Theorem iseqfveq2 9448

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

Hypotheses

Ref	Expression
iseqfveq2.1	⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘𝑀))
iseqfveq2.2	⊢ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (𝐺‘𝐾))
iseqfveq2.s	⊢ (𝜑 → 𝑆 ∈ 𝑉)
iseqfveq2.f	⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
iseqfveq2.g	⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝐾)) → (𝐺‘𝑥) ∈ 𝑆)
iseqfveq2.pl	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
iseqfveq2.3	⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝐾))
iseqfveq2.4	⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → (𝐹‘𝑘) = (𝐺‘𝑘))

Assertion

Ref	Expression
iseqfveq2	⊢ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq𝐾( + , 𝐺, 𝑆)‘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

Allowed substitution hints:   𝑉(𝑥,𝑦,𝑘)

Proof of Theorem iseqfveq2

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

Step	Hyp	Ref	Expression
1		iseqfveq2.3	. . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝐾))
2		eluzfz2 9051	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → 𝑁 ∈ (𝐾...𝑁))
3	1, 2	syl 14	. 2 ⊢ (𝜑 → 𝑁 ∈ (𝐾...𝑁))
4		eleq1 2141	. . . . . 6 ⊢ (𝑧 = 𝐾 → (𝑧 ∈ (𝐾...𝑁) ↔ 𝐾 ∈ (𝐾...𝑁)))
5		fveq2 5198	. . . . . . 7 ⊢ (𝑧 = 𝐾 → (seq𝑀( + , 𝐹, 𝑆)‘𝑧) = (seq𝑀( + , 𝐹, 𝑆)‘𝐾))
6		fveq2 5198	. . . . . . 7 ⊢ (𝑧 = 𝐾 → (seq𝐾( + , 𝐺, 𝑆)‘𝑧) = (seq𝐾( + , 𝐺, 𝑆)‘𝐾))
7	5, 6	eqeq12d 2095	. . . . . 6 ⊢ (𝑧 = 𝐾 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑧) = (seq𝐾( + , 𝐺, 𝑆)‘𝑧) ↔ (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝐾( + , 𝐺, 𝑆)‘𝐾)))
8	4, 7	imbi12d 232	. . . . 5 ⊢ (𝑧 = 𝐾 → ((𝑧 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑧) = (seq𝐾( + , 𝐺, 𝑆)‘𝑧)) ↔ (𝐾 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝐾( + , 𝐺, 𝑆)‘𝐾))))
9	8	imbi2d 228	. . . 4 ⊢ (𝑧 = 𝐾 → ((𝜑 → (𝑧 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑧) = (seq𝐾( + , 𝐺, 𝑆)‘𝑧))) ↔ (𝜑 → (𝐾 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝐾( + , 𝐺, 𝑆)‘𝐾)))))
10		eleq1 2141	. . . . . 6 ⊢ (𝑧 = 𝑤 → (𝑧 ∈ (𝐾...𝑁) ↔ 𝑤 ∈ (𝐾...𝑁)))
11		fveq2 5198	. . . . . . 7 ⊢ (𝑧 = 𝑤 → (seq𝑀( + , 𝐹, 𝑆)‘𝑧) = (seq𝑀( + , 𝐹, 𝑆)‘𝑤))
12		fveq2 5198	. . . . . . 7 ⊢ (𝑧 = 𝑤 → (seq𝐾( + , 𝐺, 𝑆)‘𝑧) = (seq𝐾( + , 𝐺, 𝑆)‘𝑤))
13	11, 12	eqeq12d 2095	. . . . . 6 ⊢ (𝑧 = 𝑤 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑧) = (seq𝐾( + , 𝐺, 𝑆)‘𝑧) ↔ (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝐾( + , 𝐺, 𝑆)‘𝑤)))
14	10, 13	imbi12d 232	. . . . 5 ⊢ (𝑧 = 𝑤 → ((𝑧 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑧) = (seq𝐾( + , 𝐺, 𝑆)‘𝑧)) ↔ (𝑤 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝐾( + , 𝐺, 𝑆)‘𝑤))))
15	14	imbi2d 228	. . . 4 ⊢ (𝑧 = 𝑤 → ((𝜑 → (𝑧 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑧) = (seq𝐾( + , 𝐺, 𝑆)‘𝑧))) ↔ (𝜑 → (𝑤 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝐾( + , 𝐺, 𝑆)‘𝑤)))))
16		eleq1 2141	. . . . . 6 ⊢ (𝑧 = (𝑤 + 1) → (𝑧 ∈ (𝐾...𝑁) ↔ (𝑤 + 1) ∈ (𝐾...𝑁)))
17		fveq2 5198	. . . . . . 7 ⊢ (𝑧 = (𝑤 + 1) → (seq𝑀( + , 𝐹, 𝑆)‘𝑧) = (seq𝑀( + , 𝐹, 𝑆)‘(𝑤 + 1)))
18		fveq2 5198	. . . . . . 7 ⊢ (𝑧 = (𝑤 + 1) → (seq𝐾( + , 𝐺, 𝑆)‘𝑧) = (seq𝐾( + , 𝐺, 𝑆)‘(𝑤 + 1)))
19	17, 18	eqeq12d 2095	. . . . . 6 ⊢ (𝑧 = (𝑤 + 1) → ((seq𝑀( + , 𝐹, 𝑆)‘𝑧) = (seq𝐾( + , 𝐺, 𝑆)‘𝑧) ↔ (seq𝑀( + , 𝐹, 𝑆)‘(𝑤 + 1)) = (seq𝐾( + , 𝐺, 𝑆)‘(𝑤 + 1))))
20	16, 19	imbi12d 232	. . . . 5 ⊢ (𝑧 = (𝑤 + 1) → ((𝑧 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑧) = (seq𝐾( + , 𝐺, 𝑆)‘𝑧)) ↔ ((𝑤 + 1) ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑤 + 1)) = (seq𝐾( + , 𝐺, 𝑆)‘(𝑤 + 1)))))
21	20	imbi2d 228	. . . 4 ⊢ (𝑧 = (𝑤 + 1) → ((𝜑 → (𝑧 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑧) = (seq𝐾( + , 𝐺, 𝑆)‘𝑧))) ↔ (𝜑 → ((𝑤 + 1) ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑤 + 1)) = (seq𝐾( + , 𝐺, 𝑆)‘(𝑤 + 1))))))
22		eleq1 2141	. . . . . 6 ⊢ (𝑧 = 𝑁 → (𝑧 ∈ (𝐾...𝑁) ↔ 𝑁 ∈ (𝐾...𝑁)))
23		fveq2 5198	. . . . . . 7 ⊢ (𝑧 = 𝑁 → (seq𝑀( + , 𝐹, 𝑆)‘𝑧) = (seq𝑀( + , 𝐹, 𝑆)‘𝑁))
24		fveq2 5198	. . . . . . 7 ⊢ (𝑧 = 𝑁 → (seq𝐾( + , 𝐺, 𝑆)‘𝑧) = (seq𝐾( + , 𝐺, 𝑆)‘𝑁))
25	23, 24	eqeq12d 2095	. . . . . 6 ⊢ (𝑧 = 𝑁 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑧) = (seq𝐾( + , 𝐺, 𝑆)‘𝑧) ↔ (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq𝐾( + , 𝐺, 𝑆)‘𝑁)))
26	22, 25	imbi12d 232	. . . . 5 ⊢ (𝑧 = 𝑁 → ((𝑧 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑧) = (seq𝐾( + , 𝐺, 𝑆)‘𝑧)) ↔ (𝑁 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq𝐾( + , 𝐺, 𝑆)‘𝑁))))
27	26	imbi2d 228	. . . 4 ⊢ (𝑧 = 𝑁 → ((𝜑 → (𝑧 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑧) = (seq𝐾( + , 𝐺, 𝑆)‘𝑧))) ↔ (𝜑 → (𝑁 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq𝐾( + , 𝐺, 𝑆)‘𝑁)))))
28		iseqfveq2.2	. . . . . . 7 ⊢ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (𝐺‘𝐾))
29		iseqfveq2.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘𝑀))
30		eluzelz 8628	. . . . . . . . 9 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → 𝐾 ∈ ℤ)
31	29, 30	syl 14	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ ℤ)
32		iseqfveq2.s	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
33		iseqfveq2.g	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝐾)) → (𝐺‘𝑥) ∈ 𝑆)
34		iseqfveq2.pl	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
35	31, 32, 33, 34	iseq1 9442	. . . . . . 7 ⊢ (𝜑 → (seq𝐾( + , 𝐺, 𝑆)‘𝐾) = (𝐺‘𝐾))
36	28, 35	eqtr4d 2116	. . . . . 6 ⊢ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝐾( + , 𝐺, 𝑆)‘𝐾))
37	36	a1d 22	. . . . 5 ⊢ (𝜑 → (𝐾 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝐾( + , 𝐺, 𝑆)‘𝐾)))
38	37	a1i 9	. . . 4 ⊢ (𝐾 ∈ ℤ → (𝜑 → (𝐾 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝐾( + , 𝐺, 𝑆)‘𝐾))))
39		peano2fzr 9056	. . . . . . . . . 10 ⊢ ((𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁)) → 𝑤 ∈ (𝐾...𝑁))
40	39	adantl 271	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → 𝑤 ∈ (𝐾...𝑁))
41	40	expr 367	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℤ≥‘𝐾)) → ((𝑤 + 1) ∈ (𝐾...𝑁) → 𝑤 ∈ (𝐾...𝑁)))
42	41	imim1d 74	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℤ≥‘𝐾)) → ((𝑤 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝐾( + , 𝐺, 𝑆)‘𝑤)) → ((𝑤 + 1) ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝐾( + , 𝐺, 𝑆)‘𝑤))))
43		oveq1 5539	. . . . . . . . . 10 ⊢ ((seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝐾( + , 𝐺, 𝑆)‘𝑤) → ((seq𝑀( + , 𝐹, 𝑆)‘𝑤) + (𝐹‘(𝑤 + 1))) = ((seq𝐾( + , 𝐺, 𝑆)‘𝑤) + (𝐹‘(𝑤 + 1))))
44		simprl 497	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → 𝑤 ∈ (ℤ≥‘𝐾))
45	29	adantr 270	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → 𝐾 ∈ (ℤ≥‘𝑀))
46		uztrn 8635	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑀)) → 𝑤 ∈ (ℤ≥‘𝑀))
47	44, 45, 46	syl2anc 403	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → 𝑤 ∈ (ℤ≥‘𝑀))
48	32	adantr 270	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → 𝑆 ∈ 𝑉)
49		iseqfveq2.f	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
50	49	adantlr 460	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
51	34	adantlr 460	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
52	47, 48, 50, 51	iseqp1 9445	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑤 + 1)) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑤) + (𝐹‘(𝑤 + 1))))
53	33	adantlr 460	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) ∧ 𝑥 ∈ (ℤ≥‘𝐾)) → (𝐺‘𝑥) ∈ 𝑆)
54	44, 48, 53, 51	iseqp1 9445	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → (seq𝐾( + , 𝐺, 𝑆)‘(𝑤 + 1)) = ((seq𝐾( + , 𝐺, 𝑆)‘𝑤) + (𝐺‘(𝑤 + 1))))
55		eluzp1p1 8644	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ (ℤ≥‘𝐾) → (𝑤 + 1) ∈ (ℤ≥‘(𝐾 + 1)))
56	55	ad2antrl 473	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → (𝑤 + 1) ∈ (ℤ≥‘(𝐾 + 1)))
57		elfzuz3 9042	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 + 1) ∈ (𝐾...𝑁) → 𝑁 ∈ (ℤ≥‘(𝑤 + 1)))
58	57	ad2antll 474	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → 𝑁 ∈ (ℤ≥‘(𝑤 + 1)))
59		elfzuzb 9039	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 + 1) ∈ ((𝐾 + 1)...𝑁) ↔ ((𝑤 + 1) ∈ (ℤ≥‘(𝐾 + 1)) ∧ 𝑁 ∈ (ℤ≥‘(𝑤 + 1))))
60	56, 58, 59	sylanbrc 408	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → (𝑤 + 1) ∈ ((𝐾 + 1)...𝑁))
61		iseqfveq2.4	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → (𝐹‘𝑘) = (𝐺‘𝑘))
62	61	ralrimiva 2434	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑘 ∈ ((𝐾 + 1)...𝑁)(𝐹‘𝑘) = (𝐺‘𝑘))
63	62	adantr 270	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → ∀𝑘 ∈ ((𝐾 + 1)...𝑁)(𝐹‘𝑘) = (𝐺‘𝑘))
64		fveq2 5198	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = (𝑤 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑤 + 1)))
65		fveq2 5198	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = (𝑤 + 1) → (𝐺‘𝑘) = (𝐺‘(𝑤 + 1)))
66	64, 65	eqeq12d 2095	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = (𝑤 + 1) → ((𝐹‘𝑘) = (𝐺‘𝑘) ↔ (𝐹‘(𝑤 + 1)) = (𝐺‘(𝑤 + 1))))
67	66	rspcv 2697	. . . . . . . . . . . . . 14 ⊢ ((𝑤 + 1) ∈ ((𝐾 + 1)...𝑁) → (∀𝑘 ∈ ((𝐾 + 1)...𝑁)(𝐹‘𝑘) = (𝐺‘𝑘) → (𝐹‘(𝑤 + 1)) = (𝐺‘(𝑤 + 1))))
68	60, 63, 67	sylc 61	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → (𝐹‘(𝑤 + 1)) = (𝐺‘(𝑤 + 1)))
69	68	oveq2d 5548	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → ((seq𝐾( + , 𝐺, 𝑆)‘𝑤) + (𝐹‘(𝑤 + 1))) = ((seq𝐾( + , 𝐺, 𝑆)‘𝑤) + (𝐺‘(𝑤 + 1))))
70	54, 69	eqtr4d 2116	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → (seq𝐾( + , 𝐺, 𝑆)‘(𝑤 + 1)) = ((seq𝐾( + , 𝐺, 𝑆)‘𝑤) + (𝐹‘(𝑤 + 1))))
71	52, 70	eqeq12d 2095	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → ((seq𝑀( + , 𝐹, 𝑆)‘(𝑤 + 1)) = (seq𝐾( + , 𝐺, 𝑆)‘(𝑤 + 1)) ↔ ((seq𝑀( + , 𝐹, 𝑆)‘𝑤) + (𝐹‘(𝑤 + 1))) = ((seq𝐾( + , 𝐺, 𝑆)‘𝑤) + (𝐹‘(𝑤 + 1)))))
72	43, 71	syl5ibr 154	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → ((seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝐾( + , 𝐺, 𝑆)‘𝑤) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑤 + 1)) = (seq𝐾( + , 𝐺, 𝑆)‘(𝑤 + 1))))
73	72	expr 367	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℤ≥‘𝐾)) → ((𝑤 + 1) ∈ (𝐾...𝑁) → ((seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝐾( + , 𝐺, 𝑆)‘𝑤) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑤 + 1)) = (seq𝐾( + , 𝐺, 𝑆)‘(𝑤 + 1)))))
74	73	a2d 26	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℤ≥‘𝐾)) → (((𝑤 + 1) ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝐾( + , 𝐺, 𝑆)‘𝑤)) → ((𝑤 + 1) ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑤 + 1)) = (seq𝐾( + , 𝐺, 𝑆)‘(𝑤 + 1)))))
75	42, 74	syld 44	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℤ≥‘𝐾)) → ((𝑤 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝐾( + , 𝐺, 𝑆)‘𝑤)) → ((𝑤 + 1) ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑤 + 1)) = (seq𝐾( + , 𝐺, 𝑆)‘(𝑤 + 1)))))
76	75	expcom 114	. . . . 5 ⊢ (𝑤 ∈ (ℤ≥‘𝐾) → (𝜑 → ((𝑤 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝐾( + , 𝐺, 𝑆)‘𝑤)) → ((𝑤 + 1) ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑤 + 1)) = (seq𝐾( + , 𝐺, 𝑆)‘(𝑤 + 1))))))
77	76	a2d 26	. . . 4 ⊢ (𝑤 ∈ (ℤ≥‘𝐾) → ((𝜑 → (𝑤 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝐾( + , 𝐺, 𝑆)‘𝑤))) → (𝜑 → ((𝑤 + 1) ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑤 + 1)) = (seq𝐾( + , 𝐺, 𝑆)‘(𝑤 + 1))))))
78	9, 15, 21, 27, 38, 77	uzind4 8676	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → (𝜑 → (𝑁 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq𝐾( + , 𝐺, 𝑆)‘𝑁))))
79	1, 78	mpcom 36	. 2 ⊢ (𝜑 → (𝑁 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq𝐾( + , 𝐺, 𝑆)‘𝑁)))
80	3, 79	mpd 13	1 ⊢ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq𝐾( + , 𝐺, 𝑆)‘𝑁))

Syntax hints:   → wi 4   ∧ wa 102   = wceq 1284   ∈ wcel 1433  ∀wral 2348  ‘cfv 4922  (class class class)co 5532  1c1 6982   + caddc 6984  ℤcz 8351  ℤ_≥cuz 8619  ...cfz 9029  seqcseq 9431

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-addcom 7076  ax-addass 7078  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-0id 7084  ax-rnegex 7085  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-ltadd 7092

This theorem depends on definitions:  df-bi 115  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-frec 6001  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-inn 8040  df-n0 8289  df-z 8352  df-uz 8620  df-fz 9030  df-iseq 9432

This theorem is referenced by:  iseqfeq2  9449  iseqfveq  9450