seqcoll2 - Metamath Proof Explorer

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Theorem seqcoll2 14522

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 13612	. . . 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 7332	. . . . . . . 8 ⊢ (𝐺 Isom < , < ((1...(♯‘𝐴)), 𝐴) → 𝐺:(1...(♯‘𝐴))–1-1-onto→𝐴)
6	4, 5	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐺:(1...(♯‘𝐴))–1-1-onto→𝐴)
7		f1of 6827	. . . . . . 7 ⊢ (𝐺:(1...(♯‘𝐴))–1-1-onto→𝐴 → 𝐺:(1...(♯‘𝐴))⟶𝐴)
8	6, 7	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐺:(1...(♯‘𝐴))⟶𝐴)
9		seqcoll2.3	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ≠ ∅)
10		fzfi 14028	. . . . . . . . . . . . 13 ⊢ (𝑀...𝑁) ∈ Fin
11		ssfi 9167	. . . . . . . . . . . . 13 ⊢ (((𝑀...𝑁) ∈ Fin ∧ 𝐴 ⊆ (𝑀...𝑁)) → 𝐴 ∈ Fin)
12	10, 3, 11	sylancr 599	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ Fin)
13		hasheq0 14419	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅))
14	12, 13	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅))
15	14	necon3bbid 2998	. . . . . . . . . 10 ⊢ (𝜑 → (¬ (♯‘𝐴) = 0 ↔ 𝐴 ≠ ∅))
16	9, 15	mpbird 260	. . . . . . . . 9 ⊢ (𝜑 → ¬ (♯‘𝐴) = 0)
17		hashcl 14412	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ₀)
18	12, 17	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝐴) ∈ ℕ₀)
19		elnn0 12524	. . . . . . . . . . 11 ⊢ ((♯‘𝐴) ∈ ℕ₀ ↔ ((♯‘𝐴) ∈ ℕ ∨ (♯‘𝐴) = 0))
20	18, 19	sylib 221	. . . . . . . . . 10 ⊢ (𝜑 → ((♯‘𝐴) ∈ ℕ ∨ (♯‘𝐴) = 0))
21	20	ord 878	. . . . . . . . 9 ⊢ (𝜑 → (¬ (♯‘𝐴) ∈ ℕ → (♯‘𝐴) = 0))
22	16, 21	mt3d 149	. . . . . . . 8 ⊢ (𝜑 → (♯‘𝐴) ∈ ℕ)
23		nnuz 12919	. . . . . . . 8 ⊢ ℕ = (ℤ_≥‘1)
24	22, 23	eleqtrdi 2876	. . . . . . 7 ⊢ (𝜑 → (♯‘𝐴) ∈ (ℤ_≥‘1))
25		eluzfz2 13578	. . . . . . 7 ⊢ ((♯‘𝐴) ∈ (ℤ_≥‘1) → (♯‘𝐴) ∈ (1...(♯‘𝐴)))
26	24, 25	syl 18	. . . . . 6 ⊢ (𝜑 → (♯‘𝐴) ∈ (1...(♯‘𝐴)))
27	8, 26	ffvelcdmd 7087	. . . . 5 ⊢ (𝜑 → (𝐺‘(♯‘𝐴)) ∈ 𝐴)
28	3, 27	sseldd 3941	. . . 4 ⊢ (𝜑 → (𝐺‘(♯‘𝐴)) ∈ (𝑀...𝑁))
29	2, 28	sselid 3938	. . 3 ⊢ (𝜑 → (𝐺‘(♯‘𝐴)) ∈ (ℤ_≥‘𝑀))
30		elfzuz3 13567	. . . 4 ⊢ ((𝐺‘(♯‘𝐴)) ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ_≥‘(𝐺‘(♯‘𝐴))))
31	28, 30	syl 18	. . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘(𝐺‘(♯‘𝐴))))
32		fzss2 13611	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘(𝐺‘(♯‘𝐴))) → (𝑀...(𝐺‘(♯‘𝐴))) ⊆ (𝑀...𝑁))
33	31, 32	syl 18	. . . . . 6 ⊢ (𝜑 → (𝑀...(𝐺‘(♯‘𝐴))) ⊆ (𝑀...𝑁))
34	33	sselda 3940	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝐺‘(♯‘𝐴)))) → 𝑘 ∈ (𝑀...𝑁))
35		seqcoll2.6	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ 𝑆)
36	34, 35	syldan 603	. . . 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 14078	. . 3 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘(𝐺‘(♯‘𝐴))) ∈ 𝑆)
39		peano2uz 12943	. . . . . . . 8 ⊢ ((𝐺‘(♯‘𝐴)) ∈ (ℤ_≥‘𝑀) → ((𝐺‘(♯‘𝐴)) + 1) ∈ (ℤ_≥‘𝑀))
40	29, 39	syl 18	. . . . . . 7 ⊢ (𝜑 → ((𝐺‘(♯‘𝐴)) + 1) ∈ (ℤ_≥‘𝑀))
41		fzss1 13610	. . . . . . 7 ⊢ (((𝐺‘(♯‘𝐴)) + 1) ∈ (ℤ_≥‘𝑀) → (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ⊆ (𝑀...𝑁))
42	40, 41	syl 18	. . . . . 6 ⊢ (𝜑 → (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ⊆ (𝑀...𝑁))
43	42	sselda 3940	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → 𝑘 ∈ (𝑀...𝑁))
44		eluzelre 12891	. . . . . . . . 9 ⊢ ((𝐺‘(♯‘𝐴)) ∈ (ℤ_≥‘𝑀) → (𝐺‘(♯‘𝐴)) ∈ ℝ)
45	29, 44	syl 18	. . . . . . . 8 ⊢ (𝜑 → (𝐺‘(♯‘𝐴)) ∈ ℝ)
46	45	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → (𝐺‘(♯‘𝐴)) ∈ ℝ)
47		peano2re 11401	. . . . . . . 8 ⊢ ((𝐺‘(♯‘𝐴)) ∈ ℝ → ((𝐺‘(♯‘𝐴)) + 1) ∈ ℝ)
48	46, 47	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → ((𝐺‘(♯‘𝐴)) + 1) ∈ ℝ)
49		elfzelz 13570	. . . . . . . . 9 ⊢ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) → 𝑘 ∈ ℤ)
50	49	zred 12718	. . . . . . . 8 ⊢ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) → 𝑘 ∈ ℝ)
51	50	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → 𝑘 ∈ ℝ)
52	46	ltp1d 12163	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → (𝐺‘(♯‘𝐴)) < ((𝐺‘(♯‘𝐴)) + 1))
53		elfzle1 13573	. . . . . . . 8 ⊢ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) → ((𝐺‘(♯‘𝐴)) + 1) ≤ 𝑘)
54	53	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → ((𝐺‘(♯‘𝐴)) + 1) ≤ 𝑘)
55	46, 48, 51, 52, 54	ltletrd 11388	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → (𝐺‘(♯‘𝐴)) < 𝑘)
56	6	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → 𝐺:(1...(♯‘𝐴))–1-1-onto→𝐴)
57		f1ocnv 6840	. . . . . . . . . . . . . 14 ⊢ (𝐺:(1...(♯‘𝐴))–1-1-onto→𝐴 → ^◡𝐺:𝐴–1-1-onto→(1...(♯‘𝐴)))
58	56, 57	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → ^◡𝐺:𝐴–1-1-onto→(1...(♯‘𝐴)))
59		f1of 6827	. . . . . . . . . . . . 13 ⊢ (^◡𝐺:𝐴–1-1-onto→(1...(♯‘𝐴)) → ^◡𝐺:𝐴⟶(1...(♯‘𝐴)))
60	58, 59	syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → ^◡𝐺:𝐴⟶(1...(♯‘𝐴)))
61		simprr 785	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → 𝑘 ∈ 𝐴)
62	60, 61	ffvelcdmd 7087	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → (^◡𝐺‘𝑘) ∈ (1...(♯‘𝐴)))
63	62	elfzelzd 13571	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → (^◡𝐺‘𝑘) ∈ ℤ)
64	63	zred 12718	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → (^◡𝐺‘𝑘) ∈ ℝ)
65	18	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → (♯‘𝐴) ∈ ℕ₀)
66	65	nn0red 12584	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → (♯‘𝐴) ∈ ℝ)
67		elfzle2 13574	. . . . . . . . . 10 ⊢ ((^◡𝐺‘𝑘) ∈ (1...(♯‘𝐴)) → (^◡𝐺‘𝑘) ≤ (♯‘𝐴))
68	62, 67	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → (^◡𝐺‘𝑘) ≤ (♯‘𝐴))
69	64, 66, 68	lensymd 11379	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → ¬ (♯‘𝐴) < (^◡𝐺‘𝑘))
70	4	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → 𝐺 Isom < , < ((1...(♯‘𝐴)), 𝐴))
71	26	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → (♯‘𝐴) ∈ (1...(♯‘𝐴)))
72		isorel 7335	. . . . . . . . . 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 850	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → ((♯‘𝐴) < (^◡𝐺‘𝑘) ↔ (𝐺‘(♯‘𝐴)) < (𝐺‘(^◡𝐺‘𝑘))))
74		f1ocnvfv2 7286	. . . . . . . . . . 11 ⊢ ((𝐺:(1...(♯‘𝐴))–1-1-onto→𝐴 ∧ 𝑘 ∈ 𝐴) → (𝐺‘(^◡𝐺‘𝑘)) = 𝑘)
75	56, 61, 74	syl2anc 596	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → (𝐺‘(^◡𝐺‘𝑘)) = 𝑘)
76	75	breq2d 5126	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → ((𝐺‘(♯‘𝐴)) < (𝐺‘(^◡𝐺‘𝑘)) ↔ (𝐺‘(♯‘𝐴)) < 𝑘))
77	73, 76	bitrd 282	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → ((♯‘𝐴) < (^◡𝐺‘𝑘) ↔ (𝐺‘(♯‘𝐴)) < 𝑘))
78	69, 77	mtbid 327	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘 ∈ 𝐴)) → ¬ (𝐺‘(♯‘𝐴)) < 𝑘)
79	78	expr 462	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → (𝑘 ∈ 𝐴 → ¬ (𝐺‘(♯‘𝐴)) < 𝑘))
80	55, 79	mt2d 137	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → ¬ 𝑘 ∈ 𝐴)
81	43, 80	eldifd 3919	. . . 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 603	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → (𝐹‘𝑘) = 𝑍)
84	1, 29, 31, 38, 83	seqid2 14104	. 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 3952	. . 3 ⊢ (𝜑 → 𝐴 ⊆ (ℤ_≥‘𝑀))
88	33	ssdifd 4102	. . . . 5 ⊢ (𝜑 → ((𝑀...(𝐺‘(♯‘𝐴))) ∖ 𝐴) ⊆ ((𝑀...𝑁) ∖ 𝐴))
89	88	sselda 3940	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀...(𝐺‘(♯‘𝐴))) ∖ 𝐴)) → 𝑘 ∈ ((𝑀...𝑁) ∖ 𝐴))
90	89, 82	syldan 603	. . 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 14521	. 2 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘(𝐺‘(♯‘𝐴))) = (seq1( + , 𝐻)‘(♯‘𝐴)))
93	84, 92	eqtr3d 2803	1 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq1( + , 𝐻)‘(♯‘𝐴)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 = wceq 1570 ∈ wcel 2146 ≠ wne 2961 ∖ cdif 3905 ⊆ wss 3908 ∅c0 4289 class class class wbr 5114 ^◡ccnv 5665 ⟶wf 6539 –1-1-onto→wf1o 6542 ‘cfv 6543 Isom wiso 6544 (class class class)co 7423 Fincfn 8952 ℝcr 11117 0cc0 11118 1c1 11119 + caddc 11121 < clt 11261 ≤ cle 11262 ℕcn 12251 ℕ₀cn0 12522 ℤ_≥cuz 12880 ...cfz 13553 seqcseq 14057 ♯chash 14386

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4878 df-int 4918 df-iun 4963 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7872 df-1st 7995 df-2nd 7996 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8703 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-card 9944 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-nn 12252 df-n0 12523 df-z 12610 df-uz 12881 df-fz 13554 df-seq 14058 df-hash 14387

This theorem is used by: isercolllem3 15744 gsumval3 20008

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