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Theorem iseqfveq2 9761
 Description: Equality of sequences. (Contributed by Jim Kingdon, 3-Jun-2020.)
Hypotheses
Ref Expression
iseqfveq2.1 (𝜑𝐾 ∈ (ℤ𝑀))
iseqfveq2.2 (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (𝐺𝐾))
iseqfveq2.f ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
iseqfveq2.g ((𝜑𝑥 ∈ (ℤ𝐾)) → (𝐺𝑥) ∈ 𝑆)
iseqfveq2.pl ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
iseqfveq2.3 (𝜑𝑁 ∈ (ℤ𝐾))
iseqfveq2.4 ((𝜑𝑘 ∈ ((𝐾 + 1)...𝑁)) → (𝐹𝑘) = (𝐺𝑘))
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

Proof of Theorem iseqfveq2
Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iseqfveq2.3 . . 3 (𝜑𝑁 ∈ (ℤ𝐾))
2 eluzfz2 9339 . . 3 (𝑁 ∈ (ℤ𝐾) → 𝑁 ∈ (𝐾...𝑁))
31, 2syl 14 . 2 (𝜑𝑁 ∈ (𝐾...𝑁))
4 eleq1 2145 . . . . . 6 (𝑧 = 𝐾 → (𝑧 ∈ (𝐾...𝑁) ↔ 𝐾 ∈ (𝐾...𝑁)))
5 fveq2 5251 . . . . . . 7 (𝑧 = 𝐾 → (seq𝑀( + , 𝐹, 𝑆)‘𝑧) = (seq𝑀( + , 𝐹, 𝑆)‘𝐾))
6 fveq2 5251 . . . . . . 7 (𝑧 = 𝐾 → (seq𝐾( + , 𝐺, 𝑆)‘𝑧) = (seq𝐾( + , 𝐺, 𝑆)‘𝐾))
75, 6eqeq12d 2097 . . . . . 6 (𝑧 = 𝐾 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑧) = (seq𝐾( + , 𝐺, 𝑆)‘𝑧) ↔ (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝐾( + , 𝐺, 𝑆)‘𝐾)))
84, 7imbi12d 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))))
98imbi2d 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 2145 . . . . . 6 (𝑧 = 𝑤 → (𝑧 ∈ (𝐾...𝑁) ↔ 𝑤 ∈ (𝐾...𝑁)))
11 fveq2 5251 . . . . . . 7 (𝑧 = 𝑤 → (seq𝑀( + , 𝐹, 𝑆)‘𝑧) = (seq𝑀( + , 𝐹, 𝑆)‘𝑤))
12 fveq2 5251 . . . . . . 7 (𝑧 = 𝑤 → (seq𝐾( + , 𝐺, 𝑆)‘𝑧) = (seq𝐾( + , 𝐺, 𝑆)‘𝑤))
1311, 12eqeq12d 2097 . . . . . 6 (𝑧 = 𝑤 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑧) = (seq𝐾( + , 𝐺, 𝑆)‘𝑧) ↔ (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝐾( + , 𝐺, 𝑆)‘𝑤)))
1410, 13imbi12d 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))))
1514imbi2d 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 2145 . . . . . 6 (𝑧 = (𝑤 + 1) → (𝑧 ∈ (𝐾...𝑁) ↔ (𝑤 + 1) ∈ (𝐾...𝑁)))
17 fveq2 5251 . . . . . . 7 (𝑧 = (𝑤 + 1) → (seq𝑀( + , 𝐹, 𝑆)‘𝑧) = (seq𝑀( + , 𝐹, 𝑆)‘(𝑤 + 1)))
18 fveq2 5251 . . . . . . 7 (𝑧 = (𝑤 + 1) → (seq𝐾( + , 𝐺, 𝑆)‘𝑧) = (seq𝐾( + , 𝐺, 𝑆)‘(𝑤 + 1)))
1917, 18eqeq12d 2097 . . . . . 6 (𝑧 = (𝑤 + 1) → ((seq𝑀( + , 𝐹, 𝑆)‘𝑧) = (seq𝐾( + , 𝐺, 𝑆)‘𝑧) ↔ (seq𝑀( + , 𝐹, 𝑆)‘(𝑤 + 1)) = (seq𝐾( + , 𝐺, 𝑆)‘(𝑤 + 1))))
2016, 19imbi12d 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)))))
2120imbi2d 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 2145 . . . . . 6 (𝑧 = 𝑁 → (𝑧 ∈ (𝐾...𝑁) ↔ 𝑁 ∈ (𝐾...𝑁)))
23 fveq2 5251 . . . . . . 7 (𝑧 = 𝑁 → (seq𝑀( + , 𝐹, 𝑆)‘𝑧) = (seq𝑀( + , 𝐹, 𝑆)‘𝑁))
24 fveq2 5251 . . . . . . 7 (𝑧 = 𝑁 → (seq𝐾( + , 𝐺, 𝑆)‘𝑧) = (seq𝐾( + , 𝐺, 𝑆)‘𝑁))
2523, 24eqeq12d 2097 . . . . . 6 (𝑧 = 𝑁 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑧) = (seq𝐾( + , 𝐺, 𝑆)‘𝑧) ↔ (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq𝐾( + , 𝐺, 𝑆)‘𝑁)))
2622, 25imbi12d 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))))
2726imbi2d 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 . . . . . 6 (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (𝐺𝐾))
29 iseqfveq2.1 . . . . . . . 8 (𝜑𝐾 ∈ (ℤ𝑀))
30 eluzelz 8921 . . . . . . . 8 (𝐾 ∈ (ℤ𝑀) → 𝐾 ∈ ℤ)
3129, 30syl 14 . . . . . . 7 (𝜑𝐾 ∈ ℤ)
32 iseqfveq2.g . . . . . . 7 ((𝜑𝑥 ∈ (ℤ𝐾)) → (𝐺𝑥) ∈ 𝑆)
33 iseqfveq2.pl . . . . . . 7 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
3431, 32, 33iseq1 9750 . . . . . 6 (𝜑 → (seq𝐾( + , 𝐺, 𝑆)‘𝐾) = (𝐺𝐾))
3528, 34eqtr4d 2118 . . . . 5 (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝐾( + , 𝐺, 𝑆)‘𝐾))
3635a1i13 24 . . . 4 (𝐾 ∈ ℤ → (𝜑 → (𝐾 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝐾( + , 𝐺, 𝑆)‘𝐾))))
37 peano2fzr 9344 . . . . . . . . . 10 ((𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁)) → 𝑤 ∈ (𝐾...𝑁))
3837adantl 271 . . . . . . . . 9 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → 𝑤 ∈ (𝐾...𝑁))
3938expr 367 . . . . . . . 8 ((𝜑𝑤 ∈ (ℤ𝐾)) → ((𝑤 + 1) ∈ (𝐾...𝑁) → 𝑤 ∈ (𝐾...𝑁)))
4039imim1d 74 . . . . . . 7 ((𝜑𝑤 ∈ (ℤ𝐾)) → ((𝑤 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝐾( + , 𝐺, 𝑆)‘𝑤)) → ((𝑤 + 1) ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝐾( + , 𝐺, 𝑆)‘𝑤))))
41 oveq1 5596 . . . . . . . . . 10 ((seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝐾( + , 𝐺, 𝑆)‘𝑤) → ((seq𝑀( + , 𝐹, 𝑆)‘𝑤) + (𝐹‘(𝑤 + 1))) = ((seq𝐾( + , 𝐺, 𝑆)‘𝑤) + (𝐹‘(𝑤 + 1))))
42 simprl 498 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → 𝑤 ∈ (ℤ𝐾))
4329adantr 270 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → 𝐾 ∈ (ℤ𝑀))
44 uztrn 8928 . . . . . . . . . . . . 13 ((𝑤 ∈ (ℤ𝐾) ∧ 𝐾 ∈ (ℤ𝑀)) → 𝑤 ∈ (ℤ𝑀))
4542, 43, 44syl2anc 403 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → 𝑤 ∈ (ℤ𝑀))
46 iseqfveq2.f . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
4746adantlr 461 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
4833adantlr 461 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
4945, 47, 48iseqp1 9755 . . . . . . . . . . 11 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑤 + 1)) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑤) + (𝐹‘(𝑤 + 1))))
5032adantlr 461 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) ∧ 𝑥 ∈ (ℤ𝐾)) → (𝐺𝑥) ∈ 𝑆)
5142, 50, 48iseqp1 9755 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → (seq𝐾( + , 𝐺, 𝑆)‘(𝑤 + 1)) = ((seq𝐾( + , 𝐺, 𝑆)‘𝑤) + (𝐺‘(𝑤 + 1))))
52 fveq2 5251 . . . . . . . . . . . . . . 15 (𝑘 = (𝑤 + 1) → (𝐹𝑘) = (𝐹‘(𝑤 + 1)))
53 fveq2 5251 . . . . . . . . . . . . . . 15 (𝑘 = (𝑤 + 1) → (𝐺𝑘) = (𝐺‘(𝑤 + 1)))
5452, 53eqeq12d 2097 . . . . . . . . . . . . . 14 (𝑘 = (𝑤 + 1) → ((𝐹𝑘) = (𝐺𝑘) ↔ (𝐹‘(𝑤 + 1)) = (𝐺‘(𝑤 + 1))))
55 iseqfveq2.4 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ ((𝐾 + 1)...𝑁)) → (𝐹𝑘) = (𝐺𝑘))
5655ralrimiva 2440 . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑘 ∈ ((𝐾 + 1)...𝑁)(𝐹𝑘) = (𝐺𝑘))
5756adantr 270 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → ∀𝑘 ∈ ((𝐾 + 1)...𝑁)(𝐹𝑘) = (𝐺𝑘))
58 eluzp1p1 8937 . . . . . . . . . . . . . . . 16 (𝑤 ∈ (ℤ𝐾) → (𝑤 + 1) ∈ (ℤ‘(𝐾 + 1)))
5958ad2antrl 474 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → (𝑤 + 1) ∈ (ℤ‘(𝐾 + 1)))
60 elfzuz3 9330 . . . . . . . . . . . . . . . 16 ((𝑤 + 1) ∈ (𝐾...𝑁) → 𝑁 ∈ (ℤ‘(𝑤 + 1)))
6160ad2antll 475 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → 𝑁 ∈ (ℤ‘(𝑤 + 1)))
62 elfzuzb 9327 . . . . . . . . . . . . . . 15 ((𝑤 + 1) ∈ ((𝐾 + 1)...𝑁) ↔ ((𝑤 + 1) ∈ (ℤ‘(𝐾 + 1)) ∧ 𝑁 ∈ (ℤ‘(𝑤 + 1))))
6359, 61, 62sylanbrc 408 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → (𝑤 + 1) ∈ ((𝐾 + 1)...𝑁))
6454, 57, 63rspcdva 2717 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → (𝐹‘(𝑤 + 1)) = (𝐺‘(𝑤 + 1)))
6564oveq2d 5605 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → ((seq𝐾( + , 𝐺, 𝑆)‘𝑤) + (𝐹‘(𝑤 + 1))) = ((seq𝐾( + , 𝐺, 𝑆)‘𝑤) + (𝐺‘(𝑤 + 1))))
6651, 65eqtr4d 2118 . . . . . . . . . . 11 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → (seq𝐾( + , 𝐺, 𝑆)‘(𝑤 + 1)) = ((seq𝐾( + , 𝐺, 𝑆)‘𝑤) + (𝐹‘(𝑤 + 1))))
6749, 66eqeq12d 2097 . . . . . . . . . 10 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (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)))))
6841, 67syl5ibr 154 . . . . . . . . 9 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → ((seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝐾( + , 𝐺, 𝑆)‘𝑤) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑤 + 1)) = (seq𝐾( + , 𝐺, 𝑆)‘(𝑤 + 1))))
6968expr 367 . . . . . . . 8 ((𝜑𝑤 ∈ (ℤ𝐾)) → ((𝑤 + 1) ∈ (𝐾...𝑁) → ((seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝐾( + , 𝐺, 𝑆)‘𝑤) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑤 + 1)) = (seq𝐾( + , 𝐺, 𝑆)‘(𝑤 + 1)))))
7069a2d 26 . . . . . . 7 ((𝜑𝑤 ∈ (ℤ𝐾)) → (((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)))))
7140, 70syld 44 . . . . . 6 ((𝜑𝑤 ∈ (ℤ𝐾)) → ((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)))))
7271expcom 114 . . . . 5 (𝑤 ∈ (ℤ𝐾) → (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))))))
7372a2d 26 . . . 4 (𝑤 ∈ (ℤ𝐾) → ((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))))))
749, 15, 21, 27, 36, 73uzind4 8969 . . 3 (𝑁 ∈ (ℤ𝐾) → (𝜑 → (𝑁 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq𝐾( + , 𝐺, 𝑆)‘𝑁))))
751, 74mpcom 36 . 2 (𝜑 → (𝑁 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq𝐾( + , 𝐺, 𝑆)‘𝑁)))
763, 75mpd 13 1 (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq𝐾( + , 𝐺, 𝑆)‘𝑁))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 102   = wceq 1285   ∈ wcel 1434  ∀wral 2353  ‘cfv 4967  (class class class)co 5589  1c1 7252   + caddc 7254  ℤcz 8644  ℤ≥cuz 8912  ...cfz 9317  seqcseq 9738 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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 3999  ax-un 4223  ax-setind 4315  ax-iinf 4365  ax-cnex 7337  ax-resscn 7338  ax-1cn 7339  ax-1re 7340  ax-icn 7341  ax-addcl 7342  ax-addrcl 7343  ax-mulcl 7344  ax-addcom 7346  ax-addass 7348  ax-distr 7350  ax-i2m1 7351  ax-0lt1 7352  ax-0id 7354  ax-rnegex 7355  ax-cnre 7357  ax-pre-ltirr 7358  ax-pre-ltwlin 7359  ax-pre-lttrn 7360  ax-pre-ltadd 7362 This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-id 4083  df-iord 4156  df-on 4158  df-ilim 4159  df-suc 4161  df-iom 4368  df-xp 4405  df-rel 4406  df-cnv 4407  df-co 4408  df-dm 4409  df-rn 4410  df-res 4411  df-ima 4412  df-iota 4932  df-fun 4969  df-fn 4970  df-f 4971  df-f1 4972  df-fo 4973  df-f1o 4974  df-fv 4975  df-riota 5545  df-ov 5592  df-oprab 5593  df-mpt2 5594  df-1st 5844  df-2nd 5845  df-recs 6000  df-frec 6086  df-pnf 7425  df-mnf 7426  df-xr 7427  df-ltxr 7428  df-le 7429  df-sub 7556  df-neg 7557  df-inn 8315  df-n0 8564  df-z 8645  df-uz 8913  df-fz 9318  df-iseq 9739 This theorem is referenced by:  iseqfeq2  9762  iseqfveq  9763
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