Theorem seqcoll2 14500

Description: The function 𝐹 contains a sparse set of nonzero values to be summed. The function 𝐺 is an order isomorphism from the set of nonzero values of 𝐹 to a 1-based finite sequence, and 𝐻 collects these nonzero values together. Under these conditions, the sum over the values in 𝐻 yields the same result as the sum over the original set 𝐹. (Contributed by Mario Carneiro, 13-Dec-2014.)

Hypotheses

Ref	Expression
seqcoll2.1	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑆) → (𝑍 + 𝑘) = 𝑘)
seqcoll2.1b	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑆) → (𝑘 + 𝑍) = 𝑘)
seqcoll2.c	⊢ ((𝜑 ∧ (𝑘 ∈ 𝑆 ∧ 𝑛 ∈ 𝑆)) → (𝑘 + 𝑛) ∈ 𝑆)
seqcoll2.a	⊢ (𝜑 → 𝑍 ∈ 𝑆)
seqcoll2.2	⊢ (𝜑 → 𝐺 Isom < , < ((1...(♯‘𝐴)), 𝐴))
seqcoll2.3	⊢ (𝜑 → 𝐴 ≠ ∅)
seqcoll2.5	⊢ (𝜑 → 𝐴 ⊆ (𝑀...𝑁))
seqcoll2.6	⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ 𝑆)
seqcoll2.7	⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀...𝑁) ∖ 𝐴)) → (𝐹‘𝑘) = 𝑍)
seqcoll2.8	⊢ ((𝜑 ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (𝐻‘𝑛) = (𝐹‘(𝐺‘𝑛)))

Assertion

Ref	Expression
seqcoll2	⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq1( + , 𝐻)‘(♯‘𝐴)))

Distinct variable groups:   𝑘,𝑛,𝐴   𝑘,𝐹,𝑛   𝑘,𝐺,𝑛   𝑛,𝐻   𝑘,𝑀,𝑛   𝜑,𝑘,𝑛   𝑘,𝑁   + ,𝑘,𝑛   𝑆,𝑘,𝑛   𝑘,𝑍

Allowed substitution hints:   𝐻(𝑘)   𝑁(𝑛)   𝑍(𝑛)

Proof of Theorem seqcoll2

Step	Hyp	Ref	Expression
1		seqcoll2.1b	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑆) → (𝑘 + 𝑍) = 𝑘)
2		fzssuz 13601	. . . 4 ⊢ (𝑀...𝑁) ⊆ (ℤ≥‘𝑀)
3		seqcoll2.5	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ (𝑀...𝑁))
4		seqcoll2.2	. . . . . . . 8 ⊢ (𝜑 → 𝐺 Isom < , < ((1...(♯‘𝐴)), 𝐴))
5		isof1o 7342	. . . . . . . 8 ⊢ (𝐺 Isom < , < ((1...(♯‘𝐴)), 𝐴) → 𝐺:(1...(♯‘𝐴))–1-1-onto→𝐴)
6	4, 5	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐺:(1...(♯‘𝐴))–1-1-onto→𝐴)
7		f1of 6848	. . . . . . 7 ⊢ (𝐺:(1...(♯‘𝐴))–1-1-onto→𝐴 → 𝐺:(1...(♯‘𝐴))⟶𝐴)
8	6, 7	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐺:(1...(♯‘𝐴))⟶𝐴)
9		seqcoll2.3	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ≠ ∅)
10		fzfi 14009	. . . . . . . . . . . . 13 ⊢ (𝑀...𝑁) ∈ Fin
11		ssfi 9211	. . . . . . . . . . . . 13 ⊢ (((𝑀...𝑁) ∈ Fin ∧ 𝐴 ⊆ (𝑀...𝑁)) → 𝐴 ∈ Fin)
12	10, 3, 11	sylancr 587	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ Fin)
13		hasheq0 14398	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅))
14	12, 13	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅))
15	14	necon3bbid 2975	. . . . . . . . . 10 ⊢ (𝜑 → (¬ (♯‘𝐴) = 0 ↔ 𝐴 ≠ ∅))
16	9, 15	mpbird 257	. . . . . . . . 9 ⊢ (𝜑 → ¬ (♯‘𝐴) = 0)
17		hashcl 14391	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0)
18	12, 17	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝐴) ∈ ℕ0)
19		elnn0 12525	. . . . . . . . . . 11 ⊢ ((♯‘𝐴) ∈ ℕ0 ↔ ((♯‘𝐴) ∈ ℕ ∨ (♯‘𝐴) = 0))
20	18, 19	sylib 218	. . . . . . . . . 10 ⊢ (𝜑 → ((♯‘𝐴) ∈ ℕ ∨ (♯‘𝐴) = 0))
21	20	ord 864	. . . . . . . . 9 ⊢ (𝜑 → (¬ (♯‘𝐴) ∈ ℕ → (♯‘𝐴) = 0))
22	16, 21	mt3d 148	. . . . . . . 8 ⊢ (𝜑 → (♯‘𝐴) ∈ ℕ)
23		nnuz 12918	. . . . . . . 8 ⊢ ℕ = (ℤ≥‘1)
24	22, 23	eleqtrdi 2848	. . . . . . 7 ⊢ (𝜑 → (♯‘𝐴) ∈ (ℤ≥‘1))
25		eluzfz2 13568	. . . . . . 7 ⊢ ((♯‘𝐴) ∈ (ℤ≥‘1) → (♯‘𝐴) ∈ (1...(♯‘𝐴)))
26	24, 25	syl 17	. . . . . 6 ⊢ (𝜑 → (♯‘𝐴) ∈ (1...(♯‘𝐴)))
27	8, 26	ffvelcdmd 7104	. . . . 5 ⊢ (𝜑 → (𝐺‘(♯‘𝐴)) ∈ 𝐴)
28	3, 27	sseldd 3995	. . . 4 ⊢ (𝜑 → (𝐺‘(♯‘𝐴)) ∈ (𝑀...𝑁))
29	2, 28	sselid 3992	. . 3 ⊢ (𝜑 → (𝐺‘(♯‘𝐴)) ∈ (ℤ≥‘𝑀))
30		elfzuz3 13557	. . . 4 ⊢ ((𝐺‘(♯‘𝐴)) ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘(𝐺‘(♯‘𝐴))))
31	28, 30	syl 17	. . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝐺‘(♯‘𝐴))))
32		fzss2 13600	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘(𝐺‘(♯‘𝐴))) → (𝑀...(𝐺‘(♯‘𝐴))) ⊆ (𝑀...𝑁))
33	31, 32	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑀...(𝐺‘(♯‘𝐴))) ⊆ (𝑀...𝑁))
34	33	sselda 3994	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝐺‘(♯‘𝐴)))) → 𝑘 ∈ (𝑀...𝑁))
35		seqcoll2.6	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ 𝑆)
36	34, 35	syldan 591	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝐺‘(♯‘𝐴)))) → (𝐹‘𝑘) ∈ 𝑆)
37		seqcoll2.c	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑆 ∧ 𝑛 ∈ 𝑆)) → (𝑘 + 𝑛) ∈ 𝑆)
38	29, 36, 37	seqcl 14059	. . 3 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘(𝐺‘(♯‘𝐴))) ∈ 𝑆)
39		peano2uz 12940	. . . . . . . 8 ⊢ ((𝐺‘(♯‘𝐴)) ∈ (ℤ≥‘𝑀) → ((𝐺‘(♯‘𝐴)) + 1) ∈ (ℤ≥‘𝑀))
40	29, 39	syl 17	. . . . . . 7 ⊢ (𝜑 → ((𝐺‘(♯‘𝐴)) + 1) ∈ (ℤ≥‘𝑀))
41		fzss1 13599	. . . . . . 7 ⊢ (((𝐺‘(♯‘𝐴)) + 1) ∈ (ℤ≥‘𝑀) → (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ⊆ (𝑀...𝑁))
42	40, 41	syl 17	. . . . . 6 ⊢ (𝜑 → (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ⊆ (𝑀...𝑁))
43	42	sselda 3994	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → 𝑘 ∈ (𝑀...𝑁))
44		eluzelre 12886	. . . . . . . . 9 ⊢ ((𝐺‘(♯‘𝐴)) ∈ (ℤ≥‘𝑀) → (𝐺‘(♯‘𝐴)) ∈ ℝ)
45	29, 44	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝐺‘(♯‘𝐴)) ∈ ℝ)
46	45	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → (𝐺‘(♯‘𝐴)) ∈ ℝ)
47		peano2re 11431	. . . . . . . 8 ⊢ ((𝐺‘(♯‘𝐴)) ∈ ℝ → ((𝐺‘(♯‘𝐴)) + 1) ∈ ℝ)
48	46, 47	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → ((𝐺‘(♯‘𝐴)) + 1) ∈ ℝ)
49		elfzelz 13560	. . . . . . . . 9 ⊢ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) → 𝑘 ∈ ℤ)
50	49	zred 12719	. . . . . . . 8 ⊢ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) → 𝑘 ∈ ℝ)
51	50	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → 𝑘 ∈ ℝ)
52	46	ltp1d 12195	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → (𝐺‘(♯‘𝐴)) < ((𝐺‘(♯‘𝐴)) + 1))
53		elfzle1 13563	. . . . . . . 8 ⊢ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) → ((𝐺‘(♯‘𝐴)) + 1) ≤ 𝑘)
54	53	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → ((𝐺‘(♯‘𝐴)) + 1) ≤ 𝑘)
55	46, 48, 51, 52, 54	ltletrd 11418	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → (𝐺‘(♯‘𝐴)) < 𝑘)
56	6	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → 𝐺:(1...(♯‘𝐴))–1-1-onto→𝐴)
57		f1ocnv 6860	. . . . . . . . . . . . . 14 ⊢ (𝐺:(1...(♯‘𝐴))–1-1-onto→𝐴 → ◡𝐺:𝐴–1-1-onto→(1...(♯‘𝐴)))
58	56, 57	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → ◡𝐺:𝐴–1-1-onto→(1...(♯‘𝐴)))
59		f1of 6848	. . . . . . . . . . . . 13 ⊢ (◡𝐺:𝐴–1-1-onto→(1...(♯‘𝐴)) → ◡𝐺:𝐴⟶(1...(♯‘𝐴)))
60	58, 59	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → ◡𝐺:𝐴⟶(1...(♯‘𝐴)))
61		simprr 773	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → 𝑘 ∈ 𝐴)
62	60, 61	ffvelcdmd 7104	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → (◡𝐺‘𝑘) ∈ (1...(♯‘𝐴)))
63	62	elfzelzd 13561	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → (◡𝐺‘𝑘) ∈ ℤ)
64	63	zred 12719	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → (◡𝐺‘𝑘) ∈ ℝ)
65	18	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → (♯‘𝐴) ∈ ℕ0)
66	65	nn0red 12585	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → (♯‘𝐴) ∈ ℝ)
67		elfzle2 13564	. . . . . . . . . 10 ⊢ ((◡𝐺‘𝑘) ∈ (1...(♯‘𝐴)) → (◡𝐺‘𝑘) ≤ (♯‘𝐴))
68	62, 67	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → (◡𝐺‘𝑘) ≤ (♯‘𝐴))
69	64, 66, 68	lensymd 11409	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → ¬ (♯‘𝐴) < (◡𝐺‘𝑘))
70	4	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → 𝐺 Isom < , < ((1...(♯‘𝐴)), 𝐴))
71	26	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → (♯‘𝐴) ∈ (1...(♯‘𝐴)))
72		isorel 7345	. . . . . . . . . 10 ⊢ ((𝐺 Isom < , < ((1...(♯‘𝐴)), 𝐴) ∧ ((♯‘𝐴) ∈ (1...(♯‘𝐴)) ∧ (◡𝐺‘𝑘) ∈ (1...(♯‘𝐴)))) → ((♯‘𝐴) < (◡𝐺‘𝑘) ↔ (𝐺‘(♯‘𝐴)) < (𝐺‘(◡𝐺‘𝑘))))
73	70, 71, 62, 72	syl12anc 837	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → ((♯‘𝐴) < (◡𝐺‘𝑘) ↔ (𝐺‘(♯‘𝐴)) < (𝐺‘(◡𝐺‘𝑘))))
74		f1ocnvfv2 7296	. . . . . . . . . . 11 ⊢ ((𝐺:(1...(♯‘𝐴))–1-1-onto→𝐴 ∧ 𝑘 ∈ 𝐴) → (𝐺‘(◡𝐺‘𝑘)) = 𝑘)
75	56, 61, 74	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → (𝐺‘(◡𝐺‘𝑘)) = 𝑘)
76	75	breq2d 5159	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → ((𝐺‘(♯‘𝐴)) < (𝐺‘(◡𝐺‘𝑘)) ↔ (𝐺‘(♯‘𝐴)) < 𝑘))
77	73, 76	bitrd 279	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → ((♯‘𝐴) < (◡𝐺‘𝑘) ↔ (𝐺‘(♯‘𝐴)) < 𝑘))
78	69, 77	mtbid 324	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → ¬ (𝐺‘(♯‘𝐴)) < 𝑘)
79	78	expr 456	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → (𝑘 ∈ 𝐴 → ¬ (𝐺‘(♯‘𝐴)) < 𝑘))
80	55, 79	mt2d 136	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → ¬ 𝑘 ∈ 𝐴)
81	43, 80	eldifd 3973	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → 𝑘 ∈ ((𝑀...𝑁) ∖ 𝐴))
82		seqcoll2.7	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀...𝑁) ∖ 𝐴)) → (𝐹‘𝑘) = 𝑍)
83	81, 82	syldan 591	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → (𝐹‘𝑘) = 𝑍)
84	1, 29, 31, 38, 83	seqid2 14085	. 2 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘(𝐺‘(♯‘𝐴))) = (seq𝑀( + , 𝐹)‘𝑁))
85		seqcoll2.1	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑆) → (𝑍 + 𝑘) = 𝑘)
86		seqcoll2.a	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑆)
87	3, 2	sstrdi 4007	. . 3 ⊢ (𝜑 → 𝐴 ⊆ (ℤ≥‘𝑀))
88	33	ssdifd 4154	. . . . 5 ⊢ (𝜑 → ((𝑀...(𝐺‘(♯‘𝐴))) ∖ 𝐴) ⊆ ((𝑀...𝑁) ∖ 𝐴))
89	88	sselda 3994	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀...(𝐺‘(♯‘𝐴))) ∖ 𝐴)) → 𝑘 ∈ ((𝑀...𝑁) ∖ 𝐴))
90	89, 82	syldan 591	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀...(𝐺‘(♯‘𝐴))) ∖ 𝐴)) → (𝐹‘𝑘) = 𝑍)
91		seqcoll2.8	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (𝐻‘𝑛) = (𝐹‘(𝐺‘𝑛)))
92	85, 1, 37, 86, 4, 26, 87, 36, 90, 91	seqcoll 14499	. 2 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘(𝐺‘(♯‘𝐴))) = (seq1( + , 𝐻)‘(♯‘𝐴)))
93	84, 92	eqtr3d 2776	1 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq1( + , 𝐻)‘(♯‘𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1536   ∈ wcel 2105   ≠ wne 2937   ∖ cdif 3959   ⊆ wss 3962  ∅c0 4338   class class class wbr 5147  ^◡ccnv 5687  ⟶wf 6558  –1-1-onto→wf1o 6561  ‘cfv 6562   Isom wiso 6563  (class class class)co 7430  Fincfn 8983  ℝcr 11151  0cc0 11152  1c1 11153   + caddc 11155   < clt 11292   ≤ cle 11293  ℕcn 12263  ℕ₀cn0 12523  ℤ_≥cuz 12875  ...cfz 13543  seqcseq 14038  ♯chash 14365

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-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  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

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-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-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-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-n0 12524  df-z 12611  df-uz 12876  df-fz 13544  df-seq 14039  df-hash 14366

This theorem is referenced by:  isercolllem3  15699  gsumval3  19939