Theorem fprodf1o 15855

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 15852	. . . 4 ⊢ ∏𝑘 ∈ ∅ 𝐵 = 1
2		fprodf1o.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴)
3	2	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 = ∅) → 𝐹:𝐶–1-1-onto→𝐴)
4		f1oeq2 6757	. . . . . . . . 9 ⊢ (𝐶 = ∅ → (𝐹:𝐶–1-1-onto→𝐴 ↔ 𝐹:∅–1-1-onto→𝐴))
5	4	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 = ∅) → (𝐹:𝐶–1-1-onto→𝐴 ↔ 𝐹:∅–1-1-onto→𝐴))
6	3, 5	mpbid 232	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 = ∅) → 𝐹:∅–1-1-onto→𝐴)
7		f1ofo 6775	. . . . . . 7 ⊢ (𝐹:∅–1-1-onto→𝐴 → 𝐹:∅–onto→𝐴)
8	6, 7	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 = ∅) → 𝐹:∅–onto→𝐴)
9		fo00 6804	. . . . . . 7 ⊢ (𝐹:∅–onto→𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
10	9	simprbi 496	. . . . . 6 ⊢ (𝐹:∅–onto→𝐴 → 𝐴 = ∅)
11	8, 10	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 = ∅) → 𝐴 = ∅)
12	11	prodeq1d 15829	. . . 4 ⊢ ((𝜑 ∧ 𝐶 = ∅) → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑘 ∈ ∅ 𝐵)
13		prodeq1 15816	. . . . . 6 ⊢ (𝐶 = ∅ → ∏𝑛 ∈ 𝐶 𝐷 = ∏𝑛 ∈ ∅ 𝐷)
14		prod0 15852	. . . . . 6 ⊢ ∏𝑛 ∈ ∅ 𝐷 = 1
15	13, 14	eqtrdi 2784	. . . . 5 ⊢ (𝐶 = ∅ → ∏𝑛 ∈ 𝐶 𝐷 = 1)
16	15	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝐶 = ∅) → ∏𝑛 ∈ 𝐶 𝐷 = 1)
17	1, 12, 16	3eqtr4a 2794	. . 3 ⊢ ((𝜑 ∧ 𝐶 = ∅) → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑛 ∈ 𝐶 𝐷)
18	17	ex 412	. 2 ⊢ (𝜑 → (𝐶 = ∅ → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑛 ∈ 𝐶 𝐷))
19		2fveq3 6833	. . . . . . . 8 ⊢ (𝑚 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚)) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘(𝑓‘𝑛))))
20		simprl 770	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) → (♯‘𝐶) ∈ ℕ)
21		simprr 772	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) → 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)
22		f1of 6768	. . . . . . . . . . . 12 ⊢ (𝐹:𝐶–1-1-onto→𝐴 → 𝐹:𝐶⟶𝐴)
23	2, 22	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝐶⟶𝐴)
24	23	ffvelcdmda 7023	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐶) → (𝐹‘𝑚) ∈ 𝐴)
25		fprodf1o.5	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
26	25	fmpttd 7054	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)
27	26	ffvelcdmda 7023	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐹‘𝑚) ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚)) ∈ ℂ)
28	24, 27	syldan 591	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐶) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚)) ∈ ℂ)
29	28	adantlr 715	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) ∧ 𝑚 ∈ 𝐶) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚)) ∈ ℂ)
30		simpr 484	. . . . . . . . . . . 12 ⊢ (((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶) → 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)
31		f1oco 6791	. . . . . . . . . . . 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 596	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) → (𝐹 ∘ 𝑓):(1...(♯‘𝐶))–1-1-onto→𝐴)
33		f1of 6768	. . . . . . . . . . 11 ⊢ ((𝐹 ∘ 𝑓):(1...(♯‘𝐶))–1-1-onto→𝐴 → (𝐹 ∘ 𝑓):(1...(♯‘𝐶))⟶𝐴)
34	32, 33	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) → (𝐹 ∘ 𝑓):(1...(♯‘𝐶))⟶𝐴)
35		fvco3 6927	. . . . . . . . . 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 580	. . . . . . . . 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 6768	. . . . . . . . . . . . 13 ⊢ (𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶 → 𝑓:(1...(♯‘𝐶))⟶𝐶)
38	37	adantl 481	. . . . . . . . . . . 12 ⊢ (((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶) → 𝑓:(1...(♯‘𝐶))⟶𝐶)
39	38	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) → 𝑓:(1...(♯‘𝐶))⟶𝐶)
40		fvco3 6927	. . . . . . . . . . 11 ⊢ ((𝑓:(1...(♯‘𝐶))⟶𝐶 ∧ 𝑛 ∈ (1...(♯‘𝐶))) → ((𝐹 ∘ 𝑓)‘𝑛) = (𝐹‘(𝑓‘𝑛)))
41	39, 40	sylan 580	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) ∧ 𝑛 ∈ (1...(♯‘𝐶))) → ((𝐹 ∘ 𝑓)‘𝑛) = (𝐹‘(𝑓‘𝑛)))
42	41	fveq2d 6832	. . . . . . . . 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 2768	. . . . . . . 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 15850	. . . . . . 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 7023	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → (𝐹‘𝑛) ∈ 𝐴)
47	45, 46	eqeltrrd 2834	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → 𝐺 ∈ 𝐴)
48		fprodf1o.1	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝐺 → 𝐵 = 𝐷)
49		eqid 2733	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵)
50	48, 49	fvmpti 6934	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ 𝐴 → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝐺) = ( I ‘𝐷))
51	47, 50	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝐺) = ( I ‘𝐷))
52	45	fveq2d 6832	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑛)) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝐺))
53		eqid 2733	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ 𝐶 ↦ 𝐷) = (𝑛 ∈ 𝐶 ↦ 𝐷)
54	53	fvmpt2i 6945	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ 𝐶 → ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑛) = ( I ‘𝐷))
55	54	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑛) = ( I ‘𝐷))
56	51, 52, 55	3eqtr4rd 2779	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑛)))
57	56	ralrimiva 3125	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑛 ∈ 𝐶 ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑛)))
58		nffvmpt1 6839	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑚)
59	58	nfeq1 2911	. . . . . . . . . . 11 ⊢ Ⅎ𝑛((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚))
60		fveq2 6828	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑛) = ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑚))
61		2fveq3 6833	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑛)) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚)))
62	60, 61	eqeq12d 2749	. . . . . . . . . . 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 3561	. . . . . . . . . 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 506	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐶) → ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚)))
65	64	adantlr 715	. . . . . . . 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 15831	. . . . . . 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 6828	. . . . . . . 8 ⊢ (𝑚 = ((𝐹 ∘ 𝑓)‘𝑛) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘((𝐹 ∘ 𝑓)‘𝑛)))
68	26	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)
69	68	ffvelcdmda 7023	. . . . . . . 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 15850	. . . . . . 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 2779	. . . . . 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 15854	. . . . . 6 ⊢ ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = ∏𝑘 ∈ 𝐴 𝐵
73		prodfc 15854	. . . . . 6 ⊢ ∏𝑚 ∈ 𝐶 ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑚) = ∏𝑛 ∈ 𝐶 𝐷
74	71, 72, 73	3eqtr3g 2791	. . . . 5 ⊢ ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑛 ∈ 𝐶 𝐷)
75	74	expr 456	. . . 4 ⊢ ((𝜑 ∧ (♯‘𝐶) ∈ ℕ) → (𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶 → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑛 ∈ 𝐶 𝐷))
76	75	exlimdv 1934	. . 3 ⊢ ((𝜑 ∧ (♯‘𝐶) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶 → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑛 ∈ 𝐶 𝐷))
77	76	expimpd 453	. 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 15619	. . 3 ⊢ (𝐶 ∈ Fin → (𝐶 = ∅ ∨ ((♯‘𝐶) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)))
80	78, 79	syl 17	. 2 ⊢ (𝜑 → (𝐶 = ∅ ∨ ((♯‘𝐶) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)))
81	18, 77, 80	mpjaod 860	1 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑛 ∈ 𝐶 𝐷)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∀wral 3048  ∅c0 4282   ↦ cmpt 5174   I cid 5513   ∘ ccom 5623  ⟶wf 6482  –onto→wfo 6484  –1-1-onto→wf1o 6485  ‘cfv 6486  (class class class)co 7352  Fincfn 8875  ℂcc 11011  1c1 11014   · cmul 11018  ℕcn 12132  ...cfz 13409  seqcseq 13910  ♯chash 14239  ∏cprod 15812

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-inf2 9538  ax-cnex 11069  ax-resscn 11070  ax-1cn 11071  ax-icn 11072  ax-addcl 11073  ax-addrcl 11074  ax-mulcl 11075  ax-mulrcl 11076  ax-mulcom 11077  ax-addass 11078  ax-mulass 11079  ax-distr 11080  ax-i2m1 11081  ax-1ne0 11082  ax-1rid 11083  ax-rnegex 11084  ax-rrecex 11085  ax-cnre 11086  ax-pre-lttri 11087  ax-pre-lttrn 11088  ax-pre-ltadd 11089  ax-pre-mulgt0 11090  ax-pre-sup 11091

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-1st 7927  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-1o 8391  df-er 8628  df-en 8876  df-dom 8877  df-sdom 8878  df-fin 8879  df-sup 9333  df-oi 9403  df-card 9839  df-pnf 11155  df-mnf 11156  df-xr 11157  df-ltxr 11158  df-le 11159  df-sub 11353  df-neg 11354  df-div 11782  df-nn 12133  df-2 12195  df-3 12196  df-n0 12389  df-z 12476  df-uz 12739  df-rp 12893  df-fz 13410  df-fzo 13557  df-seq 13911  df-exp 13971  df-hash 14240  df-cj 15008  df-re 15009  df-im 15010  df-sqrt 15144  df-abs 15145  df-clim 15397  df-prod 15813

This theorem is referenced by:  fprodss  15857  fprodshft  15885  fprodrev  15886  fprod2dlem  15889  fprodcnv  15892  gausslemma2dlem1  27305  hgt750lemg  34688