Theorem fprodf1o 15978

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 15975	. . . 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 6837	. . . . . . . . 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 6855	. . . . . . 7 ⊢ (𝐹:∅–1-1-onto→𝐴 → 𝐹:∅–onto→𝐴)
8	6, 7	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 = ∅) → 𝐹:∅–onto→𝐴)
9		fo00 6884	. . . . . . 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 15952	. . . 4 ⊢ ((𝜑 ∧ 𝐶 = ∅) → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑘 ∈ ∅ 𝐵)
13		prodeq1 15939	. . . . . 6 ⊢ (𝐶 = ∅ → ∏𝑛 ∈ 𝐶 𝐷 = ∏𝑛 ∈ ∅ 𝐷)
14		prod0 15975	. . . . . 6 ⊢ ∏𝑛 ∈ ∅ 𝐷 = 1
15	13, 14	eqtrdi 2790	. . . . 5 ⊢ (𝐶 = ∅ → ∏𝑛 ∈ 𝐶 𝐷 = 1)
16	15	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝐶 = ∅) → ∏𝑛 ∈ 𝐶 𝐷 = 1)
17	1, 12, 16	3eqtr4a 2800	. . 3 ⊢ ((𝜑 ∧ 𝐶 = ∅) → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑛 ∈ 𝐶 𝐷)
18	17	ex 412	. 2 ⊢ (𝜑 → (𝐶 = ∅ → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑛 ∈ 𝐶 𝐷))
19		2fveq3 6911	. . . . . . . 8 ⊢ (𝑚 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚)) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘(𝑓‘𝑛))))
20		simprl 771	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) → (♯‘𝐶) ∈ ℕ)
21		simprr 773	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) → 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)
22		f1of 6848	. . . . . . . . . . . 12 ⊢ (𝐹:𝐶–1-1-onto→𝐴 → 𝐹:𝐶⟶𝐴)
23	2, 22	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝐶⟶𝐴)
24	23	ffvelcdmda 7103	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐶) → (𝐹‘𝑚) ∈ 𝐴)
25		fprodf1o.5	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
26	25	fmpttd 7134	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)
27	26	ffvelcdmda 7103	. . . . . . . . . 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 6871	. . . . . . . . . . . 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 6848	. . . . . . . . . . 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 7007	. . . . . . . . . 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 6848	. . . . . . . . . . . . 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 7007	. . . . . . . . . . 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 6910	. . . . . . . . 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 2774	. . . . . . . 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 15973	. . . . . . 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 7103	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → (𝐹‘𝑛) ∈ 𝐴)
47	45, 46	eqeltrrd 2839	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → 𝐺 ∈ 𝐴)
48		fprodf1o.1	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝐺 → 𝐵 = 𝐷)
49		eqid 2734	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵)
50	48, 49	fvmpti 7014	. . . . . . . . . . . . 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 6910	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑛)) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝐺))
53		eqid 2734	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ 𝐶 ↦ 𝐷) = (𝑛 ∈ 𝐶 ↦ 𝐷)
54	53	fvmpt2i 7025	. . . . . . . . . . . . 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 2785	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑛)))
57	56	ralrimiva 3143	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑛 ∈ 𝐶 ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑛)))
58		nffvmpt1 6917	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑚)
59	58	nfeq1 2918	. . . . . . . . . . 11 ⊢ Ⅎ𝑛((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚))
60		fveq2 6906	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑛) = ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑚))
61		2fveq3 6911	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑛)) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚)))
62	60, 61	eqeq12d 2750	. . . . . . . . . . 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 3609	. . . . . . . . . 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 15954	. . . . . . 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 6906	. . . . . . . 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 7103	. . . . . . . 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 15973	. . . . . . 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 2785	. . . . . 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 15977	. . . . . 6 ⊢ ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = ∏𝑘 ∈ 𝐴 𝐵
73		prodfc 15977	. . . . . 6 ⊢ ∏𝑚 ∈ 𝐶 ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑚) = ∏𝑛 ∈ 𝐶 𝐷
74	71, 72, 73	3eqtr3g 2797	. . . . 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 1930	. . 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 15742	. . 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 1536  ∃wex 1775   ∈ wcel 2105  ∀wral 3058  ∅c0 4338   ↦ cmpt 5230   I cid 5581   ∘ ccom 5692  ⟶wf 6558  –onto→wfo 6560  –1-1-onto→wf1o 6561  ‘cfv 6562  (class class class)co 7430  Fincfn 8983  ℂcc 11150  1c1 11153   · cmul 11157  ℕcn 12263  ...cfz 13543  seqcseq 14038  ♯chash 14365  ∏cprod 15935

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-sup 9479  df-oi 9547  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-n0 12524  df-z 12611  df-uz 12876  df-rp 13032  df-fz 13544  df-fzo 13691  df-seq 14039  df-exp 14099  df-hash 14366  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-clim 15520  df-prod 15936

This theorem is referenced by:  fprodss  15980  fprodshft  16008  fprodrev  16009  fprod2dlem  16012  fprodcnv  16015  gausslemma2dlem1  27424  hgt750lemg  34647