Theorem fsumf1o 14454

Description: Re-index a finite sum using a bijection. (Contributed by Mario Carneiro, 20-Apr-2014.)

Hypotheses

Ref	Expression
fsumf1o.1	⊢ (𝑘 = 𝐺 → 𝐵 = 𝐷)
fsumf1o.2	⊢ (𝜑 → 𝐶 ∈ Fin)
fsumf1o.3	⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴)
fsumf1o.4	⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → (𝐹‘𝑛) = 𝐺)
fsumf1o.5	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)

Assertion

Ref	Expression
fsumf1o	⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑛 ∈ 𝐶 𝐷)

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

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

Proof of Theorem fsumf1o

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

Step	Hyp	Ref	Expression
1		sum0 14452	. . . 4 ⊢ Σ𝑘 ∈ ∅ 𝐵 = 0
2		fsumf1o.3	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴)
3		f1oeq2 6128	. . . . . . . 8 ⊢ (𝐶 = ∅ → (𝐹:𝐶–1-1-onto→𝐴 ↔ 𝐹:∅–1-1-onto→𝐴))
4	2, 3	syl5ibcom 235	. . . . . . 7 ⊢ (𝜑 → (𝐶 = ∅ → 𝐹:∅–1-1-onto→𝐴))
5	4	imp 445	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 = ∅) → 𝐹:∅–1-1-onto→𝐴)
6		f1ofo 6144	. . . . . 6 ⊢ (𝐹:∅–1-1-onto→𝐴 → 𝐹:∅–onto→𝐴)
7		fo00 6172	. . . . . . 7 ⊢ (𝐹:∅–onto→𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
8	7	simprbi 480	. . . . . 6 ⊢ (𝐹:∅–onto→𝐴 → 𝐴 = ∅)
9	5, 6, 8	3syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 = ∅) → 𝐴 = ∅)
10	9	sumeq1d 14431	. . . 4 ⊢ ((𝜑 ∧ 𝐶 = ∅) → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ ∅ 𝐵)
11		simpr 477	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 = ∅) → 𝐶 = ∅)
12	11	sumeq1d 14431	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 = ∅) → Σ𝑛 ∈ 𝐶 𝐷 = Σ𝑛 ∈ ∅ 𝐷)
13		sum0 14452	. . . . 5 ⊢ Σ𝑛 ∈ ∅ 𝐷 = 0
14	12, 13	syl6eq 2672	. . . 4 ⊢ ((𝜑 ∧ 𝐶 = ∅) → Σ𝑛 ∈ 𝐶 𝐷 = 0)
15	1, 10, 14	3eqtr4a 2682	. . 3 ⊢ ((𝜑 ∧ 𝐶 = ∅) → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑛 ∈ 𝐶 𝐷)
16	15	ex 450	. 2 ⊢ (𝜑 → (𝐶 = ∅ → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑛 ∈ 𝐶 𝐷))
17		fveq2 6191	. . . . . . . . 9 ⊢ (𝑚 = (𝑓‘𝑛) → (𝐹‘𝑚) = (𝐹‘(𝑓‘𝑛)))
18	17	fveq2d 6195	. . . . . . . 8 ⊢ (𝑚 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚)) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘(𝑓‘𝑛))))
19		simprl 794	. . . . . . . 8 ⊢ ((𝜑 ∧ ((#‘𝐶) ∈ ℕ ∧ 𝑓:(1...(#‘𝐶))–1-1-onto→𝐶)) → (#‘𝐶) ∈ ℕ)
20		simprr 796	. . . . . . . 8 ⊢ ((𝜑 ∧ ((#‘𝐶) ∈ ℕ ∧ 𝑓:(1...(#‘𝐶))–1-1-onto→𝐶)) → 𝑓:(1...(#‘𝐶))–1-1-onto→𝐶)
21		f1of 6137	. . . . . . . . . . . 12 ⊢ (𝐹:𝐶–1-1-onto→𝐴 → 𝐹:𝐶⟶𝐴)
22	2, 21	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝐶⟶𝐴)
23	22	ffvelrnda 6359	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐶) → (𝐹‘𝑚) ∈ 𝐴)
24		fsumf1o.5	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
25		eqid 2622	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵)
26	24, 25	fmptd 6385	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)
27	26	ffvelrnda 6359	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐹‘𝑚) ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚)) ∈ ℂ)
28	23, 27	syldan 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐶) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚)) ∈ ℂ)
29	28	adantlr 751	. . . . . . . 8 ⊢ (((𝜑 ∧ ((#‘𝐶) ∈ ℕ ∧ 𝑓:(1...(#‘𝐶))–1-1-onto→𝐶)) ∧ 𝑚 ∈ 𝐶) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚)) ∈ ℂ)
30	2	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((#‘𝐶) ∈ ℕ ∧ 𝑓:(1...(#‘𝐶))–1-1-onto→𝐶)) → 𝐹:𝐶–1-1-onto→𝐴)
31		f1oco 6159	. . . . . . . . . . . 12 ⊢ ((𝐹:𝐶–1-1-onto→𝐴 ∧ 𝑓:(1...(#‘𝐶))–1-1-onto→𝐶) → (𝐹 ∘ 𝑓):(1...(#‘𝐶))–1-1-onto→𝐴)
32	30, 20, 31	syl2anc 693	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((#‘𝐶) ∈ ℕ ∧ 𝑓:(1...(#‘𝐶))–1-1-onto→𝐶)) → (𝐹 ∘ 𝑓):(1...(#‘𝐶))–1-1-onto→𝐴)
33		f1of 6137	. . . . . . . . . . 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 6275	. . . . . . . . . 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 488	. . . . . . . . 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 6137	. . . . . . . . . . . 12 ⊢ (𝑓:(1...(#‘𝐶))–1-1-onto→𝐶 → 𝑓:(1...(#‘𝐶))⟶𝐶)
38	37	ad2antll 765	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((#‘𝐶) ∈ ℕ ∧ 𝑓:(1...(#‘𝐶))–1-1-onto→𝐶)) → 𝑓:(1...(#‘𝐶))⟶𝐶)
39		fvco3 6275	. . . . . . . . . . 11 ⊢ ((𝑓:(1...(#‘𝐶))⟶𝐶 ∧ 𝑛 ∈ (1...(#‘𝐶))) → ((𝐹 ∘ 𝑓)‘𝑛) = (𝐹‘(𝑓‘𝑛)))
40	38, 39	sylan 488	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((#‘𝐶) ∈ ℕ ∧ 𝑓:(1...(#‘𝐶))–1-1-onto→𝐶)) ∧ 𝑛 ∈ (1...(#‘𝐶))) → ((𝐹 ∘ 𝑓)‘𝑛) = (𝐹‘(𝑓‘𝑛)))
41	40	fveq2d 6195	. . . . . . . . 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))))
42	36, 41	eqtrd 2656	. . . . . . . 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))))
43	18, 19, 20, 29, 42	fsum 14451	. . . . . . 7 ⊢ ((𝜑 ∧ ((#‘𝐶) ∈ ℕ ∧ 𝑓:(1...(#‘𝐶))–1-1-onto→𝐶)) → Σ𝑚 ∈ 𝐶 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚)) = (seq1( + , ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ (𝐹 ∘ 𝑓)))‘(#‘𝐶)))
44		fsumf1o.4	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → (𝐹‘𝑛) = 𝐺)
45	22	ffvelrnda 6359	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → (𝐹‘𝑛) ∈ 𝐴)
46	44, 45	eqeltrrd 2702	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → 𝐺 ∈ 𝐴)
47		fsumf1o.1	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝐺 → 𝐵 = 𝐷)
48	47, 25	fvmpti 6281	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ 𝐴 → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝐺) = ( I ‘𝐷))
49	46, 48	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝐺) = ( I ‘𝐷))
50	44	fveq2d 6195	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑛)) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝐺))
51		eqid 2622	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ 𝐶 ↦ 𝐷) = (𝑛 ∈ 𝐶 ↦ 𝐷)
52	51	fvmpt2i 6290	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ 𝐶 → ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑛) = ( I ‘𝐷))
53	52	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑛) = ( I ‘𝐷))
54	49, 50, 53	3eqtr4rd 2667	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑛)))
55	54	ralrimiva 2966	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑛 ∈ 𝐶 ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑛)))
56		nffvmpt1 6199	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑚)
57	56	nfeq1 2778	. . . . . . . . . . 11 ⊢ Ⅎ𝑛((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚))
58		fveq2 6191	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑛) = ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑚))
59		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑚 → (𝐹‘𝑛) = (𝐹‘𝑚))
60	59	fveq2d 6195	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑛)) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚)))
61	58, 60	eqeq12d 2637	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → (((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑛)) ↔ ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚))))
62	57, 61	rspc 3303	. . . . . . . . . 10 ⊢ (𝑚 ∈ 𝐶 → (∀𝑛 ∈ 𝐶 ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑛)) → ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚))))
63	55, 62	mpan9 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐶) → ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚)))
64	63	adantlr 751	. . . . . . . 8 ⊢ (((𝜑 ∧ ((#‘𝐶) ∈ ℕ ∧ 𝑓:(1...(#‘𝐶))–1-1-onto→𝐶)) ∧ 𝑚 ∈ 𝐶) → ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚)))
65	64	sumeq2dv 14433	. . . . . . 7 ⊢ ((𝜑 ∧ ((#‘𝐶) ∈ ℕ ∧ 𝑓:(1...(#‘𝐶))–1-1-onto→𝐶)) → Σ𝑚 ∈ 𝐶 ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑚) = Σ𝑚 ∈ 𝐶 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚)))
66		fveq2 6191	. . . . . . . 8 ⊢ (𝑚 = ((𝐹 ∘ 𝑓)‘𝑛) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘((𝐹 ∘ 𝑓)‘𝑛)))
67	26	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((#‘𝐶) ∈ ℕ ∧ 𝑓:(1...(#‘𝐶))–1-1-onto→𝐶)) → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)
68	67	ffvelrnda 6359	. . . . . . . 8 ⊢ (((𝜑 ∧ ((#‘𝐶) ∈ ℕ ∧ 𝑓:(1...(#‘𝐶))–1-1-onto→𝐶)) ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) ∈ ℂ)
69	66, 19, 32, 68, 36	fsum 14451	. . . . . . 7 ⊢ ((𝜑 ∧ ((#‘𝐶) ∈ ℕ ∧ 𝑓:(1...(#‘𝐶))–1-1-onto→𝐶)) → Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = (seq1( + , ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ (𝐹 ∘ 𝑓)))‘(#‘𝐶)))
70	43, 65, 69	3eqtr4rd 2667	. . . . . 6 ⊢ ((𝜑 ∧ ((#‘𝐶) ∈ ℕ ∧ 𝑓:(1...(#‘𝐶))–1-1-onto→𝐶)) → Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = Σ𝑚 ∈ 𝐶 ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑚))
71		sumfc 14440	. . . . . 6 ⊢ Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = Σ𝑘 ∈ 𝐴 𝐵
72		sumfc 14440	. . . . . 6 ⊢ Σ𝑚 ∈ 𝐶 ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑚) = Σ𝑛 ∈ 𝐶 𝐷
73	70, 71, 72	3eqtr3g 2679	. . . . 5 ⊢ ((𝜑 ∧ ((#‘𝐶) ∈ ℕ ∧ 𝑓:(1...(#‘𝐶))–1-1-onto→𝐶)) → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑛 ∈ 𝐶 𝐷)
74	73	expr 643	. . . 4 ⊢ ((𝜑 ∧ (#‘𝐶) ∈ ℕ) → (𝑓:(1...(#‘𝐶))–1-1-onto→𝐶 → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑛 ∈ 𝐶 𝐷))
75	74	exlimdv 1861	. . 3 ⊢ ((𝜑 ∧ (#‘𝐶) ∈ ℕ) → (∃𝑓 𝑓:(1...(#‘𝐶))–1-1-onto→𝐶 → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑛 ∈ 𝐶 𝐷))
76	75	expimpd 629	. 2 ⊢ (𝜑 → (((#‘𝐶) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(#‘𝐶))–1-1-onto→𝐶) → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑛 ∈ 𝐶 𝐷))
77		fsumf1o.2	. . 3 ⊢ (𝜑 → 𝐶 ∈ Fin)
78		fz1f1o 14441	. . 3 ⊢ (𝐶 ∈ Fin → (𝐶 = ∅ ∨ ((#‘𝐶) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(#‘𝐶))–1-1-onto→𝐶)))
79	77, 78	syl 17	. 2 ⊢ (𝜑 → (𝐶 = ∅ ∨ ((#‘𝐶) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(#‘𝐶))–1-1-onto→𝐶)))
80	16, 76, 79	mpjaod 396	1 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑛 ∈ 𝐶 𝐷)

Syntax hints:   → wi 4   ∨ wo 383   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∀wral 2912  ∅c0 3915   ↦ cmpt 4729   I cid 5023   ∘ ccom 5118  ⟶wf 5884  –onto→wfo 5886  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650  Fincfn 7955  ℂcc 9934  0cc0 9936  1c1 9937   + caddc 9939  ℕcn 11020  ...cfz 12326  seqcseq 12801  #chash 13117  Σcsu 14416

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417

This theorem is referenced by:  fsumss  14456  fsum2dlem  14501  fsumcnv  14504  fsumrev  14511  fsumshft  14512  ackbijnn  14560  incexclem  14568  phisum  15495  ovoliunlem1  23270  ovolicc2lem4  23288  itg1addlem4  23466  itg1mulc  23471  basellem3  24809  basellem5  24811  fsumdvdscom  24911  dvdsflsumcom  24914  musum  24917  fsumdvdsmul  24921  sgmppw  24922  fsumvma  24938  dchrsum2  24993  sumdchr2  24995  dchrisumlem1  25178  dchrisum0flblem1  25197  dchrisum0fno1  25200  fsumiunle  29575  eulerpartlemgs2  30442  reprpmtf1o  30704  breprexplema  30708  hgt750lemb  30734  hgt750lema  30735  fsumf1of  39806  sumnnodd  39862  dvnprodlem2  40162