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Theorem iseqshft2 9086
Description: Shifting the index set of a sequence. (Contributed by Jim Kingdon, 15-Aug-2021.)
Hypotheses
Ref Expression
iseqshft2.1 (𝜑𝑁 ∈ (ℤ𝑀))
iseqshft2.2 (𝜑𝐾 ∈ ℤ)
iseqshft2.3 ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) = (𝐺‘(𝑘 + 𝐾)))
iseqshft2.s (𝜑𝑆𝑉)
iseqshft2.f ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
iseqshft2.g ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 𝐾))) → (𝐺𝑥) ∈ 𝑆)
iseqshft2.pl ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
Assertion
Ref Expression
iseqshft2 (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑁 + 𝐾)))
Distinct variable groups:   𝑥, + ,𝑦   𝑘,𝐹,𝑥   𝑦,𝐹   𝑘,𝐺,𝑥   𝑦,𝐺   𝑘,𝐾,𝑥   𝑦,𝐾   𝑘,𝑀,𝑥   𝑦,𝑀   𝑘,𝑁,𝑥   𝑦,𝑁   𝑥,𝑆,𝑦   𝜑,𝑘,𝑥   𝜑,𝑦
Allowed substitution hints:   + (𝑘)   𝑆(𝑘)   𝑉(𝑥,𝑦,𝑘)

Proof of Theorem iseqshft2
Dummy variables 𝑛 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iseqshft2.1 . . 3 (𝜑𝑁 ∈ (ℤ𝑀))
2 eluzfz2 8839 . . 3 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
31, 2syl 14 . 2 (𝜑𝑁 ∈ (𝑀...𝑁))
4 eleq1 2100 . . . . . 6 (𝑤 = 𝑀 → (𝑤 ∈ (𝑀...𝑁) ↔ 𝑀 ∈ (𝑀...𝑁)))
5 fveq2 5141 . . . . . . 7 (𝑤 = 𝑀 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘𝑀))
6 oveq1 5482 . . . . . . . 8 (𝑤 = 𝑀 → (𝑤 + 𝐾) = (𝑀 + 𝐾))
76fveq2d 5145 . . . . . . 7 (𝑤 = 𝑀 → (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑤 + 𝐾)) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑀 + 𝐾)))
85, 7eqeq12d 2054 . . . . . 6 (𝑤 = 𝑀 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑤 + 𝐾)) ↔ (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑀 + 𝐾))))
94, 8imbi12d 223 . . . . 5 (𝑤 = 𝑀 → ((𝑤 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑤 + 𝐾))) ↔ (𝑀 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑀 + 𝐾)))))
109imbi2d 219 . . . 4 (𝑤 = 𝑀 → ((𝜑 → (𝑤 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑤 + 𝐾)))) ↔ (𝜑 → (𝑀 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑀 + 𝐾))))))
11 eleq1 2100 . . . . . 6 (𝑤 = 𝑛 → (𝑤 ∈ (𝑀...𝑁) ↔ 𝑛 ∈ (𝑀...𝑁)))
12 fveq2 5141 . . . . . . 7 (𝑤 = 𝑛 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘𝑛))
13 oveq1 5482 . . . . . . . 8 (𝑤 = 𝑛 → (𝑤 + 𝐾) = (𝑛 + 𝐾))
1413fveq2d 5145 . . . . . . 7 (𝑤 = 𝑛 → (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑤 + 𝐾)) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑛 + 𝐾)))
1512, 14eqeq12d 2054 . . . . . 6 (𝑤 = 𝑛 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑤 + 𝐾)) ↔ (seq𝑀( + , 𝐹, 𝑆)‘𝑛) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑛 + 𝐾))))
1611, 15imbi12d 223 . . . . 5 (𝑤 = 𝑛 → ((𝑤 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑤 + 𝐾))) ↔ (𝑛 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑛) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑛 + 𝐾)))))
1716imbi2d 219 . . . 4 (𝑤 = 𝑛 → ((𝜑 → (𝑤 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑤 + 𝐾)))) ↔ (𝜑 → (𝑛 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑛) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑛 + 𝐾))))))
18 eleq1 2100 . . . . . 6 (𝑤 = (𝑛 + 1) → (𝑤 ∈ (𝑀...𝑁) ↔ (𝑛 + 1) ∈ (𝑀...𝑁)))
19 fveq2 5141 . . . . . . 7 (𝑤 = (𝑛 + 1) → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1)))
20 oveq1 5482 . . . . . . . 8 (𝑤 = (𝑛 + 1) → (𝑤 + 𝐾) = ((𝑛 + 1) + 𝐾))
2120fveq2d 5145 . . . . . . 7 (𝑤 = (𝑛 + 1) → (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑤 + 𝐾)) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘((𝑛 + 1) + 𝐾)))
2219, 21eqeq12d 2054 . . . . . 6 (𝑤 = (𝑛 + 1) → ((seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑤 + 𝐾)) ↔ (seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1)) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘((𝑛 + 1) + 𝐾))))
2318, 22imbi12d 223 . . . . 5 (𝑤 = (𝑛 + 1) → ((𝑤 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑤 + 𝐾))) ↔ ((𝑛 + 1) ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1)) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘((𝑛 + 1) + 𝐾)))))
2423imbi2d 219 . . . 4 (𝑤 = (𝑛 + 1) → ((𝜑 → (𝑤 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑤 + 𝐾)))) ↔ (𝜑 → ((𝑛 + 1) ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1)) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘((𝑛 + 1) + 𝐾))))))
25 eleq1 2100 . . . . . 6 (𝑤 = 𝑁 → (𝑤 ∈ (𝑀...𝑁) ↔ 𝑁 ∈ (𝑀...𝑁)))
26 fveq2 5141 . . . . . . 7 (𝑤 = 𝑁 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘𝑁))
27 oveq1 5482 . . . . . . . 8 (𝑤 = 𝑁 → (𝑤 + 𝐾) = (𝑁 + 𝐾))
2827fveq2d 5145 . . . . . . 7 (𝑤 = 𝑁 → (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑤 + 𝐾)) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑁 + 𝐾)))
2926, 28eqeq12d 2054 . . . . . 6 (𝑤 = 𝑁 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑤 + 𝐾)) ↔ (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑁 + 𝐾))))
3025, 29imbi12d 223 . . . . 5 (𝑤 = 𝑁 → ((𝑤 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑤 + 𝐾))) ↔ (𝑁 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑁 + 𝐾)))))
3130imbi2d 219 . . . 4 (𝑤 = 𝑁 → ((𝜑 → (𝑤 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑤 + 𝐾)))) ↔ (𝜑 → (𝑁 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑁 + 𝐾))))))
32 eluzfz1 8838 . . . . . . . . 9 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ (𝑀...𝑁))
331, 32syl 14 . . . . . . . 8 (𝜑𝑀 ∈ (𝑀...𝑁))
34 iseqshft2.3 . . . . . . . . 9 ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) = (𝐺‘(𝑘 + 𝐾)))
3534ralrimiva 2389 . . . . . . . 8 (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)(𝐹𝑘) = (𝐺‘(𝑘 + 𝐾)))
36 fveq2 5141 . . . . . . . . . 10 (𝑘 = 𝑀 → (𝐹𝑘) = (𝐹𝑀))
37 oveq1 5482 . . . . . . . . . . 11 (𝑘 = 𝑀 → (𝑘 + 𝐾) = (𝑀 + 𝐾))
3837fveq2d 5145 . . . . . . . . . 10 (𝑘 = 𝑀 → (𝐺‘(𝑘 + 𝐾)) = (𝐺‘(𝑀 + 𝐾)))
3936, 38eqeq12d 2054 . . . . . . . . 9 (𝑘 = 𝑀 → ((𝐹𝑘) = (𝐺‘(𝑘 + 𝐾)) ↔ (𝐹𝑀) = (𝐺‘(𝑀 + 𝐾))))
4039rspcv 2649 . . . . . . . 8 (𝑀 ∈ (𝑀...𝑁) → (∀𝑘 ∈ (𝑀...𝑁)(𝐹𝑘) = (𝐺‘(𝑘 + 𝐾)) → (𝐹𝑀) = (𝐺‘(𝑀 + 𝐾))))
4133, 35, 40sylc 56 . . . . . . 7 (𝜑 → (𝐹𝑀) = (𝐺‘(𝑀 + 𝐾)))
42 eluzel2 8426 . . . . . . . . 9 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
431, 42syl 14 . . . . . . . 8 (𝜑𝑀 ∈ ℤ)
44 iseqshft2.s . . . . . . . 8 (𝜑𝑆𝑉)
45 iseqshft2.f . . . . . . . 8 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
46 iseqshft2.pl . . . . . . . 8 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
4743, 44, 45, 46iseq1 9076 . . . . . . 7 (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (𝐹𝑀))
48 iseqshft2.2 . . . . . . . . 9 (𝜑𝐾 ∈ ℤ)
4943, 48zaddcld 8312 . . . . . . . 8 (𝜑 → (𝑀 + 𝐾) ∈ ℤ)
50 iseqshft2.g . . . . . . . 8 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 𝐾))) → (𝐺𝑥) ∈ 𝑆)
5149, 44, 50, 46iseq1 9076 . . . . . . 7 (𝜑 → (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑀 + 𝐾)) = (𝐺‘(𝑀 + 𝐾)))
5241, 47, 513eqtr4d 2082 . . . . . 6 (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑀 + 𝐾)))
5352a1d 22 . . . . 5 (𝜑 → (𝑀 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑀 + 𝐾))))
5453a1i 9 . . . 4 (𝑀 ∈ ℤ → (𝜑 → (𝑀 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑀 + 𝐾)))))
55 peano2fzr 8844 . . . . . . . . . 10 ((𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → 𝑛 ∈ (𝑀...𝑁))
5655adantl 262 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ (𝑀...𝑁))
5756expr 357 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → ((𝑛 + 1) ∈ (𝑀...𝑁) → 𝑛 ∈ (𝑀...𝑁)))
5857imim1d 69 . . . . . . 7 ((𝜑𝑛 ∈ (ℤ𝑀)) → ((𝑛 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑛) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑛 + 𝐾))) → ((𝑛 + 1) ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑛) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑛 + 𝐾)))))
59 oveq1 5482 . . . . . . . . . 10 ((seq𝑀( + , 𝐹, 𝑆)‘𝑛) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑛 + 𝐾)) → ((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1))) = ((seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑛 + 𝐾)) + (𝐹‘(𝑛 + 1))))
60 simprl 483 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ (ℤ𝑀))
6144adantr 261 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑆𝑉)
6245adantlr 446 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
6346adantlr 446 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
6460, 61, 62, 63iseqp1 9079 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1))))
6548adantr 261 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝐾 ∈ ℤ)
66 eluzadd 8449 . . . . . . . . . . . . . 14 ((𝑛 ∈ (ℤ𝑀) ∧ 𝐾 ∈ ℤ) → (𝑛 + 𝐾) ∈ (ℤ‘(𝑀 + 𝐾)))
6760, 65, 66syl2anc 391 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝑛 + 𝐾) ∈ (ℤ‘(𝑀 + 𝐾)))
6850adantlr 446 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) ∧ 𝑥 ∈ (ℤ‘(𝑀 + 𝐾))) → (𝐺𝑥) ∈ 𝑆)
6967, 61, 68, 63iseqp1 9079 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘((𝑛 + 𝐾) + 1)) = ((seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑛 + 𝐾)) + (𝐺‘((𝑛 + 𝐾) + 1))))
70 eluzelz 8430 . . . . . . . . . . . . . . 15 (𝑛 ∈ (ℤ𝑀) → 𝑛 ∈ ℤ)
7160, 70syl 14 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ ℤ)
72 zcn 8198 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℤ → 𝑛 ∈ ℂ)
73 zcn 8198 . . . . . . . . . . . . . . 15 (𝐾 ∈ ℤ → 𝐾 ∈ ℂ)
74 ax-1cn 6934 . . . . . . . . . . . . . . . 16 1 ∈ ℂ
75 add32 7126 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝐾 ∈ ℂ) → ((𝑛 + 1) + 𝐾) = ((𝑛 + 𝐾) + 1))
7674, 75mp3an2 1220 . . . . . . . . . . . . . . 15 ((𝑛 ∈ ℂ ∧ 𝐾 ∈ ℂ) → ((𝑛 + 1) + 𝐾) = ((𝑛 + 𝐾) + 1))
7772, 73, 76syl2an 273 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℤ ∧ 𝐾 ∈ ℤ) → ((𝑛 + 1) + 𝐾) = ((𝑛 + 𝐾) + 1))
7871, 65, 77syl2anc 391 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ((𝑛 + 1) + 𝐾) = ((𝑛 + 𝐾) + 1))
7978fveq2d 5145 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘((𝑛 + 1) + 𝐾)) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘((𝑛 + 𝐾) + 1)))
80 simprr 484 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝑛 + 1) ∈ (𝑀...𝑁))
8135adantr 261 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ∀𝑘 ∈ (𝑀...𝑁)(𝐹𝑘) = (𝐺‘(𝑘 + 𝐾)))
82 fveq2 5141 . . . . . . . . . . . . . . . . 17 (𝑘 = (𝑛 + 1) → (𝐹𝑘) = (𝐹‘(𝑛 + 1)))
83 oveq1 5482 . . . . . . . . . . . . . . . . . 18 (𝑘 = (𝑛 + 1) → (𝑘 + 𝐾) = ((𝑛 + 1) + 𝐾))
8483fveq2d 5145 . . . . . . . . . . . . . . . . 17 (𝑘 = (𝑛 + 1) → (𝐺‘(𝑘 + 𝐾)) = (𝐺‘((𝑛 + 1) + 𝐾)))
8582, 84eqeq12d 2054 . . . . . . . . . . . . . . . 16 (𝑘 = (𝑛 + 1) → ((𝐹𝑘) = (𝐺‘(𝑘 + 𝐾)) ↔ (𝐹‘(𝑛 + 1)) = (𝐺‘((𝑛 + 1) + 𝐾))))
8685rspcv 2649 . . . . . . . . . . . . . . 15 ((𝑛 + 1) ∈ (𝑀...𝑁) → (∀𝑘 ∈ (𝑀...𝑁)(𝐹𝑘) = (𝐺‘(𝑘 + 𝐾)) → (𝐹‘(𝑛 + 1)) = (𝐺‘((𝑛 + 1) + 𝐾))))
8780, 81, 86sylc 56 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐹‘(𝑛 + 1)) = (𝐺‘((𝑛 + 1) + 𝐾)))
8878fveq2d 5145 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐺‘((𝑛 + 1) + 𝐾)) = (𝐺‘((𝑛 + 𝐾) + 1)))
8987, 88eqtrd 2072 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐹‘(𝑛 + 1)) = (𝐺‘((𝑛 + 𝐾) + 1)))
9089oveq2d 5491 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ((seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑛 + 𝐾)) + (𝐹‘(𝑛 + 1))) = ((seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑛 + 𝐾)) + (𝐺‘((𝑛 + 𝐾) + 1))))
9169, 79, 903eqtr4d 2082 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘((𝑛 + 1) + 𝐾)) = ((seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑛 + 𝐾)) + (𝐹‘(𝑛 + 1))))
9264, 91eqeq12d 2054 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ((seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1)) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘((𝑛 + 1) + 𝐾)) ↔ ((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1))) = ((seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑛 + 𝐾)) + (𝐹‘(𝑛 + 1)))))
9359, 92syl5ibr 145 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ((seq𝑀( + , 𝐹, 𝑆)‘𝑛) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑛 + 𝐾)) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1)) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘((𝑛 + 1) + 𝐾))))
9493expr 357 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → ((𝑛 + 1) ∈ (𝑀...𝑁) → ((seq𝑀( + , 𝐹, 𝑆)‘𝑛) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑛 + 𝐾)) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1)) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘((𝑛 + 1) + 𝐾)))))
9594a2d 23 . . . . . . 7 ((𝜑𝑛 ∈ (ℤ𝑀)) → (((𝑛 + 1) ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑛) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑛 + 𝐾))) → ((𝑛 + 1) ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1)) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘((𝑛 + 1) + 𝐾)))))
9658, 95syld 40 . . . . . 6 ((𝜑𝑛 ∈ (ℤ𝑀)) → ((𝑛 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑛) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑛 + 𝐾))) → ((𝑛 + 1) ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1)) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘((𝑛 + 1) + 𝐾)))))
9796expcom 109 . . . . 5 (𝑛 ∈ (ℤ𝑀) → (𝜑 → ((𝑛 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑛) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑛 + 𝐾))) → ((𝑛 + 1) ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1)) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘((𝑛 + 1) + 𝐾))))))
9897a2d 23 . . . 4 (𝑛 ∈ (ℤ𝑀) → ((𝜑 → (𝑛 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑛) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑛 + 𝐾)))) → (𝜑 → ((𝑛 + 1) ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1)) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘((𝑛 + 1) + 𝐾))))))
9910, 17, 24, 31, 54, 98uzind4 8479 . . 3 (𝑁 ∈ (ℤ𝑀) → (𝜑 → (𝑁 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑁 + 𝐾)))))
1001, 99mpcom 32 . 2 (𝜑 → (𝑁 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑁 + 𝐾))))
1013, 100mpd 13 1 (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑁 + 𝐾)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97   = wceq 1243  wcel 1393  wral 2303  cfv 4865  (class class class)co 5475  cc 6844  1c1 6847   + caddc 6849  cz 8193  cuz 8421  ...cfz 8817  seqcseq 9065
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 3869  ax-sep 3872  ax-nul 3880  ax-pow 3924  ax-pr 3941  ax-un 4142  ax-setind 4232  ax-iinf 4274  ax-cnex 6932  ax-resscn 6933  ax-1cn 6934  ax-1re 6935  ax-icn 6936  ax-addcl 6937  ax-addrcl 6938  ax-mulcl 6939  ax-addcom 6941  ax-addass 6943  ax-distr 6945  ax-i2m1 6946  ax-0id 6949  ax-rnegex 6950  ax-cnre 6952  ax-pre-ltirr 6953  ax-pre-ltwlin 6954  ax-pre-lttrn 6955  ax-pre-ltadd 6957
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 2308  df-rex 2309  df-reu 2310  df-rab 2312  df-v 2556  df-sbc 2762  df-csb 2850  df-dif 2917  df-un 2919  df-in 2921  df-ss 2928  df-nul 3222  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3578  df-int 3613  df-iun 3656  df-br 3762  df-opab 3816  df-mpt 3817  df-tr 3852  df-eprel 4023  df-id 4027  df-po 4030  df-iso 4031  df-iord 4075  df-on 4077  df-suc 4080  df-iom 4277  df-xp 4314  df-rel 4315  df-cnv 4316  df-co 4317  df-dm 4318  df-rn 4319  df-res 4320  df-ima 4321  df-iota 4830  df-fun 4867  df-fn 4868  df-f 4869  df-f1 4870  df-fo 4871  df-f1o 4872  df-fv 4873  df-riota 5431  df-ov 5478  df-oprab 5479  df-mpt2 5480  df-1st 5730  df-2nd 5731  df-recs 5883  df-irdg 5920  df-frec 5941  df-1o 5964  df-2o 5965  df-oadd 5968  df-omul 5969  df-er 6069  df-ec 6071  df-qs 6075  df-ni 6359  df-pli 6360  df-mi 6361  df-lti 6362  df-plpq 6399  df-mpq 6400  df-enq 6402  df-nqqs 6403  df-plqqs 6404  df-mqqs 6405  df-1nqqs 6406  df-rq 6407  df-ltnqqs 6408  df-enq0 6479  df-nq0 6480  df-0nq0 6481  df-plq0 6482  df-mq0 6483  df-inp 6521  df-i1p 6522  df-iplp 6523  df-iltp 6525  df-enr 6768  df-nr 6769  df-ltr 6772  df-0r 6773  df-1r 6774  df-0 6853  df-1 6854  df-r 6856  df-lt 6859  df-pnf 7018  df-mnf 7019  df-xr 7020  df-ltxr 7021  df-le 7022  df-sub 7140  df-neg 7141  df-inn 7867  df-n0 8130  df-z 8194  df-uz 8422  df-fz 8818  df-iseq 9066
This theorem is referenced by: (None)
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