Theorem seqcoll2 13249

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 12382	. . . 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 6573	. . . . . . . 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 6137	. . . . . . 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 12771	. . . . . . . . . . . . 13 ⊢ (𝑀...𝑁) ∈ Fin
11		ssfi 8180	. . . . . . . . . . . . 13 ⊢ (((𝑀...𝑁) ∈ Fin ∧ 𝐴 ⊆ (𝑀...𝑁)) → 𝐴 ∈ Fin)
12	10, 3, 11	sylancr 695	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ Fin)
13		hasheq0 13154	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Fin → ((#‘𝐴) = 0 ↔ 𝐴 = ∅))
14	12, 13	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ((#‘𝐴) = 0 ↔ 𝐴 = ∅))
15	14	necon3bbid 2831	. . . . . . . . . 10 ⊢ (𝜑 → (¬ (#‘𝐴) = 0 ↔ 𝐴 ≠ ∅))
16	9, 15	mpbird 247	. . . . . . . . 9 ⊢ (𝜑 → ¬ (#‘𝐴) = 0)
17		hashcl 13147	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Fin → (#‘𝐴) ∈ ℕ0)
18	12, 17	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (#‘𝐴) ∈ ℕ0)
19		elnn0 11294	. . . . . . . . . . 11 ⊢ ((#‘𝐴) ∈ ℕ0 ↔ ((#‘𝐴) ∈ ℕ ∨ (#‘𝐴) = 0))
20	18, 19	sylib 208	. . . . . . . . . 10 ⊢ (𝜑 → ((#‘𝐴) ∈ ℕ ∨ (#‘𝐴) = 0))
21	20	ord 392	. . . . . . . . 9 ⊢ (𝜑 → (¬ (#‘𝐴) ∈ ℕ → (#‘𝐴) = 0))
22	16, 21	mt3d 140	. . . . . . . 8 ⊢ (𝜑 → (#‘𝐴) ∈ ℕ)
23		nnuz 11723	. . . . . . . 8 ⊢ ℕ = (ℤ≥‘1)
24	22, 23	syl6eleq 2711	. . . . . . 7 ⊢ (𝜑 → (#‘𝐴) ∈ (ℤ≥‘1))
25		eluzfz2 12349	. . . . . . 7 ⊢ ((#‘𝐴) ∈ (ℤ≥‘1) → (#‘𝐴) ∈ (1...(#‘𝐴)))
26	24, 25	syl 17	. . . . . 6 ⊢ (𝜑 → (#‘𝐴) ∈ (1...(#‘𝐴)))
27	8, 26	ffvelrnd 6360	. . . . 5 ⊢ (𝜑 → (𝐺‘(#‘𝐴)) ∈ 𝐴)
28	3, 27	sseldd 3604	. . . 4 ⊢ (𝜑 → (𝐺‘(#‘𝐴)) ∈ (𝑀...𝑁))
29	2, 28	sseldi 3601	. . 3 ⊢ (𝜑 → (𝐺‘(#‘𝐴)) ∈ (ℤ≥‘𝑀))
30		elfzuz3 12339	. . . 4 ⊢ ((𝐺‘(#‘𝐴)) ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘(𝐺‘(#‘𝐴))))
31	28, 30	syl 17	. . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝐺‘(#‘𝐴))))
32		fzss2 12381	. . . . . . 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 3603	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝐺‘(#‘𝐴)))) → 𝑘 ∈ (𝑀...𝑁))
35		seqcoll2.6	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ 𝑆)
36	34, 35	syldan 487	. . . 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 12821	. . 3 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘(𝐺‘(#‘𝐴))) ∈ 𝑆)
39		peano2uz 11741	. . . . . . . 8 ⊢ ((𝐺‘(#‘𝐴)) ∈ (ℤ≥‘𝑀) → ((𝐺‘(#‘𝐴)) + 1) ∈ (ℤ≥‘𝑀))
40	29, 39	syl 17	. . . . . . 7 ⊢ (𝜑 → ((𝐺‘(#‘𝐴)) + 1) ∈ (ℤ≥‘𝑀))
41		fzss1 12380	. . . . . . 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 3603	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁)) → 𝑘 ∈ (𝑀...𝑁))
44		eluzelre 11698	. . . . . . . . 9 ⊢ ((𝐺‘(#‘𝐴)) ∈ (ℤ≥‘𝑀) → (𝐺‘(#‘𝐴)) ∈ ℝ)
45	29, 44	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝐺‘(#‘𝐴)) ∈ ℝ)
46	45	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁)) → (𝐺‘(#‘𝐴)) ∈ ℝ)
47		peano2re 10209	. . . . . . . 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 12342	. . . . . . . . 9 ⊢ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) → 𝑘 ∈ ℤ)
50	49	zred 11482	. . . . . . . 8 ⊢ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) → 𝑘 ∈ ℝ)
51	50	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁)) → 𝑘 ∈ ℝ)
52	46	ltp1d 10954	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁)) → (𝐺‘(#‘𝐴)) < ((𝐺‘(#‘𝐴)) + 1))
53		elfzle1 12344	. . . . . . . 8 ⊢ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) → ((𝐺‘(#‘𝐴)) + 1) ≤ 𝑘)
54	53	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁)) → ((𝐺‘(#‘𝐴)) + 1) ≤ 𝑘)
55	46, 48, 51, 52, 54	ltletrd 10197	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁)) → (𝐺‘(#‘𝐴)) < 𝑘)
56	6	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → 𝐺:(1...(#‘𝐴))–1-1-onto→𝐴)
57		f1ocnv 6149	. . . . . . . . . . . . 13 ⊢ (𝐺:(1...(#‘𝐴))–1-1-onto→𝐴 → ◡𝐺:𝐴–1-1-onto→(1...(#‘𝐴)))
58	56, 57	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → ◡𝐺:𝐴–1-1-onto→(1...(#‘𝐴)))
59		f1of 6137	. . . . . . . . . . . 12 ⊢ (◡𝐺:𝐴–1-1-onto→(1...(#‘𝐴)) → ◡𝐺:𝐴⟶(1...(#‘𝐴)))
60	58, 59	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → ◡𝐺:𝐴⟶(1...(#‘𝐴)))
61		simprr 796	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → 𝑘 ∈ 𝐴)
62	60, 61	ffvelrnd 6360	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → (◡𝐺‘𝑘) ∈ (1...(#‘𝐴)))
63		elfzle2 12345	. . . . . . . . . 10 ⊢ ((◡𝐺‘𝑘) ∈ (1...(#‘𝐴)) → (◡𝐺‘𝑘) ≤ (#‘𝐴))
64	62, 63	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → (◡𝐺‘𝑘) ≤ (#‘𝐴))
65		elfzelz 12342	. . . . . . . . . . . 12 ⊢ ((◡𝐺‘𝑘) ∈ (1...(#‘𝐴)) → (◡𝐺‘𝑘) ∈ ℤ)
66	62, 65	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → (◡𝐺‘𝑘) ∈ ℤ)
67	66	zred 11482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → (◡𝐺‘𝑘) ∈ ℝ)
68	18	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → (#‘𝐴) ∈ ℕ0)
69	68	nn0red 11352	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → (#‘𝐴) ∈ ℝ)
70	67, 69	lenltd 10183	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → ((◡𝐺‘𝑘) ≤ (#‘𝐴) ↔ ¬ (#‘𝐴) < (◡𝐺‘𝑘)))
71	64, 70	mpbid 222	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → ¬ (#‘𝐴) < (◡𝐺‘𝑘))
72	4	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → 𝐺 Isom < , < ((1...(#‘𝐴)), 𝐴))
73	26	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → (#‘𝐴) ∈ (1...(#‘𝐴)))
74		isorel 6576	. . . . . . . . . 10 ⊢ ((𝐺 Isom < , < ((1...(#‘𝐴)), 𝐴) ∧ ((#‘𝐴) ∈ (1...(#‘𝐴)) ∧ (◡𝐺‘𝑘) ∈ (1...(#‘𝐴)))) → ((#‘𝐴) < (◡𝐺‘𝑘) ↔ (𝐺‘(#‘𝐴)) < (𝐺‘(◡𝐺‘𝑘))))
75	72, 73, 62, 74	syl12anc 1324	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → ((#‘𝐴) < (◡𝐺‘𝑘) ↔ (𝐺‘(#‘𝐴)) < (𝐺‘(◡𝐺‘𝑘))))
76		f1ocnvfv2 6533	. . . . . . . . . . 11 ⊢ ((𝐺:(1...(#‘𝐴))–1-1-onto→𝐴 ∧ 𝑘 ∈ 𝐴) → (𝐺‘(◡𝐺‘𝑘)) = 𝑘)
77	56, 61, 76	syl2anc 693	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → (𝐺‘(◡𝐺‘𝑘)) = 𝑘)
78	77	breq2d 4665	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → ((𝐺‘(#‘𝐴)) < (𝐺‘(◡𝐺‘𝑘)) ↔ (𝐺‘(#‘𝐴)) < 𝑘))
79	75, 78	bitrd 268	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → ((#‘𝐴) < (◡𝐺‘𝑘) ↔ (𝐺‘(#‘𝐴)) < 𝑘))
80	71, 79	mtbid 314	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → ¬ (𝐺‘(#‘𝐴)) < 𝑘)
81	80	expr 643	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁)) → (𝑘 ∈ 𝐴 → ¬ (𝐺‘(#‘𝐴)) < 𝑘))
82	55, 81	mt2d 131	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁)) → ¬ 𝑘 ∈ 𝐴)
83	43, 82	eldifd 3585	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁)) → 𝑘 ∈ ((𝑀...𝑁) ∖ 𝐴))
84		seqcoll2.7	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀...𝑁) ∖ 𝐴)) → (𝐹‘𝑘) = 𝑍)
85	83, 84	syldan 487	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁)) → (𝐹‘𝑘) = 𝑍)
86	1, 29, 31, 38, 85	seqid2 12847	. 2 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘(𝐺‘(#‘𝐴))) = (seq𝑀( + , 𝐹)‘𝑁))
87		seqcoll2.1	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑆) → (𝑍 + 𝑘) = 𝑘)
88		seqcoll2.a	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑆)
89	3, 2	syl6ss 3615	. . 3 ⊢ (𝜑 → 𝐴 ⊆ (ℤ≥‘𝑀))
90	33	ssdifd 3746	. . . . 5 ⊢ (𝜑 → ((𝑀...(𝐺‘(#‘𝐴))) ∖ 𝐴) ⊆ ((𝑀...𝑁) ∖ 𝐴))
91	90	sselda 3603	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀...(𝐺‘(#‘𝐴))) ∖ 𝐴)) → 𝑘 ∈ ((𝑀...𝑁) ∖ 𝐴))
92	91, 84	syldan 487	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀...(𝐺‘(#‘𝐴))) ∖ 𝐴)) → (𝐹‘𝑘) = 𝑍)
93		seqcoll2.8	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(#‘𝐴))) → (𝐻‘𝑛) = (𝐹‘(𝐺‘𝑛)))
94	87, 1, 37, 88, 4, 26, 89, 36, 92, 93	seqcoll 13248	. 2 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘(𝐺‘(#‘𝐴))) = (seq1( + , 𝐻)‘(#‘𝐴)))
95	86, 94	eqtr3d 2658	1 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq1( + , 𝐻)‘(#‘𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794   ∖ cdif 3571   ⊆ wss 3574  ∅c0 3915   class class class wbr 4653  ^◡ccnv 5113  ⟶wf 5884  –1-1-onto→wf1o 5887  ‘cfv 5888   Isom wiso 5889  (class class class)co 6650  Fincfn 7955  ℝcr 9935  0cc0 9936  1c1 9937   + caddc 9939   < clt 10074   ≤ cle 10075  ℕcn 11020  ℕ₀cn0 11292  ℤcz 11377  ℤ_≥cuz 11687  ...cfz 12326  seqcseq 12801  #chash 13117

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  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-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-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-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  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-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-seq 12802  df-hash 13118

This theorem is referenced by:  isercolllem3  14397  gsumval3  18308