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Theorem iseqfveq2 9228
 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.s (𝜑𝑆𝑉)
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
Allowed substitution hints:   𝑉(𝑥,𝑦,𝑘)

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 8896 . . 3 (𝑁 ∈ (ℤ𝐾) → 𝑁 ∈ (𝐾...𝑁))
31, 2syl 14 . 2 (𝜑𝑁 ∈ (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))
75, 6eqeq12d 2054 . . . . . 6 (𝑧 = 𝐾 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑧) = (seq𝐾( + , 𝐺, 𝑆)‘𝑧) ↔ (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝐾( + , 𝐺, 𝑆)‘𝐾)))
84, 7imbi12d 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))))
98imbi2d 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))
1311, 12eqeq12d 2054 . . . . . 6 (𝑧 = 𝑤 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑧) = (seq𝐾( + , 𝐺, 𝑆)‘𝑧) ↔ (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝐾( + , 𝐺, 𝑆)‘𝑤)))
1410, 13imbi12d 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))))
1514imbi2d 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)))
1917, 18eqeq12d 2054 . . . . . 6 (𝑧 = (𝑤 + 1) → ((seq𝑀( + , 𝐹, 𝑆)‘𝑧) = (seq𝐾( + , 𝐺, 𝑆)‘𝑧) ↔ (seq𝑀( + , 𝐹, 𝑆)‘(𝑤 + 1)) = (seq𝐾( + , 𝐺, 𝑆)‘(𝑤 + 1))))
2016, 19imbi12d 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)))))
2120imbi2d 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))
2523, 24eqeq12d 2054 . . . . . 6 (𝑧 = 𝑁 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑧) = (seq𝐾( + , 𝐺, 𝑆)‘𝑧) ↔ (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq𝐾( + , 𝐺, 𝑆)‘𝑁)))
2622, 25imbi12d 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))))
2726imbi2d 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) = (𝐺𝐾))
29 iseqfveq2.1 . . . . . . . . 9 (𝜑𝐾 ∈ (ℤ𝑀))
30 eluzelz 8482 . . . . . . . . 9 (𝐾 ∈ (ℤ𝑀) → 𝐾 ∈ ℤ)
3129, 30syl 14 . . . . . . . 8 (𝜑𝐾 ∈ ℤ)
32 iseqfveq2.s . . . . . . . 8 (𝜑𝑆𝑉)
33 iseqfveq2.g . . . . . . . 8 ((𝜑𝑥 ∈ (ℤ𝐾)) → (𝐺𝑥) ∈ 𝑆)
34 iseqfveq2.pl . . . . . . . 8 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
3531, 32, 33, 34iseq1 9222 . . . . . . 7 (𝜑 → (seq𝐾( + , 𝐺, 𝑆)‘𝐾) = (𝐺𝐾))
3628, 35eqtr4d 2075 . . . . . 6 (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝐾( + , 𝐺, 𝑆)‘𝐾))
3736a1d 22 . . . . 5 (𝜑 → (𝐾 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝐾( + , 𝐺, 𝑆)‘𝐾)))
3837a1i 9 . . . 4 (𝐾 ∈ ℤ → (𝜑 → (𝐾 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝐾( + , 𝐺, 𝑆)‘𝐾))))
39 peano2fzr 8901 . . . . . . . . . 10 ((𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁)) → 𝑤 ∈ (𝐾...𝑁))
4039adantl 262 . . . . . . . . 9 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → 𝑤 ∈ (𝐾...𝑁))
4140expr 357 . . . . . . . 8 ((𝜑𝑤 ∈ (ℤ𝐾)) → ((𝑤 + 1) ∈ (𝐾...𝑁) → 𝑤 ∈ (𝐾...𝑁)))
4241imim1d 69 . . . . . . 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))))
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 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → 𝑤 ∈ (ℤ𝐾))
4529adantr 261 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → 𝐾 ∈ (ℤ𝑀))
46 uztrn 8489 . . . . . . . . . . . . 13 ((𝑤 ∈ (ℤ𝐾) ∧ 𝐾 ∈ (ℤ𝑀)) → 𝑤 ∈ (ℤ𝑀))
4744, 45, 46syl2anc 391 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → 𝑤 ∈ (ℤ𝑀))
4832adantr 261 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → 𝑆𝑉)
49 iseqfveq2.f . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
5049adantlr 446 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
5134adantlr 446 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
5247, 48, 50, 51iseqp1 9225 . . . . . . . . . . 11 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑤 + 1)) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑤) + (𝐹‘(𝑤 + 1))))
5333adantlr 446 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) ∧ 𝑥 ∈ (ℤ𝐾)) → (𝐺𝑥) ∈ 𝑆)
5444, 48, 53, 51iseqp1 9225 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → (seq𝐾( + , 𝐺, 𝑆)‘(𝑤 + 1)) = ((seq𝐾( + , 𝐺, 𝑆)‘𝑤) + (𝐺‘(𝑤 + 1))))
55 eluzp1p1 8498 . . . . . . . . . . . . . . . 16 (𝑤 ∈ (ℤ𝐾) → (𝑤 + 1) ∈ (ℤ‘(𝐾 + 1)))
5655ad2antrl 459 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → (𝑤 + 1) ∈ (ℤ‘(𝐾 + 1)))
57 elfzuz3 8887 . . . . . . . . . . . . . . . 16 ((𝑤 + 1) ∈ (𝐾...𝑁) → 𝑁 ∈ (ℤ‘(𝑤 + 1)))
5857ad2antll 460 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (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))))
6056, 58, 59sylanbrc 394 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → (𝑤 + 1) ∈ ((𝐾 + 1)...𝑁))
61 iseqfveq2.4 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ ((𝐾 + 1)...𝑁)) → (𝐹𝑘) = (𝐺𝑘))
6261ralrimiva 2392 . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑘 ∈ ((𝐾 + 1)...𝑁)(𝐹𝑘) = (𝐺𝑘))
6362adantr 261 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → ∀𝑘 ∈ ((𝐾 + 1)...𝑁)(𝐹𝑘) = (𝐺𝑘))
64 fveq2 5178 . . . . . . . . . . . . . . . 16 (𝑘 = (𝑤 + 1) → (𝐹𝑘) = (𝐹‘(𝑤 + 1)))
65 fveq2 5178 . . . . . . . . . . . . . . . 16 (𝑘 = (𝑤 + 1) → (𝐺𝑘) = (𝐺‘(𝑤 + 1)))
6664, 65eqeq12d 2054 . . . . . . . . . . . . . . 15 (𝑘 = (𝑤 + 1) → ((𝐹𝑘) = (𝐺𝑘) ↔ (𝐹‘(𝑤 + 1)) = (𝐺‘(𝑤 + 1))))
6766rspcv 2652 . . . . . . . . . . . . . 14 ((𝑤 + 1) ∈ ((𝐾 + 1)...𝑁) → (∀𝑘 ∈ ((𝐾 + 1)...𝑁)(𝐹𝑘) = (𝐺𝑘) → (𝐹‘(𝑤 + 1)) = (𝐺‘(𝑤 + 1))))
6860, 63, 67sylc 56 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → (𝐹‘(𝑤 + 1)) = (𝐺‘(𝑤 + 1)))
6968oveq2d 5528 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → ((seq𝐾( + , 𝐺, 𝑆)‘𝑤) + (𝐹‘(𝑤 + 1))) = ((seq𝐾( + , 𝐺, 𝑆)‘𝑤) + (𝐺‘(𝑤 + 1))))
7054, 69eqtr4d 2075 . . . . . . . . . . 11 ((𝜑 ∧ (𝑤 ∈ (ℤ𝐾) ∧ (𝑤 + 1) ∈ (𝐾...𝑁))) → (seq𝐾( + , 𝐺, 𝑆)‘(𝑤 + 1)) = ((seq𝐾( + , 𝐺, 𝑆)‘𝑤) + (𝐹‘(𝑤 + 1))))
7152, 70eqeq12d 2054 . . . . . . . . . 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)))))
7243, 71syl5ibr 145 . . . . . . . . 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))))
7372expr 357 . . . . . . . 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)))))
7473a2d 23 . . . . . . 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)))))
7542, 74syld 40 . . . . . 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)))))
7675expcom 109 . . . . 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))))))
7776a2d 23 . . . 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))))))
789, 15, 21, 27, 38, 77uzind4 8531 . . 3 (𝑁 ∈ (ℤ𝐾) → (𝜑 → (𝑁 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq𝐾( + , 𝐺, 𝑆)‘𝑁))))
791, 78mpcom 32 . 2 (𝜑 → (𝑁 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq𝐾( + , 𝐺, 𝑆)‘𝑁)))
803, 79mpd 13 1 (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq𝐾( + , 𝐺, 𝑆)‘𝑁))
 Colors of variables: wff set class 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
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