Theorem iseqfveq2 9228

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 8896	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → 𝑁 ∈ (𝐾...𝑁))
3	1, 2	syl 14	. 2 ⊢ (𝜑 → 𝑁 ∈ (𝐾...𝑁))
4		eleq1 2100	. . . . . 6 ⊢ (𝑧 = 𝐾 → (𝑧 ∈ (𝐾...𝑁) ↔ 𝐾 ∈ (𝐾...𝑁)))
5		fveq2 5178	. . . . . . 7 ⊢ (𝑧 = 𝐾 → (seq𝑀( + , 𝐹, 𝑆)‘𝑧) = (seq𝑀( + , 𝐹, 𝑆)‘𝐾))
6		fveq2 5178	. . . . . . 7 ⊢ (𝑧 = 𝐾 → (seq𝐾( + , 𝐺, 𝑆)‘𝑧) = (seq𝐾( + , 𝐺, 𝑆)‘𝐾))
7	5, 6	eqeq12d 2054	. . . . . 6 ⊢ (𝑧 = 𝐾 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑧) = (seq𝐾( + , 𝐺, 𝑆)‘𝑧) ↔ (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝐾( + , 𝐺, 𝑆)‘𝐾)))
8	4, 7	imbi12d 223	. . . . 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 219	. . . 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 2100	. . . . . 6 ⊢ (𝑧 = 𝑤 → (𝑧 ∈ (𝐾...𝑁) ↔ 𝑤 ∈ (𝐾...𝑁)))
11		fveq2 5178	. . . . . . 7 ⊢ (𝑧 = 𝑤 → (seq𝑀( + , 𝐹, 𝑆)‘𝑧) = (seq𝑀( + , 𝐹, 𝑆)‘𝑤))
12		fveq2 5178	. . . . . . 7 ⊢ (𝑧 = 𝑤 → (seq𝐾( + , 𝐺, 𝑆)‘𝑧) = (seq𝐾( + , 𝐺, 𝑆)‘𝑤))
13	11, 12	eqeq12d 2054	. . . . . 6 ⊢ (𝑧 = 𝑤 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑧) = (seq𝐾( + , 𝐺, 𝑆)‘𝑧) ↔ (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝐾( + , 𝐺, 𝑆)‘𝑤)))
14	10, 13	imbi12d 223	. . . . 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 219	. . . 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 2100	. . . . . 6 ⊢ (𝑧 = (𝑤 + 1) → (𝑧 ∈ (𝐾...𝑁) ↔ (𝑤 + 1) ∈ (𝐾...𝑁)))
17		fveq2 5178	. . . . . . 7 ⊢ (𝑧 = (𝑤 + 1) → (seq𝑀( + , 𝐹, 𝑆)‘𝑧) = (seq𝑀( + , 𝐹, 𝑆)‘(𝑤 + 1)))
18		fveq2 5178	. . . . . . 7 ⊢ (𝑧 = (𝑤 + 1) → (seq𝐾( + , 𝐺, 𝑆)‘𝑧) = (seq𝐾( + , 𝐺, 𝑆)‘(𝑤 + 1)))
19	17, 18	eqeq12d 2054	. . . . . 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 223	. . . . 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 219	. . . 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 2100	. . . . . 6 ⊢ (𝑧 = 𝑁 → (𝑧 ∈ (𝐾...𝑁) ↔ 𝑁 ∈ (𝐾...𝑁)))
23		fveq2 5178	. . . . . . 7 ⊢ (𝑧 = 𝑁 → (seq𝑀( + , 𝐹, 𝑆)‘𝑧) = (seq𝑀( + , 𝐹, 𝑆)‘𝑁))
24		fveq2 5178	. . . . . . 7 ⊢ (𝑧 = 𝑁 → (seq𝐾( + , 𝐺, 𝑆)‘𝑧) = (seq𝐾( + , 𝐺, 𝑆)‘𝑁))
25	23, 24	eqeq12d 2054	. . . . . 6 ⊢ (𝑧 = 𝑁 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑧) = (seq𝐾( + , 𝐺, 𝑆)‘𝑧) ↔ (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq𝐾( + , 𝐺, 𝑆)‘𝑁)))
26	22, 25	imbi12d 223	. . . . 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 219	. . . 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 8482	. . . . . . . . 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 9222	. . . . . . 7 ⊢ (𝜑 → (seq𝐾( + , 𝐺, 𝑆)‘𝐾) = (𝐺‘𝐾))
36	28, 35	eqtr4d 2075	. . . . . 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 8901	. . . . . . . . . 10 ⊢ ((𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁)) → 𝑤 ∈ (𝐾...𝑁))
40	39	adantl 262	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → 𝑤 ∈ (𝐾...𝑁))
41	40	expr 357	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℤ≥‘𝐾)) → ((𝑤 + 1) ∈ (𝐾...𝑁) → 𝑤 ∈ (𝐾...𝑁)))
42	41	imim1d 69	. . . . . . 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 5519	. . . . . . . . . 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 483	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → 𝑤 ∈ (ℤ≥‘𝐾))
45	29	adantr 261	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → 𝐾 ∈ (ℤ≥‘𝑀))
46		uztrn 8489	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑀)) → 𝑤 ∈ (ℤ≥‘𝑀))
47	44, 45, 46	syl2anc 391	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → 𝑤 ∈ (ℤ≥‘𝑀))
48	32	adantr 261	. . . . . . . . . . . 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 446	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
51	34	adantlr 446	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
52	47, 48, 50, 51	iseqp1 9225	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑤 + 1)) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑤) + (𝐹‘(𝑤 + 1))))
53	33	adantlr 446	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) ∧ 𝑥 ∈ (ℤ≥‘𝐾)) → (𝐺‘𝑥) ∈ 𝑆)
54	44, 48, 53, 51	iseqp1 9225	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → (seq𝐾( + , 𝐺, 𝑆)‘(𝑤 + 1)) = ((seq𝐾( + , 𝐺, 𝑆)‘𝑤) + (𝐺‘(𝑤 + 1))))
55		eluzp1p1 8498	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ (ℤ≥‘𝐾) → (𝑤 + 1) ∈ (ℤ≥‘(𝐾 + 1)))
56	55	ad2antrl 459	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → (𝑤 + 1) ∈ (ℤ≥‘(𝐾 + 1)))
57		elfzuz3 8887	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 + 1) ∈ (𝐾...𝑁) → 𝑁 ∈ (ℤ≥‘(𝑤 + 1)))
58	57	ad2antll 460	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → 𝑁 ∈ (ℤ≥‘(𝑤 + 1)))
59		elfzuzb 8884	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 + 1) ∈ ((𝐾 + 1)...𝑁) ↔ ((𝑤 + 1) ∈ (ℤ≥‘(𝐾 + 1)) ∧ 𝑁 ∈ (ℤ≥‘(𝑤 + 1))))
60	56, 58, 59	sylanbrc 394	. . . . . . . . . . . . . 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 2392	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑘 ∈ ((𝐾 + 1)...𝑁)(𝐹‘𝑘) = (𝐺‘𝑘))
63	62	adantr 261	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → ∀𝑘 ∈ ((𝐾 + 1)...𝑁)(𝐹‘𝑘) = (𝐺‘𝑘))
64		fveq2 5178	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = (𝑤 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑤 + 1)))
65		fveq2 5178	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = (𝑤 + 1) → (𝐺‘𝑘) = (𝐺‘(𝑤 + 1)))
66	64, 65	eqeq12d 2054	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = (𝑤 + 1) → ((𝐹‘𝑘) = (𝐺‘𝑘) ↔ (𝐹‘(𝑤 + 1)) = (𝐺‘(𝑤 + 1))))
67	66	rspcv 2652	. . . . . . . . . . . . . 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 56	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑤 ∈ (ℤ≥‘𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → (𝐹‘(𝑤 + 1)) = (𝐺‘(𝑤 + 1)))
69	68	oveq2d 5528	. . . . . . . . . . . 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 2075	. . . . . . . . . . 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 2054	. . . . . . . . . 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 145	. . . . . . . . 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 357	. . . . . . . 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 23	. . . . . . 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 40	. . . . . 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 109	. . . . 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 23	. . . 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 8531	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → (𝜑 → (𝑁 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq𝐾( + , 𝐺, 𝑆)‘𝑁))))
79	1, 78	mpcom 32	. 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 97   = wceq 1243   ∈ wcel 1393  ∀wral 2306  ‘cfv 4902  (class class class)co 5512  1c1 6890   + caddc 6892  ℤcz 8245  ℤ_≥cuz 8473  ...cfz 8874  seqcseq 9211

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311  ax-cnex 6975  ax-resscn 6976  ax-1cn 6977  ax-1re 6978  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-addcom 6984  ax-addass 6986  ax-distr 6988  ax-i2m1 6989  ax-0id 6992  ax-rnegex 6993  ax-cnre 6995  ax-pre-ltirr 6996  ax-pre-ltwlin 6997  ax-pre-lttrn 6998  ax-pre-ltadd 7000

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-frec 5978  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-i1p 6565  df-iplp 6566  df-iltp 6568  df-enr 6811  df-nr 6812  df-ltr 6815  df-0r 6816  df-1r 6817  df-0 6896  df-1 6897  df-r 6899  df-lt 6902  df-pnf 7062  df-mnf 7063  df-xr 7064  df-ltxr 7065  df-le 7066  df-sub 7184  df-neg 7185  df-inn 7915  df-n0 8182  df-z 8246  df-uz 8474  df-fz 8875  df-iseq 9212

This theorem is referenced by:  iseqfeq2  9229  iseqfveq  9230