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Theorem seqcoll2 14369

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 13462	. . . 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 7257	. . . . . . . 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 6763	. . . . . . 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 13876	. . . . . . . . . . . . 13 ⊢ (𝑀...𝑁) ∈ Fin
11		ssfi 9082	. . . . . . . . . . . . 13 ⊢ (((𝑀...𝑁) ∈ Fin ∧ 𝐴 ⊆ (𝑀...𝑁)) → 𝐴 ∈ Fin)
12	10, 3, 11	sylancr 587	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ Fin)
13		hasheq0 14267	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅))
14	12, 13	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅))
15	14	necon3bbid 2965	. . . . . . . . . 10 ⊢ (𝜑 → (¬ (♯‘𝐴) = 0 ↔ 𝐴 ≠ ∅))
16	9, 15	mpbird 257	. . . . . . . . 9 ⊢ (𝜑 → ¬ (♯‘𝐴) = 0)
17		hashcl 14260	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ₀)
18	12, 17	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝐴) ∈ ℕ₀)
19		elnn0 12380	. . . . . . . . . . 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 12772	. . . . . . . 8 ⊢ ℕ = (ℤ_≥‘1)
24	22, 23	eleqtrdi 2841	. . . . . . 7 ⊢ (𝜑 → (♯‘𝐴) ∈ (ℤ_≥‘1))
25		eluzfz2 13429	. . . . . . 7 ⊢ ((♯‘𝐴) ∈ (ℤ_≥‘1) → (♯‘𝐴) ∈ (1...(♯‘𝐴)))
26	24, 25	syl 17	. . . . . 6 ⊢ (𝜑 → (♯‘𝐴) ∈ (1...(♯‘𝐴)))
27	8, 26	ffvelcdmd 7018	. . . . 5 ⊢ (𝜑 → (𝐺‘(♯‘𝐴)) ∈ 𝐴)
28	3, 27	sseldd 3935	. . . 4 ⊢ (𝜑 → (𝐺‘(♯‘𝐴)) ∈ (𝑀...𝑁))
29	2, 28	sselid 3932	. . 3 ⊢ (𝜑 → (𝐺‘(♯‘𝐴)) ∈ (ℤ_≥‘𝑀))
30		elfzuz3 13418	. . . 4 ⊢ ((𝐺‘(♯‘𝐴)) ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ_≥‘(𝐺‘(♯‘𝐴))))
31	28, 30	syl 17	. . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘(𝐺‘(♯‘𝐴))))
32		fzss2 13461	. . . . . . 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 3934	. . . . 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 13926	. . 3 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘(𝐺‘(♯‘𝐴))) ∈ 𝑆)
39		peano2uz 12796	. . . . . . . 8 ⊢ ((𝐺‘(♯‘𝐴)) ∈ (ℤ_≥‘𝑀) → ((𝐺‘(♯‘𝐴)) + 1) ∈ (ℤ_≥‘𝑀))
40	29, 39	syl 17	. . . . . . 7 ⊢ (𝜑 → ((𝐺‘(♯‘𝐴)) + 1) ∈ (ℤ_≥‘𝑀))
41		fzss1 13460	. . . . . . 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 3934	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → 𝑘 ∈ (𝑀...𝑁))
44		eluzelre 12740	. . . . . . . . 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 11283	. . . . . . . 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 13421	. . . . . . . . 9 ⊢ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) → 𝑘 ∈ ℤ)
50	49	zred 12574	. . . . . . . 8 ⊢ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) → 𝑘 ∈ ℝ)
51	50	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → 𝑘 ∈ ℝ)
52	46	ltp1d 12049	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → (𝐺‘(♯‘𝐴)) < ((𝐺‘(♯‘𝐴)) + 1))
53		elfzle1 13424	. . . . . . . 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 11270	. . . . . 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 6775	. . . . . . . . . . . . . 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 6763	. . . . . . . . . . . . 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 7018	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → (^◡𝐺‘𝑘) ∈ (1...(♯‘𝐴)))
63	62	elfzelzd 13422	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → (^◡𝐺‘𝑘) ∈ ℤ)
64	63	zred 12574	. . . . . . . . 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 12440	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → (♯‘𝐴) ∈ ℝ)
67		elfzle2 13425	. . . . . . . . . 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 11261	. . . . . . . 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 7260	. . . . . . . . . 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 7211	. . . . . . . . . . 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 5103	. . . . . . . . 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 3913	. . . 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 13952	. 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 3947	. . 3 ⊢ (𝜑 → 𝐴 ⊆ (ℤ_≥‘𝑀))
88	33	ssdifd 4095	. . . . 5 ⊢ (𝜑 → ((𝑀...(𝐺‘(♯‘𝐴))) ∖ 𝐴) ⊆ ((𝑀...𝑁) ∖ 𝐴))
89	88	sselda 3934	. . . 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 14368	. 2 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘(𝐺‘(♯‘𝐴))) = (seq1( + , 𝐻)‘(♯‘𝐴)))
93	84, 92	eqtr3d 2768	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 1541 ∈ wcel 2111 ≠ wne 2928 ∖ cdif 3899 ⊆ wss 3902 ∅c0 4283 class class class wbr 5091 ^◡ccnv 5615 ⟶wf 6477 –1-1-onto→wf1o 6480 ‘cfv 6481 Isom wiso 6482 (class class class)co 7346 Fincfn 8869 ℝcr 11002 0cc0 11003 1c1 11004 + caddc 11006 < clt 11143 ≤ cle 11144 ℕcn 12122 ℕ₀cn0 12378 ℤ_≥cuz 12729 ...cfz 13404 seqcseq 13905 ♯chash 14234

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-cnex 11059 ax-resscn 11060 ax-1cn 11061 ax-icn 11062 ax-addcl 11063 ax-addrcl 11064 ax-mulcl 11065 ax-mulrcl 11066 ax-mulcom 11067 ax-addass 11068 ax-mulass 11069 ax-distr 11070 ax-i2m1 11071 ax-1ne0 11072 ax-1rid 11073 ax-rnegex 11074 ax-rrecex 11075 ax-cnre 11076 ax-pre-lttri 11077 ax-pre-lttrn 11078 ax-pre-ltadd 11079 ax-pre-mulgt0 11080

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-op 4583 df-uni 4860 df-int 4898 df-iun 4943 df-br 5092 df-opab 5154 df-mpt 5173 df-tr 5199 df-id 5511 df-eprel 5516 df-po 5524 df-so 5525 df-fr 5569 df-we 5571 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-er 8622 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-card 9829 df-pnf 11145 df-mnf 11146 df-xr 11147 df-ltxr 11148 df-le 11149 df-sub 11343 df-neg 11344 df-nn 12123 df-n0 12379 df-z 12466 df-uz 12730 df-fz 13405 df-seq 13906 df-hash 14235

This theorem is referenced by: isercolllem3 15571 gsumval3 19817

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