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Theorem iseqshft2 9396
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 8998 . . 3 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
31, 2syl 14 . 2 (𝜑𝑁 ∈ (𝑀...𝑁))
4 eleq1 2116 . . . . . 6 (𝑤 = 𝑀 → (𝑤 ∈ (𝑀...𝑁) ↔ 𝑀 ∈ (𝑀...𝑁)))
5 fveq2 5206 . . . . . . 7 (𝑤 = 𝑀 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘𝑀))
6 oveq1 5547 . . . . . . . 8 (𝑤 = 𝑀 → (𝑤 + 𝐾) = (𝑀 + 𝐾))
76fveq2d 5210 . . . . . . 7 (𝑤 = 𝑀 → (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑤 + 𝐾)) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑀 + 𝐾)))
85, 7eqeq12d 2070 . . . . . 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 227 . . . . 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 223 . . . 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 2116 . . . . . 6 (𝑤 = 𝑛 → (𝑤 ∈ (𝑀...𝑁) ↔ 𝑛 ∈ (𝑀...𝑁)))
12 fveq2 5206 . . . . . . 7 (𝑤 = 𝑛 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘𝑛))
13 oveq1 5547 . . . . . . . 8 (𝑤 = 𝑛 → (𝑤 + 𝐾) = (𝑛 + 𝐾))
1413fveq2d 5210 . . . . . . 7 (𝑤 = 𝑛 → (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑤 + 𝐾)) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑛 + 𝐾)))
1512, 14eqeq12d 2070 . . . . . 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 227 . . . . 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 223 . . . 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 2116 . . . . . 6 (𝑤 = (𝑛 + 1) → (𝑤 ∈ (𝑀...𝑁) ↔ (𝑛 + 1) ∈ (𝑀...𝑁)))
19 fveq2 5206 . . . . . . 7 (𝑤 = (𝑛 + 1) → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1)))
20 oveq1 5547 . . . . . . . 8 (𝑤 = (𝑛 + 1) → (𝑤 + 𝐾) = ((𝑛 + 1) + 𝐾))
2120fveq2d 5210 . . . . . . 7 (𝑤 = (𝑛 + 1) → (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑤 + 𝐾)) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘((𝑛 + 1) + 𝐾)))
2219, 21eqeq12d 2070 . . . . . 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 227 . . . . 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 223 . . . 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 2116 . . . . . 6 (𝑤 = 𝑁 → (𝑤 ∈ (𝑀...𝑁) ↔ 𝑁 ∈ (𝑀...𝑁)))
26 fveq2 5206 . . . . . . 7 (𝑤 = 𝑁 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘𝑁))
27 oveq1 5547 . . . . . . . 8 (𝑤 = 𝑁 → (𝑤 + 𝐾) = (𝑁 + 𝐾))
2827fveq2d 5210 . . . . . . 7 (𝑤 = 𝑁 → (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑤 + 𝐾)) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑁 + 𝐾)))
2926, 28eqeq12d 2070 . . . . . 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 227 . . . . 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 223 . . . 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 8997 . . . . . . . . 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 2409 . . . . . . . 8 (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)(𝐹𝑘) = (𝐺‘(𝑘 + 𝐾)))
36 fveq2 5206 . . . . . . . . . 10 (𝑘 = 𝑀 → (𝐹𝑘) = (𝐹𝑀))
37 oveq1 5547 . . . . . . . . . . 11 (𝑘 = 𝑀 → (𝑘 + 𝐾) = (𝑀 + 𝐾))
3837fveq2d 5210 . . . . . . . . . 10 (𝑘 = 𝑀 → (𝐺‘(𝑘 + 𝐾)) = (𝐺‘(𝑀 + 𝐾)))
3936, 38eqeq12d 2070 . . . . . . . . 9 (𝑘 = 𝑀 → ((𝐹𝑘) = (𝐺‘(𝑘 + 𝐾)) ↔ (𝐹𝑀) = (𝐺‘(𝑀 + 𝐾))))
4039rspcv 2669 . . . . . . . 8 (𝑀 ∈ (𝑀...𝑁) → (∀𝑘 ∈ (𝑀...𝑁)(𝐹𝑘) = (𝐺‘(𝑘 + 𝐾)) → (𝐹𝑀) = (𝐺‘(𝑀 + 𝐾))))
4133, 35, 40sylc 60 . . . . . . 7 (𝜑 → (𝐹𝑀) = (𝐺‘(𝑀 + 𝐾)))
42 eluzel2 8574 . . . . . . . . 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 9386 . . . . . . 7 (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (𝐹𝑀))
48 iseqshft2.2 . . . . . . . . 9 (𝜑𝐾 ∈ ℤ)
4943, 48zaddcld 8423 . . . . . . . 8 (𝜑 → (𝑀 + 𝐾) ∈ ℤ)
50 iseqshft2.g . . . . . . . 8 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 𝐾))) → (𝐺𝑥) ∈ 𝑆)
5149, 44, 50, 46iseq1 9386 . . . . . . 7 (𝜑 → (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑀 + 𝐾)) = (𝐺‘(𝑀 + 𝐾)))
5241, 47, 513eqtr4d 2098 . . . . . 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 9003 . . . . . . . . . 10 ((𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → 𝑛 ∈ (𝑀...𝑁))
5655adantl 266 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ (𝑀...𝑁))
5756expr 361 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → ((𝑛 + 1) ∈ (𝑀...𝑁) → 𝑛 ∈ (𝑀...𝑁)))
5857imim1d 73 . . . . . . 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 5547 . . . . . . . . . 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 491 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ (ℤ𝑀))
6144adantr 265 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑆𝑉)
6245adantlr 454 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
6346adantlr 454 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
6460, 61, 62, 63iseqp1 9389 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1))))
6548adantr 265 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝐾 ∈ ℤ)
66 eluzadd 8597 . . . . . . . . . . . . . 14 ((𝑛 ∈ (ℤ𝑀) ∧ 𝐾 ∈ ℤ) → (𝑛 + 𝐾) ∈ (ℤ‘(𝑀 + 𝐾)))
6760, 65, 66syl2anc 397 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝑛 + 𝐾) ∈ (ℤ‘(𝑀 + 𝐾)))
6850adantlr 454 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) ∧ 𝑥 ∈ (ℤ‘(𝑀 + 𝐾))) → (𝐺𝑥) ∈ 𝑆)
6967, 61, 68, 63iseqp1 9389 . . . . . . . . . . . 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 8578 . . . . . . . . . . . . . . 15 (𝑛 ∈ (ℤ𝑀) → 𝑛 ∈ ℤ)
7160, 70syl 14 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ ℤ)
72 zcn 8307 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℤ → 𝑛 ∈ ℂ)
73 zcn 8307 . . . . . . . . . . . . . . 15 (𝐾 ∈ ℤ → 𝐾 ∈ ℂ)
74 ax-1cn 7035 . . . . . . . . . . . . . . . 16 1 ∈ ℂ
75 add32 7233 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝐾 ∈ ℂ) → ((𝑛 + 1) + 𝐾) = ((𝑛 + 𝐾) + 1))
7674, 75mp3an2 1231 . . . . . . . . . . . . . . 15 ((𝑛 ∈ ℂ ∧ 𝐾 ∈ ℂ) → ((𝑛 + 1) + 𝐾) = ((𝑛 + 𝐾) + 1))
7772, 73, 76syl2an 277 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℤ ∧ 𝐾 ∈ ℤ) → ((𝑛 + 1) + 𝐾) = ((𝑛 + 𝐾) + 1))
7871, 65, 77syl2anc 397 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ((𝑛 + 1) + 𝐾) = ((𝑛 + 𝐾) + 1))
7978fveq2d 5210 . . . . . . . . . . . 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 492 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝑛 + 1) ∈ (𝑀...𝑁))
8135adantr 265 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ∀𝑘 ∈ (𝑀...𝑁)(𝐹𝑘) = (𝐺‘(𝑘 + 𝐾)))
82 fveq2 5206 . . . . . . . . . . . . . . . . 17 (𝑘 = (𝑛 + 1) → (𝐹𝑘) = (𝐹‘(𝑛 + 1)))
83 oveq1 5547 . . . . . . . . . . . . . . . . . 18 (𝑘 = (𝑛 + 1) → (𝑘 + 𝐾) = ((𝑛 + 1) + 𝐾))
8483fveq2d 5210 . . . . . . . . . . . . . . . . 17 (𝑘 = (𝑛 + 1) → (𝐺‘(𝑘 + 𝐾)) = (𝐺‘((𝑛 + 1) + 𝐾)))
8582, 84eqeq12d 2070 . . . . . . . . . . . . . . . 16 (𝑘 = (𝑛 + 1) → ((𝐹𝑘) = (𝐺‘(𝑘 + 𝐾)) ↔ (𝐹‘(𝑛 + 1)) = (𝐺‘((𝑛 + 1) + 𝐾))))
8685rspcv 2669 . . . . . . . . . . . . . . 15 ((𝑛 + 1) ∈ (𝑀...𝑁) → (∀𝑘 ∈ (𝑀...𝑁)(𝐹𝑘) = (𝐺‘(𝑘 + 𝐾)) → (𝐹‘(𝑛 + 1)) = (𝐺‘((𝑛 + 1) + 𝐾))))
8780, 81, 86sylc 60 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐹‘(𝑛 + 1)) = (𝐺‘((𝑛 + 1) + 𝐾)))
8878fveq2d 5210 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐺‘((𝑛 + 1) + 𝐾)) = (𝐺‘((𝑛 + 𝐾) + 1)))
8987, 88eqtrd 2088 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐹‘(𝑛 + 1)) = (𝐺‘((𝑛 + 𝐾) + 1)))
9089oveq2d 5556 . . . . . . . . . . . 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 2098 . . . . . . . . . . 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 2070 . . . . . . . . . 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 149 . . . . . . . . 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 361 . . . . . . . 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 26 . . . . . . 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 44 . . . . . 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 113 . . . . 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 26 . . . 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 8627 . . 3 (𝑁 ∈ (ℤ𝑀) → (𝜑 → (𝑁 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑁 + 𝐾)))))
1001, 99mpcom 36 . 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 101   = wceq 1259  wcel 1409  wral 2323  cfv 4930  (class class class)co 5540  cc 6945  1c1 6948   + caddc 6950  cz 8302  cuz 8569  ...cfz 8976  seqcseq 9375
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339  ax-cnex 7033  ax-resscn 7034  ax-1cn 7035  ax-1re 7036  ax-icn 7037  ax-addcl 7038  ax-addrcl 7039  ax-mulcl 7040  ax-addcom 7042  ax-addass 7044  ax-distr 7046  ax-i2m1 7047  ax-0id 7050  ax-rnegex 7051  ax-cnre 7053  ax-pre-ltirr 7054  ax-pre-ltwlin 7055  ax-pre-lttrn 7056  ax-pre-ltadd 7058
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-nel 2315  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-riota 5496  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-frec 6009  df-1o 6032  df-2o 6033  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-enq0 6580  df-nq0 6581  df-0nq0 6582  df-plq0 6583  df-mq0 6584  df-inp 6622  df-i1p 6623  df-iplp 6624  df-iltp 6626  df-enr 6869  df-nr 6870  df-ltr 6873  df-0r 6874  df-1r 6875  df-0 6954  df-1 6955  df-r 6957  df-lt 6960  df-pnf 7121  df-mnf 7122  df-xr 7123  df-ltxr 7124  df-le 7125  df-sub 7247  df-neg 7248  df-inn 7991  df-n0 8240  df-z 8303  df-uz 8570  df-fz 8977  df-iseq 9376
This theorem is referenced by: (None)
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