Theorem seq3fveq2 10425

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 9988	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → 𝑁 ∈ (𝐾...𝑁))
3	1, 2	syl 14	. 2 ⊢ (𝜑 → 𝑁 ∈ (𝐾...𝑁))
4		eleq1 2233	. . . . . 6 ⊢ (𝑧 = 𝐾 → (𝑧 ∈ (𝐾...𝑁) ↔ 𝐾 ∈ (𝐾...𝑁)))
5		fveq2 5496	. . . . . . 7 ⊢ (𝑧 = 𝐾 → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝑀( + , 𝐹)‘𝐾))
6		fveq2 5496	. . . . . . 7 ⊢ (𝑧 = 𝐾 → (seq𝐾( + , 𝐺)‘𝑧) = (seq𝐾( + , 𝐺)‘𝐾))
7	5, 6	eqeq12d 2185	. . . . . 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 2233	. . . . . 6 ⊢ (𝑧 = 𝑤 → (𝑧 ∈ (𝐾...𝑁) ↔ 𝑤 ∈ (𝐾...𝑁)))
11		fveq2 5496	. . . . . . 7 ⊢ (𝑧 = 𝑤 → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝑀( + , 𝐹)‘𝑤))
12		fveq2 5496	. . . . . . 7 ⊢ (𝑧 = 𝑤 → (seq𝐾( + , 𝐺)‘𝑧) = (seq𝐾( + , 𝐺)‘𝑤))
13	11, 12	eqeq12d 2185	. . . . . 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 2233	. . . . . 6 ⊢ (𝑧 = (𝑤 + 1) → (𝑧 ∈ (𝐾...𝑁) ↔ (𝑤 + 1) ∈ (𝐾...𝑁)))
17		fveq2 5496	. . . . . . 7 ⊢ (𝑧 = (𝑤 + 1) → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝑀( + , 𝐹)‘(𝑤 + 1)))
18		fveq2 5496	. . . . . . 7 ⊢ (𝑧 = (𝑤 + 1) → (seq𝐾( + , 𝐺)‘𝑧) = (seq𝐾( + , 𝐺)‘(𝑤 + 1)))
19	17, 18	eqeq12d 2185	. . . . . 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 2233	. . . . . 6 ⊢ (𝑧 = 𝑁 → (𝑧 ∈ (𝐾...𝑁) ↔ 𝑁 ∈ (𝐾...𝑁)))
23		fveq2 5496	. . . . . . 7 ⊢ (𝑧 = 𝑁 → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝑀( + , 𝐹)‘𝑁))
24		fveq2 5496	. . . . . . 7 ⊢ (𝑧 = 𝑁 → (seq𝐾( + , 𝐺)‘𝑧) = (seq𝐾( + , 𝐺)‘𝑁))
25	23, 24	eqeq12d 2185	. . . . . 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 9496	. . . . . . . 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 10416	. . . . . 6 ⊢ (𝜑 → (seq𝐾( + , 𝐺)‘𝐾) = (𝐺‘𝐾))
35	28, 34	eqtr4d 2206	. . . . 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 9993	. . . . . . . 8 ⊢ ((𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁)) → 𝑤 ∈ (𝐾...𝑁))
38	37	adantl 275	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → 𝑤 ∈ (𝐾...𝑁))
39	38	expr 373	. . . . . 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 5860	. . . . . 6 ⊢ ((seq𝑀( + , 𝐹)‘𝑤) = (seq𝐾( + , 𝐺)‘𝑤) → ((seq𝑀( + , 𝐹)‘𝑤) + (𝐹‘(𝑤 + 1))) = ((seq𝐾( + , 𝐺)‘𝑤) + (𝐹‘(𝑤 + 1))))
42		simprl 526	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → 𝑤 ∈ (ℤ≥‘𝐾))
43	29	adantr 274	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → 𝐾 ∈ (ℤ≥‘𝑀))
44		uztrn 9503	. . . . . . . . 9 ⊢ ((𝑤 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑀)) → 𝑤 ∈ (ℤ≥‘𝑀))
45	42, 43, 44	syl2anc 409	. . . . . . . 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 474	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
48	33	adantlr 474	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
49	45, 47, 48	seq3p1 10418	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → (seq𝑀( + , 𝐹)‘(𝑤 + 1)) = ((seq𝑀( + , 𝐹)‘𝑤) + (𝐹‘(𝑤 + 1))))
50	32	adantlr 474	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) ∧ 𝑥 ∈ (ℤ≥‘𝐾)) → (𝐺‘𝑥) ∈ 𝑆)
51	42, 50, 48	seq3p1 10418	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → (seq𝐾( + , 𝐺)‘(𝑤 + 1)) = ((seq𝐾( + , 𝐺)‘𝑤) + (𝐺‘(𝑤 + 1))))
52		fveq2 5496	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑤 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑤 + 1)))
53		fveq2 5496	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑤 + 1) → (𝐺‘𝑘) = (𝐺‘(𝑤 + 1)))
54	52, 53	eqeq12d 2185	. . . . . . . . . 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 2543	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑘 ∈ ((𝐾 + 1)...𝑁)(𝐹‘𝑘) = (𝐺‘𝑘))
57	56	adantr 274	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → ∀𝑘 ∈ ((𝐾 + 1)...𝑁)(𝐹‘𝑘) = (𝐺‘𝑘))
58		eluzp1p1 9512	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ (ℤ≥‘𝐾) → (𝑤 + 1) ∈ (ℤ≥‘(𝐾 + 1)))
59	58	ad2antrl 487	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → (𝑤 + 1) ∈ (ℤ≥‘(𝐾 + 1)))
60		elfzuz3 9978	. . . . . . . . . . . 12 ⊢ ((𝑤 + 1) ∈ (𝐾...𝑁) → 𝑁 ∈ (ℤ≥‘(𝑤 + 1)))
61	60	ad2antll 488	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → 𝑁 ∈ (ℤ≥‘(𝑤 + 1)))
62		elfzuzb 9975	. . . . . . . . . . 11 ⊢ ((𝑤 + 1) ∈ ((𝐾 + 1)...𝑁) ↔ ((𝑤 + 1) ∈ (ℤ≥‘(𝐾 + 1)) ∧ 𝑁 ∈ (ℤ≥‘(𝑤 + 1))))
63	59, 61, 62	sylanbrc 415	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → (𝑤 + 1) ∈ ((𝐾 + 1)...𝑁))
64	54, 57, 63	rspcdva 2839	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → (𝐹‘(𝑤 + 1)) = (𝐺‘(𝑤 + 1)))
65	64	oveq2d 5869	. . . . . . . 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 2206	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → (seq𝐾( + , 𝐺)‘(𝑤 + 1)) = ((seq𝐾( + , 𝐺)‘𝑤) + (𝐹‘(𝑤 + 1))))
67	49, 66	eqeq12d 2185	. . . . . 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 554	. . . 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 9547	. . 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 1348   ∈ wcel 2141  ∀wral 2448  ‘cfv 5198  (class class class)co 5853  1c1 7775   + caddc 7777  ℤcz 9212  ℤ_≥cuz 9487  ...cfz 9965  seqcseq 10401

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-ltadd 7890

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-inn 8879  df-n0 9136  df-z 9213  df-uz 9488  df-fz 9966  df-seqfrec 10402

This theorem is referenced by:  seq3feq2  10426  seq3fveq  10427