Theorem seq3fveq2 10642

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 10174	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → 𝑁 ∈ (𝐾...𝑁))
3	1, 2	syl 14	. 2 ⊢ (𝜑 → 𝑁 ∈ (𝐾...𝑁))
4		eleq1 2269	. . . . . 6 ⊢ (𝑧 = 𝐾 → (𝑧 ∈ (𝐾...𝑁) ↔ 𝐾 ∈ (𝐾...𝑁)))
5		fveq2 5589	. . . . . . 7 ⊢ (𝑧 = 𝐾 → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝑀( + , 𝐹)‘𝐾))
6		fveq2 5589	. . . . . . 7 ⊢ (𝑧 = 𝐾 → (seq𝐾( + , 𝐺)‘𝑧) = (seq𝐾( + , 𝐺)‘𝐾))
7	5, 6	eqeq12d 2221	. . . . . 6 ⊢ (𝑧 = 𝐾 → ((seq𝑀( + , 𝐹)‘𝑧) = (seq𝐾( + , 𝐺)‘𝑧) ↔ (seq𝑀( + , 𝐹)‘𝐾) = (seq𝐾( + , 𝐺)‘𝐾)))
8	4, 7	imbi12d 234	. . . . 5 ⊢ (𝑧 = 𝐾 → ((𝑧 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝐾( + , 𝐺)‘𝑧)) ↔ (𝐾 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝐾( + , 𝐺)‘𝐾))))
9	8	imbi2d 230	. . . 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 2269	. . . . . 6 ⊢ (𝑧 = 𝑤 → (𝑧 ∈ (𝐾...𝑁) ↔ 𝑤 ∈ (𝐾...𝑁)))
11		fveq2 5589	. . . . . . 7 ⊢ (𝑧 = 𝑤 → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝑀( + , 𝐹)‘𝑤))
12		fveq2 5589	. . . . . . 7 ⊢ (𝑧 = 𝑤 → (seq𝐾( + , 𝐺)‘𝑧) = (seq𝐾( + , 𝐺)‘𝑤))
13	11, 12	eqeq12d 2221	. . . . . 6 ⊢ (𝑧 = 𝑤 → ((seq𝑀( + , 𝐹)‘𝑧) = (seq𝐾( + , 𝐺)‘𝑧) ↔ (seq𝑀( + , 𝐹)‘𝑤) = (seq𝐾( + , 𝐺)‘𝑤)))
14	10, 13	imbi12d 234	. . . . 5 ⊢ (𝑧 = 𝑤 → ((𝑧 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝐾( + , 𝐺)‘𝑧)) ↔ (𝑤 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑤) = (seq𝐾( + , 𝐺)‘𝑤))))
15	14	imbi2d 230	. . . 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 2269	. . . . . 6 ⊢ (𝑧 = (𝑤 + 1) → (𝑧 ∈ (𝐾...𝑁) ↔ (𝑤 + 1) ∈ (𝐾...𝑁)))
17		fveq2 5589	. . . . . . 7 ⊢ (𝑧 = (𝑤 + 1) → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝑀( + , 𝐹)‘(𝑤 + 1)))
18		fveq2 5589	. . . . . . 7 ⊢ (𝑧 = (𝑤 + 1) → (seq𝐾( + , 𝐺)‘𝑧) = (seq𝐾( + , 𝐺)‘(𝑤 + 1)))
19	17, 18	eqeq12d 2221	. . . . . 6 ⊢ (𝑧 = (𝑤 + 1) → ((seq𝑀( + , 𝐹)‘𝑧) = (seq𝐾( + , 𝐺)‘𝑧) ↔ (seq𝑀( + , 𝐹)‘(𝑤 + 1)) = (seq𝐾( + , 𝐺)‘(𝑤 + 1))))
20	16, 19	imbi12d 234	. . . . 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 230	. . . 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 2269	. . . . . 6 ⊢ (𝑧 = 𝑁 → (𝑧 ∈ (𝐾...𝑁) ↔ 𝑁 ∈ (𝐾...𝑁)))
23		fveq2 5589	. . . . . . 7 ⊢ (𝑧 = 𝑁 → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝑀( + , 𝐹)‘𝑁))
24		fveq2 5589	. . . . . . 7 ⊢ (𝑧 = 𝑁 → (seq𝐾( + , 𝐺)‘𝑧) = (seq𝐾( + , 𝐺)‘𝑁))
25	23, 24	eqeq12d 2221	. . . . . 6 ⊢ (𝑧 = 𝑁 → ((seq𝑀( + , 𝐹)‘𝑧) = (seq𝐾( + , 𝐺)‘𝑧) ↔ (seq𝑀( + , 𝐹)‘𝑁) = (seq𝐾( + , 𝐺)‘𝑁)))
26	22, 25	imbi12d 234	. . . . 5 ⊢ (𝑧 = 𝑁 → ((𝑧 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝐾( + , 𝐺)‘𝑧)) ↔ (𝑁 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑁) = (seq𝐾( + , 𝐺)‘𝑁))))
27	26	imbi2d 230	. . . 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 9677	. . . . . . . 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 10629	. . . . . 6 ⊢ (𝜑 → (seq𝐾( + , 𝐺)‘𝐾) = (𝐺‘𝐾))
35	28, 34	eqtr4d 2242	. . . . 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 10179	. . . . . . . 8 ⊢ ((𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁)) → 𝑤 ∈ (𝐾...𝑁))
38	37	adantl 277	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → 𝑤 ∈ (𝐾...𝑁))
39	38	expr 375	. . . . . 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 5964	. . . . . 6 ⊢ ((seq𝑀( + , 𝐹)‘𝑤) = (seq𝐾( + , 𝐺)‘𝑤) → ((seq𝑀( + , 𝐹)‘𝑤) + (𝐹‘(𝑤 + 1))) = ((seq𝐾( + , 𝐺)‘𝑤) + (𝐹‘(𝑤 + 1))))
42		simprl 529	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → 𝑤 ∈ (ℤ≥‘𝐾))
43	29	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → 𝐾 ∈ (ℤ≥‘𝑀))
44		uztrn 9685	. . . . . . . . 9 ⊢ ((𝑤 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑀)) → 𝑤 ∈ (ℤ≥‘𝑀))
45	42, 43, 44	syl2anc 411	. . . . . . . 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 477	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
48	33	adantlr 477	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
49	45, 47, 48	seq3p1 10632	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → (seq𝑀( + , 𝐹)‘(𝑤 + 1)) = ((seq𝑀( + , 𝐹)‘𝑤) + (𝐹‘(𝑤 + 1))))
50	32	adantlr 477	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) ∧ 𝑥 ∈ (ℤ≥‘𝐾)) → (𝐺‘𝑥) ∈ 𝑆)
51	42, 50, 48	seq3p1 10632	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → (seq𝐾( + , 𝐺)‘(𝑤 + 1)) = ((seq𝐾( + , 𝐺)‘𝑤) + (𝐺‘(𝑤 + 1))))
52		fveq2 5589	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑤 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑤 + 1)))
53		fveq2 5589	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑤 + 1) → (𝐺‘𝑘) = (𝐺‘(𝑤 + 1)))
54	52, 53	eqeq12d 2221	. . . . . . . . . 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 2580	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑘 ∈ ((𝐾 + 1)...𝑁)(𝐹‘𝑘) = (𝐺‘𝑘))
57	56	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → ∀𝑘 ∈ ((𝐾 + 1)...𝑁)(𝐹‘𝑘) = (𝐺‘𝑘))
58		eluzp1p1 9694	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ (ℤ≥‘𝐾) → (𝑤 + 1) ∈ (ℤ≥‘(𝐾 + 1)))
59	58	ad2antrl 490	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → (𝑤 + 1) ∈ (ℤ≥‘(𝐾 + 1)))
60		elfzuz3 10164	. . . . . . . . . . . 12 ⊢ ((𝑤 + 1) ∈ (𝐾...𝑁) → 𝑁 ∈ (ℤ≥‘(𝑤 + 1)))
61	60	ad2antll 491	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → 𝑁 ∈ (ℤ≥‘(𝑤 + 1)))
62		elfzuzb 10161	. . . . . . . . . . 11 ⊢ ((𝑤 + 1) ∈ ((𝐾 + 1)...𝑁) ↔ ((𝑤 + 1) ∈ (ℤ≥‘(𝐾 + 1)) ∧ 𝑁 ∈ (ℤ≥‘(𝑤 + 1))))
63	59, 61, 62	sylanbrc 417	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → (𝑤 + 1) ∈ ((𝐾 + 1)...𝑁))
64	54, 57, 63	rspcdva 2886	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → (𝐹‘(𝑤 + 1)) = (𝐺‘(𝑤 + 1)))
65	64	oveq2d 5973	. . . . . . . 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 2242	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → (seq𝐾( + , 𝐺)‘(𝑤 + 1)) = ((seq𝐾( + , 𝐺)‘𝑤) + (𝐹‘(𝑤 + 1))))
67	49, 66	eqeq12d 2221	. . . . . 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	imbitrrid 156	. . . . 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 559	. . . 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 9729	. . 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 104   = wceq 1373   ∈ wcel 2177  ∀wral 2485  ‘cfv 5280  (class class class)co 5957  1c1 7946   + caddc 7948  ℤcz 9392  ℤ_≥cuz 9668  ...cfz 10150  seqcseq 10614

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-iinf 4644  ax-cnex 8036  ax-resscn 8037  ax-1cn 8038  ax-1re 8039  ax-icn 8040  ax-addcl 8041  ax-addrcl 8042  ax-mulcl 8043  ax-addcom 8045  ax-addass 8047  ax-distr 8049  ax-i2m1 8050  ax-0lt1 8051  ax-0id 8053  ax-rnegex 8054  ax-cnre 8056  ax-pre-ltirr 8057  ax-pre-ltwlin 8058  ax-pre-lttrn 8059  ax-pre-ltadd 8061

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-tr 4151  df-id 4348  df-iord 4421  df-on 4423  df-ilim 4424  df-suc 4426  df-iom 4647  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-riota 5912  df-ov 5960  df-oprab 5961  df-mpo 5962  df-1st 6239  df-2nd 6240  df-recs 6404  df-frec 6490  df-pnf 8129  df-mnf 8130  df-xr 8131  df-ltxr 8132  df-le 8133  df-sub 8265  df-neg 8266  df-inn 9057  df-n0 9316  df-z 9393  df-uz 9669  df-fz 10151  df-seqfrec 10615

This theorem is referenced by:  seq3feq2  10643  seq3fveq  10646  gsumsplit1r  13305