2 ⊢ (𝜑 → ∏𝑚 ∈ 𝐵 ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚) = ∏𝑘 ∈ 𝐵 𝐶)
127	121, 124, 126	3eqtr3d 2279	1 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 720  DECID wdc 846   = wceq 1402  ∃wex 1545   ∈ wcel 2209  ∀wral 2528  ∃wrex 2529  ⦋csb 3147   ∖ cdif 3217   ⊆ wss 3220  ifcif 3638   class class class wbr 4128   ↦ cmpt 4190  ‘cfv 5375  ℂcc 8171  0cc0 8173  1c1 8174   · cmul 8178   # cap 8903  ℤcz 9627  ℤ_≥cuz 9904  seqcseq 10867   ⇝ cli 12027  ∏cprod 12300

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-mulrcl 8272  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-precex 8283  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289  ax-pre-mulgt0 8290  ax-pre-mulext 8291

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-isom 5384  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-irdg 6635  df-frec 6656  df-1o 6681  df-oadd 6685  df-er 6801  df-en 7017  df-dom 7018  df-fin 7019  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-reap 8897  df-ap 8904  df-div 8997  df-inn 9288  df-2 9346  df-n0 9547  df-z 9628  df-uz 9905  df-q 10003  df-rp 10038  df-fz 10395  df-fzo 10533  df-seqfrec 10868  df-exp 10959  df-ihash 11198  df-cj 11590  df-rsqrt 11747  df-abs 11748  df-clim 12028  df-proddc 12301

This theorem is referenced by:  fprodssdc  12340