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Theorem seqcoll2 14437

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 13533	. . . 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 7301	. . . . . . . 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 6803	. . . . . . 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 13944	. . . . . . . . . . . . 13 ⊢ (𝑀...𝑁) ∈ Fin
11		ssfi 9143	. . . . . . . . . . . . 13 ⊢ (((𝑀...𝑁) ∈ Fin ∧ 𝐴 ⊆ (𝑀...𝑁)) → 𝐴 ∈ Fin)
12	10, 3, 11	sylancr 587	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ Fin)
13		hasheq0 14335	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅))
14	12, 13	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅))
15	14	necon3bbid 2963	. . . . . . . . . 10 ⊢ (𝜑 → (¬ (♯‘𝐴) = 0 ↔ 𝐴 ≠ ∅))
16	9, 15	mpbird 257	. . . . . . . . 9 ⊢ (𝜑 → ¬ (♯‘𝐴) = 0)
17		hashcl 14328	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ₀)
18	12, 17	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝐴) ∈ ℕ₀)
19		elnn0 12451	. . . . . . . . . . 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 12843	. . . . . . . 8 ⊢ ℕ = (ℤ_≥‘1)
24	22, 23	eleqtrdi 2839	. . . . . . 7 ⊢ (𝜑 → (♯‘𝐴) ∈ (ℤ_≥‘1))
25		eluzfz2 13500	. . . . . . 7 ⊢ ((♯‘𝐴) ∈ (ℤ_≥‘1) → (♯‘𝐴) ∈ (1...(♯‘𝐴)))
26	24, 25	syl 17	. . . . . 6 ⊢ (𝜑 → (♯‘𝐴) ∈ (1...(♯‘𝐴)))
27	8, 26	ffvelcdmd 7060	. . . . 5 ⊢ (𝜑 → (𝐺‘(♯‘𝐴)) ∈ 𝐴)
28	3, 27	sseldd 3950	. . . 4 ⊢ (𝜑 → (𝐺‘(♯‘𝐴)) ∈ (𝑀...𝑁))
29	2, 28	sselid 3947	. . 3 ⊢ (𝜑 → (𝐺‘(♯‘𝐴)) ∈ (ℤ_≥‘𝑀))
30		elfzuz3 13489	. . . 4 ⊢ ((𝐺‘(♯‘𝐴)) ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ_≥‘(𝐺‘(♯‘𝐴))))
31	28, 30	syl 17	. . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘(𝐺‘(♯‘𝐴))))
32		fzss2 13532	. . . . . . 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 3949	. . . . 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 13994	. . 3 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘(𝐺‘(♯‘𝐴))) ∈ 𝑆)
39		peano2uz 12867	. . . . . . . 8 ⊢ ((𝐺‘(♯‘𝐴)) ∈ (ℤ_≥‘𝑀) → ((𝐺‘(♯‘𝐴)) + 1) ∈ (ℤ_≥‘𝑀))
40	29, 39	syl 17	. . . . . . 7 ⊢ (𝜑 → ((𝐺‘(♯‘𝐴)) + 1) ∈ (ℤ_≥‘𝑀))
41		fzss1 13531	. . . . . . 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 3949	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → 𝑘 ∈ (𝑀...𝑁))
44		eluzelre 12811	. . . . . . . . 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 11354	. . . . . . . 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 13492	. . . . . . . . 9 ⊢ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) → 𝑘 ∈ ℤ)
50	49	zred 12645	. . . . . . . 8 ⊢ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) → 𝑘 ∈ ℝ)
51	50	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → 𝑘 ∈ ℝ)
52	46	ltp1d 12120	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → (𝐺‘(♯‘𝐴)) < ((𝐺‘(♯‘𝐴)) + 1))
53		elfzle1 13495	. . . . . . . 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 11341	. . . . . 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 6815	. . . . . . . . . . . . . 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 6803	. . . . . . . . . . . . 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 772	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → 𝑘 ∈ 𝐴)
62	60, 61	ffvelcdmd 7060	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → (^◡𝐺‘𝑘) ∈ (1...(♯‘𝐴)))
63	62	elfzelzd 13493	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → (^◡𝐺‘𝑘) ∈ ℤ)
64	63	zred 12645	. . . . . . . . 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 12511	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → (♯‘𝐴) ∈ ℝ)
67		elfzle2 13496	. . . . . . . . . 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 11332	. . . . . . . 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 7304	. . . . . . . . . 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 836	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → ((♯‘𝐴) < (^◡𝐺‘𝑘) ↔ (𝐺‘(♯‘𝐴)) < (𝐺‘(^◡𝐺‘𝑘))))
74		f1ocnvfv2 7255	. . . . . . . . . . 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 5122	. . . . . . . . 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 3928	. . . 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 14020	. 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 3962	. . 3 ⊢ (𝜑 → 𝐴 ⊆ (ℤ_≥‘𝑀))
88	33	ssdifd 4111	. . . . 5 ⊢ (𝜑 → ((𝑀...(𝐺‘(♯‘𝐴))) ∖ 𝐴) ⊆ ((𝑀...𝑁) ∖ 𝐴))
89	88	sselda 3949	. . . 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 14436	. 2 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘(𝐺‘(♯‘𝐴))) = (seq1( + , 𝐻)‘(♯‘𝐴)))
93	84, 92	eqtr3d 2767	1 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq1( + , 𝐻)‘(♯‘𝐴)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1540 ∈ wcel 2109 ≠ wne 2926 ∖ cdif 3914 ⊆ wss 3917 ∅c0 4299 class class class wbr 5110 ^◡ccnv 5640 ⟶wf 6510 –1-1-onto→wf1o 6513 ‘cfv 6514 Isom wiso 6515 (class class class)co 7390 Fincfn 8921 ℝcr 11074 0cc0 11075 1c1 11076 + caddc 11078 < clt 11215 ≤ cle 11216 ℕcn 12193 ℕ₀cn0 12449 ℤ_≥cuz 12800 ...cfz 13475 seqcseq 13973 ♯chash 14302

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-isom 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-er 8674 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-nn 12194 df-n0 12450 df-z 12537 df-uz 12801 df-fz 13476 df-seq 13974 df-hash 14303

This theorem is referenced by: isercolllem3 15640 gsumval3 19844

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