Theorem fprodf1o 16026

Description: Re-index a finite product using a bijection. (Contributed by Scott Fenton, 7-Dec-2017.)

Hypotheses

Ref	Expression
fprodf1o.1	⊢ (𝑘 = 𝐺 → 𝐵 = 𝐷)
fprodf1o.2	⊢ (𝜑 → 𝐶 ∈ Fin)
fprodf1o.3	⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴)
fprodf1o.4	⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → (𝐹‘𝑛) = 𝐺)
fprodf1o.5	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)

Assertion

Ref	Expression
fprodf1o	⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑛 ∈ 𝐶 𝐷)

Distinct variable groups:   𝐴,𝑘,𝑛   𝐵,𝑛   𝐶,𝑛   𝐷,𝑘   𝑛,𝐹   𝑘,𝐺   𝑘,𝑛,𝜑

Allowed substitution hints:   𝐵(𝑘)   𝐶(𝑘)   𝐷(𝑛)   𝐹(𝑘)   𝐺(𝑛)

Proof of Theorem fprodf1o

Dummy variables 𝑓 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prod0 16023	. . . 4 ⊢ ∏𝑘 ∈ ∅ 𝐵 = 1
2		fprodf1o.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴)
3	2	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 = ∅) → 𝐹:𝐶–1-1-onto→𝐴)
4		f1oeq2 6816	. . . . . . . . 9 ⊢ (𝐶 = ∅ → (𝐹:𝐶–1-1-onto→𝐴 ↔ 𝐹:∅–1-1-onto→𝐴))
5	4	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 = ∅) → (𝐹:𝐶–1-1-onto→𝐴 ↔ 𝐹:∅–1-1-onto→𝐴))
6	3, 5	mpbid 235	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 = ∅) → 𝐹:∅–1-1-onto→𝐴)
7		f1ofo 6835	. . . . . . 7 ⊢ (𝐹:∅–1-1-onto→𝐴 → 𝐹:∅–onto→𝐴)
8	6, 7	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 = ∅) → 𝐹:∅–onto→𝐴)
9		fo00 6864	. . . . . . 7 ⊢ (𝐹:∅–onto→𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
10	9	simprbi 503	. . . . . 6 ⊢ (𝐹:∅–onto→𝐴 → 𝐴 = ∅)
11	8, 10	syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 = ∅) → 𝐴 = ∅)
12	11	prodeq1d 16000	. . . 4 ⊢ ((𝜑 ∧ 𝐶 = ∅) → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑘 ∈ ∅ 𝐵)
13		prodeq1 15987	. . . . . 6 ⊢ (𝐶 = ∅ → ∏𝑛 ∈ 𝐶 𝐷 = ∏𝑛 ∈ ∅ 𝐷)
14		prod0 16023	. . . . . 6 ⊢ ∏𝑛 ∈ ∅ 𝐷 = 1
15	13, 14	eqtrdi 2817	. . . . 5 ⊢ (𝐶 = ∅ → ∏𝑛 ∈ 𝐶 𝐷 = 1)
16	15	adantl 487	. . . 4 ⊢ ((𝜑 ∧ 𝐶 = ∅) → ∏𝑛 ∈ 𝐶 𝐷 = 1)
17	1, 12, 16	3eqtr4a 2827	. . 3 ⊢ ((𝜑 ∧ 𝐶 = ∅) → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑛 ∈ 𝐶 𝐷)
18	17	ex 418	. 2 ⊢ (𝜑 → (𝐶 = ∅ → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑛 ∈ 𝐶 𝐷))
19		2fveq3 6893	. . . . . . . 8 ⊢ (𝑚 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚)) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘(𝑓‘𝑛))))
20		simprl 783	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) → (♯‘𝐶) ∈ ℕ)
21		simprr 785	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) → 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)
22		f1of 6827	. . . . . . . . . . . 12 ⊢ (𝐹:𝐶–1-1-onto→𝐴 → 𝐹:𝐶⟶𝐴)
23	2, 22	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝐶⟶𝐴)
24	23	ffvelcdmda 7086	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐶) → (𝐹‘𝑚) ∈ 𝐴)
25		fprodf1o.5	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
26	25	fmpttd 7117	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)
27	26	ffvelcdmda 7086	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐹‘𝑚) ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚)) ∈ ℂ)
28	24, 27	syldan 603	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐶) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚)) ∈ ℂ)
29	28	adantlr 728	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) ∧ 𝑚 ∈ 𝐶) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚)) ∈ ℂ)
30		simpr 490	. . . . . . . . . . . 12 ⊢ (((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶) → 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)
31		f1oco 6851	. . . . . . . . . . . 12 ⊢ ((𝐹:𝐶–1-1-onto→𝐴 ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶) → (𝐹 ∘ 𝑓):(1...(♯‘𝐶))–1-1-onto→𝐴)
32	2, 30, 31	syl2an 608	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) → (𝐹 ∘ 𝑓):(1...(♯‘𝐶))–1-1-onto→𝐴)
33		f1of 6827	. . . . . . . . . . 11 ⊢ ((𝐹 ∘ 𝑓):(1...(♯‘𝐶))–1-1-onto→𝐴 → (𝐹 ∘ 𝑓):(1...(♯‘𝐶))⟶𝐴)
34	32, 33	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) → (𝐹 ∘ 𝑓):(1...(♯‘𝐶))⟶𝐴)
35		fvco3 6988	. . . . . . . . . 10 ⊢ (((𝐹 ∘ 𝑓):(1...(♯‘𝐶))⟶𝐴 ∧ 𝑛 ∈ (1...(♯‘𝐶))) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ (𝐹 ∘ 𝑓))‘𝑛) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘((𝐹 ∘ 𝑓)‘𝑛)))
36	34, 35	sylan 592	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) ∧ 𝑛 ∈ (1...(♯‘𝐶))) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ (𝐹 ∘ 𝑓))‘𝑛) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘((𝐹 ∘ 𝑓)‘𝑛)))
37		f1of 6827	. . . . . . . . . . . . 13 ⊢ (𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶 → 𝑓:(1...(♯‘𝐶))⟶𝐶)
38	37	adantl 487	. . . . . . . . . . . 12 ⊢ (((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶) → 𝑓:(1...(♯‘𝐶))⟶𝐶)
39	38	adantl 487	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) → 𝑓:(1...(♯‘𝐶))⟶𝐶)
40		fvco3 6988	. . . . . . . . . . 11 ⊢ ((𝑓:(1...(♯‘𝐶))⟶𝐶 ∧ 𝑛 ∈ (1...(♯‘𝐶))) → ((𝐹 ∘ 𝑓)‘𝑛) = (𝐹‘(𝑓‘𝑛)))
41	39, 40	sylan 592	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) ∧ 𝑛 ∈ (1...(♯‘𝐶))) → ((𝐹 ∘ 𝑓)‘𝑛) = (𝐹‘(𝑓‘𝑛)))
42	41	fveq2d 6892	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) ∧ 𝑛 ∈ (1...(♯‘𝐶))) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘((𝐹 ∘ 𝑓)‘𝑛)) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘(𝑓‘𝑛))))
43	36, 42	eqtrd 2801	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) ∧ 𝑛 ∈ (1...(♯‘𝐶))) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ (𝐹 ∘ 𝑓))‘𝑛) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘(𝑓‘𝑛))))
44	19, 20, 21, 29, 43	fprod 16021	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) → ∏𝑚 ∈ 𝐶 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚)) = (seq1( · , ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ (𝐹 ∘ 𝑓)))‘(♯‘𝐶)))
45		fprodf1o.4	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → (𝐹‘𝑛) = 𝐺)
46	23	ffvelcdmda 7086	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → (𝐹‘𝑛) ∈ 𝐴)
47	45, 46	eqeltrrd 2867	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → 𝐺 ∈ 𝐴)
48		fprodf1o.1	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝐺 → 𝐵 = 𝐷)
49		eqid 2766	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵)
50	48, 49	fvmpti 6995	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ 𝐴 → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝐺) = ( I ‘𝐷))
51	47, 50	syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝐺) = ( I ‘𝐷))
52	45	fveq2d 6892	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑛)) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝐺))
53		eqid 2766	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ 𝐶 ↦ 𝐷) = (𝑛 ∈ 𝐶 ↦ 𝐷)
54	53	fvmpt2i 7007	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ 𝐶 → ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑛) = ( I ‘𝐷))
55	54	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑛) = ( I ‘𝐷))
56	51, 52, 55	3eqtr4rd 2812	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑛)))
57	56	ralrimiva 3160	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑛 ∈ 𝐶 ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑛)))
58		nffvmpt1 6899	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑚)
59	58	nfeq1 2943	. . . . . . . . . . 11 ⊢ Ⅎ𝑛((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚))
60		fveq2 6888	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑛) = ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑚))
61		2fveq3 6893	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑛)) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚)))
62	60, 61	eqeq12d 2782	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → (((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑛)) ↔ ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚))))
63	59, 62	rspc 3572	. . . . . . . . . 10 ⊢ (𝑚 ∈ 𝐶 → (∀𝑛 ∈ 𝐶 ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑛)) → ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚))))
64	57, 63	mpan9 516	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐶) → ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚)))
65	64	adantlr 728	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) ∧ 𝑚 ∈ 𝐶) → ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚)))
66	65	prodeq2dv 16002	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) → ∏𝑚 ∈ 𝐶 ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑚) = ∏𝑚 ∈ 𝐶 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚)))
67		fveq2 6888	. . . . . . . 8 ⊢ (𝑚 = ((𝐹 ∘ 𝑓)‘𝑛) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘((𝐹 ∘ 𝑓)‘𝑛)))
68	26	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)
69	68	ffvelcdmda 7086	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) ∈ ℂ)
70	67, 20, 32, 69, 36	fprod 16021	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) → ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = (seq1( · , ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ (𝐹 ∘ 𝑓)))‘(♯‘𝐶)))
71	44, 66, 70	3eqtr4rd 2812	. . . . . 6 ⊢ ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) → ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = ∏𝑚 ∈ 𝐶 ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑚))
72		prodfc 16025	. . . . . 6 ⊢ ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = ∏𝑘 ∈ 𝐴 𝐵
73		prodfc 16025	. . . . . 6 ⊢ ∏𝑚 ∈ 𝐶 ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑚) = ∏𝑛 ∈ 𝐶 𝐷
74	71, 72, 73	3eqtr3g 2824	. . . . 5 ⊢ ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑛 ∈ 𝐶 𝐷)
75	74	expr 462	. . . 4 ⊢ ((𝜑 ∧ (♯‘𝐶) ∈ ℕ) → (𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶 → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑛 ∈ 𝐶 𝐷))
76	75	exlimdv 1966	. . 3 ⊢ ((𝜑 ∧ (♯‘𝐶) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶 → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑛 ∈ 𝐶 𝐷))
77	76	expimpd 459	. 2 ⊢ (𝜑 → (((♯‘𝐶) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶) → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑛 ∈ 𝐶 𝐷))
78		fprodf1o.2	. . 3 ⊢ (𝜑 → 𝐶 ∈ Fin)
79		fz1f1o 15787	. . 3 ⊢ (𝐶 ∈ Fin → (𝐶 = ∅ ∨ ((♯‘𝐶) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)))
80	78, 79	syl 18	. 2 ⊢ (𝜑 → (𝐶 = ∅ ∨ ((♯‘𝐶) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)))
81	18, 77, 80	mpjaod 874	1 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑛 ∈ 𝐶 𝐷)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570  ∃wex 1812   ∈ wcel 2146  ∀wral 3082  ∅c0 4289   ↦ cmpt 5197   I cid 5560   ∘ ccom 5670  ⟶wf 6539  –onto→wfo 6541  –1-1-onto→wf1o 6542  ‘cfv 6543  (class class class)co 7423  Fincfn 8952  ℂcc 11116  1c1 11119   · cmul 11123  ℕcn 12251  ...cfz 13553  seqcseq 14057  ♯chash 14386  ∏cprod 15983

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745  ax-inf2 9620  ax-cnex 11174  ax-resscn 11175  ax-1cn 11176  ax-icn 11177  ax-addcl 11178  ax-addrcl 11179  ax-mulcl 11180  ax-mulrcl 11181  ax-mulcom 11182  ax-addass 11183  ax-mulass 11184  ax-distr 11185  ax-i2m1 11186  ax-1ne0 11187  ax-1rid 11188  ax-rnegex 11189  ax-rrecex 11190  ax-cnre 11191  ax-pre-lttri 11192  ax-pre-lttrn 11193  ax-pre-ltadd 11194  ax-pre-mulgt0 11195  ax-pre-sup 11196

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-nel 3068  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-int 4918  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-se 5620  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-om 7872  df-1st 7995  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8703  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-sup 9412  df-oi 9482  df-card 9944  df-pnf 11263  df-mnf 11264  df-xr 11265  df-ltxr 11266  df-le 11267  df-sub 11461  df-neg 11462  df-div 11890  df-nn 12252  df-2 12321  df-3 12322  df-n0 12523  df-z 12610  df-uz 12881  df-rp 13035  df-fz 13554  df-fzo 13702  df-seq 14058  df-exp 14118  df-hash 14387  df-cj 15176  df-re 15177  df-im 15178  df-sqrt 15312  df-abs 15313  df-clim 15565  df-prod 15984

This theorem is used by:  fprodss  16028  fprodshft  16056  fprodrev  16057  fprod2dlem  16060  fprodcnv  16063  gausslemma2dlem1  27567  hgt750lemg  35073