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Theorem iseqshft2 9767
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 9341 . . 3 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
31, 2syl 14 . 2 (𝜑𝑁 ∈ (𝑀...𝑁))
4 eleq1 2145 . . . . . 6 (𝑤 = 𝑀 → (𝑤 ∈ (𝑀...𝑁) ↔ 𝑀 ∈ (𝑀...𝑁)))
5 fveq2 5253 . . . . . . 7 (𝑤 = 𝑀 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘𝑀))
6 oveq1 5598 . . . . . . . 8 (𝑤 = 𝑀 → (𝑤 + 𝐾) = (𝑀 + 𝐾))
76fveq2d 5257 . . . . . . 7 (𝑤 = 𝑀 → (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑤 + 𝐾)) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑀 + 𝐾)))
85, 7eqeq12d 2097 . . . . . 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 232 . . . . 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 228 . . . 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 2145 . . . . . 6 (𝑤 = 𝑛 → (𝑤 ∈ (𝑀...𝑁) ↔ 𝑛 ∈ (𝑀...𝑁)))
12 fveq2 5253 . . . . . . 7 (𝑤 = 𝑛 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘𝑛))
13 oveq1 5598 . . . . . . . 8 (𝑤 = 𝑛 → (𝑤 + 𝐾) = (𝑛 + 𝐾))
1413fveq2d 5257 . . . . . . 7 (𝑤 = 𝑛 → (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑤 + 𝐾)) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑛 + 𝐾)))
1512, 14eqeq12d 2097 . . . . . 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 232 . . . . 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 228 . . . 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 2145 . . . . . 6 (𝑤 = (𝑛 + 1) → (𝑤 ∈ (𝑀...𝑁) ↔ (𝑛 + 1) ∈ (𝑀...𝑁)))
19 fveq2 5253 . . . . . . 7 (𝑤 = (𝑛 + 1) → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1)))
20 oveq1 5598 . . . . . . . 8 (𝑤 = (𝑛 + 1) → (𝑤 + 𝐾) = ((𝑛 + 1) + 𝐾))
2120fveq2d 5257 . . . . . . 7 (𝑤 = (𝑛 + 1) → (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑤 + 𝐾)) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘((𝑛 + 1) + 𝐾)))
2219, 21eqeq12d 2097 . . . . . 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 232 . . . . 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 228 . . . 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 2145 . . . . . 6 (𝑤 = 𝑁 → (𝑤 ∈ (𝑀...𝑁) ↔ 𝑁 ∈ (𝑀...𝑁)))
26 fveq2 5253 . . . . . . 7 (𝑤 = 𝑁 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘𝑁))
27 oveq1 5598 . . . . . . . 8 (𝑤 = 𝑁 → (𝑤 + 𝐾) = (𝑁 + 𝐾))
2827fveq2d 5257 . . . . . . 7 (𝑤 = 𝑁 → (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑤 + 𝐾)) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑁 + 𝐾)))
2926, 28eqeq12d 2097 . . . . . 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 232 . . . . 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 228 . . . 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 fveq2 5253 . . . . . . . 8 (𝑘 = 𝑀 → (𝐹𝑘) = (𝐹𝑀))
33 oveq1 5598 . . . . . . . . 9 (𝑘 = 𝑀 → (𝑘 + 𝐾) = (𝑀 + 𝐾))
3433fveq2d 5257 . . . . . . . 8 (𝑘 = 𝑀 → (𝐺‘(𝑘 + 𝐾)) = (𝐺‘(𝑀 + 𝐾)))
3532, 34eqeq12d 2097 . . . . . . 7 (𝑘 = 𝑀 → ((𝐹𝑘) = (𝐺‘(𝑘 + 𝐾)) ↔ (𝐹𝑀) = (𝐺‘(𝑀 + 𝐾))))
36 iseqshft2.3 . . . . . . . 8 ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) = (𝐺‘(𝑘 + 𝐾)))
3736ralrimiva 2440 . . . . . . 7 (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)(𝐹𝑘) = (𝐺‘(𝑘 + 𝐾)))
38 eluzfz1 9340 . . . . . . . 8 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ (𝑀...𝑁))
391, 38syl 14 . . . . . . 7 (𝜑𝑀 ∈ (𝑀...𝑁))
4035, 37, 39rspcdva 2717 . . . . . 6 (𝜑 → (𝐹𝑀) = (𝐺‘(𝑀 + 𝐾)))
41 eluzel2 8919 . . . . . . . 8 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
421, 41syl 14 . . . . . . 7 (𝜑𝑀 ∈ ℤ)
43 iseqshft2.f . . . . . . 7 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
44 iseqshft2.pl . . . . . . 7 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
4542, 43, 44iseq1 9752 . . . . . 6 (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (𝐹𝑀))
46 iseqshft2.2 . . . . . . . 8 (𝜑𝐾 ∈ ℤ)
4742, 46zaddcld 8768 . . . . . . 7 (𝜑 → (𝑀 + 𝐾) ∈ ℤ)
48 iseqshft2.g . . . . . . 7 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 𝐾))) → (𝐺𝑥) ∈ 𝑆)
4947, 48, 44iseq1 9752 . . . . . 6 (𝜑 → (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑀 + 𝐾)) = (𝐺‘(𝑀 + 𝐾)))
5040, 45, 493eqtr4d 2125 . . . . 5 (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑀 + 𝐾)))
5150a1i13 24 . . . 4 (𝑀 ∈ ℤ → (𝜑 → (𝑀 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑀 + 𝐾)))))
52 peano2fzr 9346 . . . . . . . . . 10 ((𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → 𝑛 ∈ (𝑀...𝑁))
5352adantl 271 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ (𝑀...𝑁))
5453expr 367 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → ((𝑛 + 1) ∈ (𝑀...𝑁) → 𝑛 ∈ (𝑀...𝑁)))
5554imim1d 74 . . . . . . 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)))))
56 oveq1 5598 . . . . . . . . . 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))))
57 simprl 498 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ (ℤ𝑀))
5843adantlr 461 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
5944adantlr 461 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
6057, 58, 59iseqp1 9757 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1))))
6146adantr 270 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝐾 ∈ ℤ)
62 eluzadd 8942 . . . . . . . . . . . . . 14 ((𝑛 ∈ (ℤ𝑀) ∧ 𝐾 ∈ ℤ) → (𝑛 + 𝐾) ∈ (ℤ‘(𝑀 + 𝐾)))
6357, 61, 62syl2anc 403 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝑛 + 𝐾) ∈ (ℤ‘(𝑀 + 𝐾)))
6448adantlr 461 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) ∧ 𝑥 ∈ (ℤ‘(𝑀 + 𝐾))) → (𝐺𝑥) ∈ 𝑆)
6563, 64, 59iseqp1 9757 . . . . . . . . . . . 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))))
66 eluzelz 8923 . . . . . . . . . . . . . . 15 (𝑛 ∈ (ℤ𝑀) → 𝑛 ∈ ℤ)
6757, 66syl 14 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ ℤ)
68 zcn 8651 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℤ → 𝑛 ∈ ℂ)
69 zcn 8651 . . . . . . . . . . . . . . 15 (𝐾 ∈ ℤ → 𝐾 ∈ ℂ)
70 ax-1cn 7341 . . . . . . . . . . . . . . . 16 1 ∈ ℂ
71 add32 7544 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝐾 ∈ ℂ) → ((𝑛 + 1) + 𝐾) = ((𝑛 + 𝐾) + 1))
7270, 71mp3an2 1257 . . . . . . . . . . . . . . 15 ((𝑛 ∈ ℂ ∧ 𝐾 ∈ ℂ) → ((𝑛 + 1) + 𝐾) = ((𝑛 + 𝐾) + 1))
7368, 69, 72syl2an 283 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℤ ∧ 𝐾 ∈ ℤ) → ((𝑛 + 1) + 𝐾) = ((𝑛 + 𝐾) + 1))
7467, 61, 73syl2anc 403 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ((𝑛 + 1) + 𝐾) = ((𝑛 + 𝐾) + 1))
7574fveq2d 5257 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘((𝑛 + 1) + 𝐾)) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘((𝑛 + 𝐾) + 1)))
76 fveq2 5253 . . . . . . . . . . . . . . . 16 (𝑘 = (𝑛 + 1) → (𝐹𝑘) = (𝐹‘(𝑛 + 1)))
77 oveq1 5598 . . . . . . . . . . . . . . . . 17 (𝑘 = (𝑛 + 1) → (𝑘 + 𝐾) = ((𝑛 + 1) + 𝐾))
7877fveq2d 5257 . . . . . . . . . . . . . . . 16 (𝑘 = (𝑛 + 1) → (𝐺‘(𝑘 + 𝐾)) = (𝐺‘((𝑛 + 1) + 𝐾)))
7976, 78eqeq12d 2097 . . . . . . . . . . . . . . 15 (𝑘 = (𝑛 + 1) → ((𝐹𝑘) = (𝐺‘(𝑘 + 𝐾)) ↔ (𝐹‘(𝑛 + 1)) = (𝐺‘((𝑛 + 1) + 𝐾))))
8037adantr 270 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ∀𝑘 ∈ (𝑀...𝑁)(𝐹𝑘) = (𝐺‘(𝑘 + 𝐾)))
81 simprr 499 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝑛 + 1) ∈ (𝑀...𝑁))
8279, 80, 81rspcdva 2717 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐹‘(𝑛 + 1)) = (𝐺‘((𝑛 + 1) + 𝐾)))
8374fveq2d 5257 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐺‘((𝑛 + 1) + 𝐾)) = (𝐺‘((𝑛 + 𝐾) + 1)))
8482, 83eqtrd 2115 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐹‘(𝑛 + 1)) = (𝐺‘((𝑛 + 𝐾) + 1)))
8584oveq2d 5607 . . . . . . . . . . . 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))))
8665, 75, 853eqtr4d 2125 . . . . . . . . . . 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))))
8760, 86eqeq12d 2097 . . . . . . . . . 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)))))
8856, 87syl5ibr 154 . . . . . . . . 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))))
8988expr 367 . . . . . . . 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)))))
9089a2d 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)))))
9155, 90syld 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)))))
9291expcom 114 . . . . 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))))))
9392a2d 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))))))
9410, 17, 24, 31, 51, 93uzind4 8971 . . 3 (𝑁 ∈ (ℤ𝑀) → (𝜑 → (𝑁 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑁 + 𝐾)))))
951, 94mpcom 36 . 2 (𝜑 → (𝑁 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑁 + 𝐾))))
963, 95mpd 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 102   = wceq 1285  wcel 1434  wral 2353  cfv 4969  (class class class)co 5591  cc 7251  1c1 7254   + caddc 7256  cz 8646  cuz 8914  ...cfz 9319  seqcseq 9740
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 4000  ax-un 4224  ax-setind 4316  ax-iinf 4366  ax-cnex 7339  ax-resscn 7340  ax-1cn 7341  ax-1re 7342  ax-icn 7343  ax-addcl 7344  ax-addrcl 7345  ax-mulcl 7346  ax-addcom 7348  ax-addass 7350  ax-distr 7352  ax-i2m1 7353  ax-0lt1 7354  ax-0id 7356  ax-rnegex 7357  ax-cnre 7359  ax-pre-ltirr 7360  ax-pre-ltwlin 7361  ax-pre-lttrn 7362  ax-pre-ltadd 7364
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 4084  df-iord 4157  df-on 4159  df-ilim 4160  df-suc 4162  df-iom 4369  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-fv 4977  df-riota 5547  df-ov 5594  df-oprab 5595  df-mpt2 5596  df-1st 5846  df-2nd 5847  df-recs 6002  df-frec 6088  df-pnf 7427  df-mnf 7428  df-xr 7429  df-ltxr 7430  df-le 7431  df-sub 7558  df-neg 7559  df-inn 8317  df-n0 8566  df-z 8647  df-uz 8915  df-fz 9320  df-iseq 9741
This theorem is referenced by: (None)
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