Theorem seq3fveq2 9953

Description: Equality of sequences. (Contributed by Jim Kingdon, 3-Jun-2020.)

Hypotheses

Ref	Expression
seq3fveq2.1	⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘𝑀))
seq3fveq2.2	⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = (𝐺‘𝐾))
seq3fveq2.f	⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
seq3fveq2.g	⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝐾)) → (𝐺‘𝑥) ∈ 𝑆)
seq3fveq2.pl	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
seq3fveq2.3	⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝐾))
seq3fveq2.4	⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → (𝐹‘𝑘) = (𝐺‘𝑘))

Assertion

Ref	Expression
seq3fveq2	⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq𝐾( + , 𝐺)‘𝑁))

Distinct variable groups:   𝑥,𝑘,𝑦,𝐹   𝑘,𝐺,𝑥,𝑦   𝑘,𝐾,𝑥,𝑦   𝑘,𝑁,𝑥,𝑦   𝜑,𝑘,𝑥,𝑦   𝑘,𝑀,𝑥,𝑦   + ,𝑘,𝑥,𝑦   𝑆,𝑘,𝑥,𝑦

Proof of Theorem seq3fveq2

Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		seq3fveq2.3	. . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝐾))
2		eluzfz2 9507	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → 𝑁 ∈ (𝐾...𝑁))
3	1, 2	syl 14	. 2 ⊢ (𝜑 → 𝑁 ∈ (𝐾...𝑁))
4		eleq1 2151	. . . . . 6 ⊢ (𝑧 = 𝐾 → (𝑧 ∈ (𝐾...𝑁) ↔ 𝐾 ∈ (𝐾...𝑁)))
5		fveq2 5318	. . . . . . 7 ⊢ (𝑧 = 𝐾 → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝑀( + , 𝐹)‘𝐾))
6		fveq2 5318	. . . . . . 7 ⊢ (𝑧 = 𝐾 → (seq𝐾( + , 𝐺)‘𝑧) = (seq𝐾( + , 𝐺)‘𝐾))
7	5, 6	eqeq12d 2103	. . . . . 6 ⊢ (𝑧 = 𝐾 → ((seq𝑀( + , 𝐹)‘𝑧) = (seq𝐾( + , 𝐺)‘𝑧) ↔ (seq𝑀( + , 𝐹)‘𝐾) = (seq𝐾( + , 𝐺)‘𝐾)))
8	4, 7	imbi12d 233	. . . . 5 ⊢ (𝑧 = 𝐾 → ((𝑧 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝐾( + , 𝐺)‘𝑧)) ↔ (𝐾 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝐾( + , 𝐺)‘𝐾))))
9	8	imbi2d 229	. . . 4 ⊢ (𝑧 = 𝐾 → ((𝜑 → (𝑧 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝐾( + , 𝐺)‘𝑧))) ↔ (𝜑 → (𝐾 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝐾( + , 𝐺)‘𝐾)))))
10		eleq1 2151	. . . . . 6 ⊢ (𝑧 = 𝑤 → (𝑧 ∈ (𝐾...𝑁) ↔ 𝑤 ∈ (𝐾...𝑁)))
11		fveq2 5318	. . . . . . 7 ⊢ (𝑧 = 𝑤 → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝑀( + , 𝐹)‘𝑤))
12		fveq2 5318	. . . . . . 7 ⊢ (𝑧 = 𝑤 → (seq𝐾( + , 𝐺)‘𝑧) = (seq𝐾( + , 𝐺)‘𝑤))
13	11, 12	eqeq12d 2103	. . . . . 6 ⊢ (𝑧 = 𝑤 → ((seq𝑀( + , 𝐹)‘𝑧) = (seq𝐾( + , 𝐺)‘𝑧) ↔ (seq𝑀( + , 𝐹)‘𝑤) = (seq𝐾( + , 𝐺)‘𝑤)))
14	10, 13	imbi12d 233	. . . . 5 ⊢ (𝑧 = 𝑤 → ((𝑧 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝐾( + , 𝐺)‘𝑧)) ↔ (𝑤 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑤) = (seq𝐾( + , 𝐺)‘𝑤))))
15	14	imbi2d 229	. . . 4 ⊢ (𝑧 = 𝑤 → ((𝜑 → (𝑧 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝐾( + , 𝐺)‘𝑧))) ↔ (𝜑 → (𝑤 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑤) = (seq𝐾( + , 𝐺)‘𝑤)))))
16		eleq1 2151	. . . . . 6 ⊢ (𝑧 = (𝑤 + 1) → (𝑧 ∈ (𝐾...𝑁) ↔ (𝑤 + 1) ∈ (𝐾...𝑁)))
17		fveq2 5318	. . . . . . 7 ⊢ (𝑧 = (𝑤 + 1) → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝑀( + , 𝐹)‘(𝑤 + 1)))
18		fveq2 5318	. . . . . . 7 ⊢ (𝑧 = (𝑤 + 1) → (seq𝐾( + , 𝐺)‘𝑧) = (seq𝐾( + , 𝐺)‘(𝑤 + 1)))
19	17, 18	eqeq12d 2103	. . . . . 6 ⊢ (𝑧 = (𝑤 + 1) → ((seq𝑀( + , 𝐹)‘𝑧) = (seq𝐾( + , 𝐺)‘𝑧) ↔ (seq𝑀( + , 𝐹)‘(𝑤 + 1)) = (seq𝐾( + , 𝐺)‘(𝑤 + 1))))
20	16, 19	imbi12d 233	. . . . 5 ⊢ (𝑧 = (𝑤 + 1) → ((𝑧 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝐾( + , 𝐺)‘𝑧)) ↔ ((𝑤 + 1) ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘(𝑤 + 1)) = (seq𝐾( + , 𝐺)‘(𝑤 + 1)))))
21	20	imbi2d 229	. . . 4 ⊢ (𝑧 = (𝑤 + 1) → ((𝜑 → (𝑧 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝐾( + , 𝐺)‘𝑧))) ↔ (𝜑 → ((𝑤 + 1) ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘(𝑤 + 1)) = (seq𝐾( + , 𝐺)‘(𝑤 + 1))))))
22		eleq1 2151	. . . . . 6 ⊢ (𝑧 = 𝑁 → (𝑧 ∈ (𝐾...𝑁) ↔ 𝑁 ∈ (𝐾...𝑁)))
23		fveq2 5318	. . . . . . 7 ⊢ (𝑧 = 𝑁 → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝑀( + , 𝐹)‘𝑁))
24		fveq2 5318	. . . . . . 7 ⊢ (𝑧 = 𝑁 → (seq𝐾( + , 𝐺)‘𝑧) = (seq𝐾( + , 𝐺)‘𝑁))
25	23, 24	eqeq12d 2103	. . . . . 6 ⊢ (𝑧 = 𝑁 → ((seq𝑀( + , 𝐹)‘𝑧) = (seq𝐾( + , 𝐺)‘𝑧) ↔ (seq𝑀( + , 𝐹)‘𝑁) = (seq𝐾( + , 𝐺)‘𝑁)))
26	22, 25	imbi12d 233	. . . . 5 ⊢ (𝑧 = 𝑁 → ((𝑧 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝐾( + , 𝐺)‘𝑧)) ↔ (𝑁 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑁) = (seq𝐾( + , 𝐺)‘𝑁))))
27	26	imbi2d 229	. . . 4 ⊢ (𝑧 = 𝑁 → ((𝜑 → (𝑧 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝐾( + , 𝐺)‘𝑧))) ↔ (𝜑 → (𝑁 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑁) = (seq𝐾( + , 𝐺)‘𝑁)))))
28		seq3fveq2.2	. . . . . 6 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = (𝐺‘𝐾))
29		seq3fveq2.1	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘𝑀))
30		eluzelz 9089	. . . . . . . 8 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → 𝐾 ∈ ℤ)
31	29, 30	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ ℤ)
32		seq3fveq2.g	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝐾)) → (𝐺‘𝑥) ∈ 𝑆)
33		seq3fveq2.pl	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
34	31, 32, 33	seq3-1 9938	. . . . . 6 ⊢ (𝜑 → (seq𝐾( + , 𝐺)‘𝐾) = (𝐺‘𝐾))
35	28, 34	eqtr4d 2124	. . . . 5 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝐾( + , 𝐺)‘𝐾))
36	35	a1i13 24	. . . 4 ⊢ (𝐾 ∈ ℤ → (𝜑 → (𝐾 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝐾( + , 𝐺)‘𝐾))))
37		peano2fzr 9512	. . . . . . . 8 ⊢ ((𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁)) → 𝑤 ∈ (𝐾...𝑁))
38	37	adantl 272	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → 𝑤 ∈ (𝐾...𝑁))
39	38	expr 368	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℤ≥‘𝐾)) → ((𝑤 + 1) ∈ (𝐾...𝑁) → 𝑤 ∈ (𝐾...𝑁)))
40	39	imim1d 75	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℤ≥‘𝐾)) → ((𝑤 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑤) = (seq𝐾( + , 𝐺)‘𝑤)) → ((𝑤 + 1) ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑤) = (seq𝐾( + , 𝐺)‘𝑤))))
41		oveq1 5673	. . . . . 6 ⊢ ((seq𝑀( + , 𝐹)‘𝑤) = (seq𝐾( + , 𝐺)‘𝑤) → ((seq𝑀( + , 𝐹)‘𝑤) + (𝐹‘(𝑤 + 1))) = ((seq𝐾( + , 𝐺)‘𝑤) + (𝐹‘(𝑤 + 1))))
42		simprl 499	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → 𝑤 ∈ (ℤ≥‘𝐾))
43	29	adantr 271	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → 𝐾 ∈ (ℤ≥‘𝑀))
44		uztrn 9096	. . . . . . . . 9 ⊢ ((𝑤 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑀)) → 𝑤 ∈ (ℤ≥‘𝑀))
45	42, 43, 44	syl2anc 404	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → 𝑤 ∈ (ℤ≥‘𝑀))
46		seq3fveq2.f	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
47	46	adantlr 462	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
48	33	adantlr 462	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
49	45, 47, 48	seq3p1 9945	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → (seq𝑀( + , 𝐹)‘(𝑤 + 1)) = ((seq𝑀( + , 𝐹)‘𝑤) + (𝐹‘(𝑤 + 1))))
50	32	adantlr 462	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) ∧ 𝑥 ∈ (ℤ≥‘𝐾)) → (𝐺‘𝑥) ∈ 𝑆)
51	42, 50, 48	seq3p1 9945	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → (seq𝐾( + , 𝐺)‘(𝑤 + 1)) = ((seq𝐾( + , 𝐺)‘𝑤) + (𝐺‘(𝑤 + 1))))
52		fveq2 5318	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑤 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑤 + 1)))
53		fveq2 5318	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑤 + 1) → (𝐺‘𝑘) = (𝐺‘(𝑤 + 1)))
54	52, 53	eqeq12d 2103	. . . . . . . . . 10 ⊢ (𝑘 = (𝑤 + 1) → ((𝐹‘𝑘) = (𝐺‘𝑘) ↔ (𝐹‘(𝑤 + 1)) = (𝐺‘(𝑤 + 1))))
55		seq3fveq2.4	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → (𝐹‘𝑘) = (𝐺‘𝑘))
56	55	ralrimiva 2447	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑘 ∈ ((𝐾 + 1)...𝑁)(𝐹‘𝑘) = (𝐺‘𝑘))
57	56	adantr 271	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → ∀𝑘 ∈ ((𝐾 + 1)...𝑁)(𝐹‘𝑘) = (𝐺‘𝑘))
58		eluzp1p1 9105	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ (ℤ≥‘𝐾) → (𝑤 + 1) ∈ (ℤ≥‘(𝐾 + 1)))
59	58	ad2antrl 475	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → (𝑤 + 1) ∈ (ℤ≥‘(𝐾 + 1)))
60		elfzuz3 9498	. . . . . . . . . . . 12 ⊢ ((𝑤 + 1) ∈ (𝐾...𝑁) → 𝑁 ∈ (ℤ≥‘(𝑤 + 1)))
61	60	ad2antll 476	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → 𝑁 ∈ (ℤ≥‘(𝑤 + 1)))
62		elfzuzb 9495	. . . . . . . . . . 11 ⊢ ((𝑤 + 1) ∈ ((𝐾 + 1)...𝑁) ↔ ((𝑤 + 1) ∈ (ℤ≥‘(𝐾 + 1)) ∧ 𝑁 ∈ (ℤ≥‘(𝑤 + 1))))
63	59, 61, 62	sylanbrc 409	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → (𝑤 + 1) ∈ ((𝐾 + 1)...𝑁))
64	54, 57, 63	rspcdva 2728	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → (𝐹‘(𝑤 + 1)) = (𝐺‘(𝑤 + 1)))
65	64	oveq2d 5682	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → ((seq𝐾( + , 𝐺)‘𝑤) + (𝐹‘(𝑤 + 1))) = ((seq𝐾( + , 𝐺)‘𝑤) + (𝐺‘(𝑤 + 1))))
66	51, 65	eqtr4d 2124	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → (seq𝐾( + , 𝐺)‘(𝑤 + 1)) = ((seq𝐾( + , 𝐺)‘𝑤) + (𝐹‘(𝑤 + 1))))
67	49, 66	eqeq12d 2103	. . . . . 6 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → ((seq𝑀( + , 𝐹)‘(𝑤 + 1)) = (seq𝐾( + , 𝐺)‘(𝑤 + 1)) ↔ ((seq𝑀( + , 𝐹)‘𝑤) + (𝐹‘(𝑤 + 1))) = ((seq𝐾( + , 𝐺)‘𝑤) + (𝐹‘(𝑤 + 1)))))
68	41, 67	syl5ibr 155	. . . . 5 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → ((seq𝑀( + , 𝐹)‘𝑤) = (seq𝐾( + , 𝐺)‘𝑤) → (seq𝑀( + , 𝐹)‘(𝑤 + 1)) = (seq𝐾( + , 𝐺)‘(𝑤 + 1))))
69	40, 68	animpimp2impd 527	. . . 4 ⊢ (𝑤 ∈ (ℤ≥‘𝐾) → ((𝜑 → (𝑤 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑤) = (seq𝐾( + , 𝐺)‘𝑤))) → (𝜑 → ((𝑤 + 1) ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘(𝑤 + 1)) = (seq𝐾( + , 𝐺)‘(𝑤 + 1))))))
70	9, 15, 21, 27, 36, 69	uzind4 9137	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → (𝜑 → (𝑁 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑁) = (seq𝐾( + , 𝐺)‘𝑁))))
71	1, 70	mpcom 36	. 2 ⊢ (𝜑 → (𝑁 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑁) = (seq𝐾( + , 𝐺)‘𝑁)))
72	3, 71	mpd 13	1 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq𝐾( + , 𝐺)‘𝑁))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1290   ∈ wcel 1439  ∀wral 2360  ‘cfv 5028  (class class class)co 5666  1c1 7412   + caddc 7414  ℤcz 8811  ℤ_≥cuz 9080  ...cfz 9485  seqcseq 9913

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416  ax-cnex 7497  ax-resscn 7498  ax-1cn 7499  ax-1re 7500  ax-icn 7501  ax-addcl 7502  ax-addrcl 7503  ax-mulcl 7504  ax-addcom 7506  ax-addass 7508  ax-distr 7510  ax-i2m1 7511  ax-0lt1 7512  ax-0id 7514  ax-rnegex 7515  ax-cnre 7517  ax-pre-ltirr 7518  ax-pre-ltwlin 7519  ax-pre-lttrn 7520  ax-pre-ltadd 7522

This theorem depends on definitions:  df-bi 116  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-id 4129  df-iord 4202  df-on 4204  df-ilim 4205  df-suc 4207  df-iom 4419  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-recs 6084  df-frec 6170  df-pnf 7585  df-mnf 7586  df-xr 7587  df-ltxr 7588  df-le 7589  df-sub 7716  df-neg 7717  df-inn 8484  df-n0 8735  df-z 8812  df-uz 9081  df-fz 9486  df-iseq 9914  df-seq3 9915

This theorem is referenced by:  seq3feq2  9954  seq3fveq  9956