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Theorem seq3shft2 10239
 Description: Shifting the index set of a sequence. (Contributed by Jim Kingdon, 15-Aug-2021.) (Revised by Jim Kingdon, 7-Apr-2023.)
Hypotheses
Ref Expression
seq3shft2.1 (𝜑𝑁 ∈ (ℤ𝑀))
seq3shft2.2 (𝜑𝐾 ∈ ℤ)
seq3shft2.3 ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) = (𝐺‘(𝑘 + 𝐾)))
seq3shft2.f ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
seq3shft2.g ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 𝐾))) → (𝐺𝑥) ∈ 𝑆)
seq3shft2.pl ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
Assertion
Ref Expression
seq3shft2 (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(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)

Proof of Theorem seq3shft2
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 seq3shft2.1 . . 3 (𝜑𝑁 ∈ (ℤ𝑀))
2 eluzfz2 9805 . . 3 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
31, 2syl 14 . 2 (𝜑𝑁 ∈ (𝑀...𝑁))
4 eleq1 2200 . . . . . 6 (𝑥 = 𝑀 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑀 ∈ (𝑀...𝑁)))
5 fveq2 5414 . . . . . . 7 (𝑥 = 𝑀 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝑀))
6 fvoveq1 5790 . . . . . . 7 (𝑥 = 𝑀 → (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑥 + 𝐾)) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑀 + 𝐾)))
75, 6eqeq12d 2152 . . . . . 6 (𝑥 = 𝑀 → ((seq𝑀( + , 𝐹)‘𝑥) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑥 + 𝐾)) ↔ (seq𝑀( + , 𝐹)‘𝑀) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑀 + 𝐾))))
84, 7imbi12d 233 . . . . 5 (𝑥 = 𝑀 → ((𝑥 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑥) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑥 + 𝐾))) ↔ (𝑀 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑀) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑀 + 𝐾)))))
98imbi2d 229 . . . 4 (𝑥 = 𝑀 → ((𝜑 → (𝑥 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑥) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑥 + 𝐾)))) ↔ (𝜑 → (𝑀 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑀) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑀 + 𝐾))))))
10 eleq1 2200 . . . . . 6 (𝑥 = 𝑛 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑛 ∈ (𝑀...𝑁)))
11 fveq2 5414 . . . . . . 7 (𝑥 = 𝑛 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝑛))
12 fvoveq1 5790 . . . . . . 7 (𝑥 = 𝑛 → (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑥 + 𝐾)) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑛 + 𝐾)))
1311, 12eqeq12d 2152 . . . . . 6 (𝑥 = 𝑛 → ((seq𝑀( + , 𝐹)‘𝑥) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑥 + 𝐾)) ↔ (seq𝑀( + , 𝐹)‘𝑛) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑛 + 𝐾))))
1410, 13imbi12d 233 . . . . 5 (𝑥 = 𝑛 → ((𝑥 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑥) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑥 + 𝐾))) ↔ (𝑛 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑛) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑛 + 𝐾)))))
1514imbi2d 229 . . . 4 (𝑥 = 𝑛 → ((𝜑 → (𝑥 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑥) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑥 + 𝐾)))) ↔ (𝜑 → (𝑛 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑛) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑛 + 𝐾))))))
16 eleq1 2200 . . . . . 6 (𝑥 = (𝑛 + 1) → (𝑥 ∈ (𝑀...𝑁) ↔ (𝑛 + 1) ∈ (𝑀...𝑁)))
17 fveq2 5414 . . . . . . 7 (𝑥 = (𝑛 + 1) → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘(𝑛 + 1)))
18 fvoveq1 5790 . . . . . . 7 (𝑥 = (𝑛 + 1) → (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑥 + 𝐾)) = (seq(𝑀 + 𝐾)( + , 𝐺)‘((𝑛 + 1) + 𝐾)))
1917, 18eqeq12d 2152 . . . . . 6 (𝑥 = (𝑛 + 1) → ((seq𝑀( + , 𝐹)‘𝑥) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑥 + 𝐾)) ↔ (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = (seq(𝑀 + 𝐾)( + , 𝐺)‘((𝑛 + 1) + 𝐾))))
2016, 19imbi12d 233 . . . . 5 (𝑥 = (𝑛 + 1) → ((𝑥 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑥) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑥 + 𝐾))) ↔ ((𝑛 + 1) ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = (seq(𝑀 + 𝐾)( + , 𝐺)‘((𝑛 + 1) + 𝐾)))))
2120imbi2d 229 . . . 4 (𝑥 = (𝑛 + 1) → ((𝜑 → (𝑥 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑥) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑥 + 𝐾)))) ↔ (𝜑 → ((𝑛 + 1) ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = (seq(𝑀 + 𝐾)( + , 𝐺)‘((𝑛 + 1) + 𝐾))))))
22 eleq1 2200 . . . . . 6 (𝑥 = 𝑁 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑁 ∈ (𝑀...𝑁)))
23 fveq2 5414 . . . . . . 7 (𝑥 = 𝑁 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝑁))
24 fvoveq1 5790 . . . . . . 7 (𝑥 = 𝑁 → (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑥 + 𝐾)) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑁 + 𝐾)))
2523, 24eqeq12d 2152 . . . . . 6 (𝑥 = 𝑁 → ((seq𝑀( + , 𝐹)‘𝑥) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑥 + 𝐾)) ↔ (seq𝑀( + , 𝐹)‘𝑁) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑁 + 𝐾))))
2622, 25imbi12d 233 . . . . 5 (𝑥 = 𝑁 → ((𝑥 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑥) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑥 + 𝐾))) ↔ (𝑁 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑁) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑁 + 𝐾)))))
2726imbi2d 229 . . . 4 (𝑥 = 𝑁 → ((𝜑 → (𝑥 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑥) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑥 + 𝐾)))) ↔ (𝜑 → (𝑁 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑁) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑁 + 𝐾))))))
28 fveq2 5414 . . . . . . . 8 (𝑘 = 𝑀 → (𝐹𝑘) = (𝐹𝑀))
29 fvoveq1 5790 . . . . . . . 8 (𝑘 = 𝑀 → (𝐺‘(𝑘 + 𝐾)) = (𝐺‘(𝑀 + 𝐾)))
3028, 29eqeq12d 2152 . . . . . . 7 (𝑘 = 𝑀 → ((𝐹𝑘) = (𝐺‘(𝑘 + 𝐾)) ↔ (𝐹𝑀) = (𝐺‘(𝑀 + 𝐾))))
31 seq3shft2.3 . . . . . . . 8 ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) = (𝐺‘(𝑘 + 𝐾)))
3231ralrimiva 2503 . . . . . . 7 (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)(𝐹𝑘) = (𝐺‘(𝑘 + 𝐾)))
33 eluzfz1 9804 . . . . . . . 8 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ (𝑀...𝑁))
341, 33syl 14 . . . . . . 7 (𝜑𝑀 ∈ (𝑀...𝑁))
3530, 32, 34rspcdva 2789 . . . . . 6 (𝜑 → (𝐹𝑀) = (𝐺‘(𝑀 + 𝐾)))
36 eluzel2 9324 . . . . . . . 8 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
371, 36syl 14 . . . . . . 7 (𝜑𝑀 ∈ ℤ)
38 seq3shft2.f . . . . . . 7 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
39 seq3shft2.pl . . . . . . 7 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
4037, 38, 39seq3-1 10226 . . . . . 6 (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹𝑀))
41 seq3shft2.2 . . . . . . . 8 (𝜑𝐾 ∈ ℤ)
4237, 41zaddcld 9170 . . . . . . 7 (𝜑 → (𝑀 + 𝐾) ∈ ℤ)
43 seq3shft2.g . . . . . . 7 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 𝐾))) → (𝐺𝑥) ∈ 𝑆)
4442, 43, 39seq3-1 10226 . . . . . 6 (𝜑 → (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑀 + 𝐾)) = (𝐺‘(𝑀 + 𝐾)))
4535, 40, 443eqtr4d 2180 . . . . 5 (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑀 + 𝐾)))
4645a1i13 24 . . . 4 (𝑀 ∈ ℤ → (𝜑 → (𝑀 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑀) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑀 + 𝐾)))))
47 peano2fzr 9810 . . . . . . . 8 ((𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → 𝑛 ∈ (𝑀...𝑁))
4847adantl 275 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ (𝑀...𝑁))
4948expr 372 . . . . . 6 ((𝜑𝑛 ∈ (ℤ𝑀)) → ((𝑛 + 1) ∈ (𝑀...𝑁) → 𝑛 ∈ (𝑀...𝑁)))
5049imim1d 75 . . . . 5 ((𝜑𝑛 ∈ (ℤ𝑀)) → ((𝑛 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑛) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑛 + 𝐾))) → ((𝑛 + 1) ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑛) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑛 + 𝐾)))))
51 oveq1 5774 . . . . . 6 ((seq𝑀( + , 𝐹)‘𝑛) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑛 + 𝐾)) → ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))) = ((seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑛 + 𝐾)) + (𝐹‘(𝑛 + 1))))
52 simprl 520 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ (ℤ𝑀))
5338adantlr 468 . . . . . . . 8 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
5439adantlr 468 . . . . . . . 8 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
5552, 53, 54seq3p1 10228 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
5641adantr 274 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝐾 ∈ ℤ)
57 eluzadd 9347 . . . . . . . . . 10 ((𝑛 ∈ (ℤ𝑀) ∧ 𝐾 ∈ ℤ) → (𝑛 + 𝐾) ∈ (ℤ‘(𝑀 + 𝐾)))
5852, 56, 57syl2anc 408 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝑛 + 𝐾) ∈ (ℤ‘(𝑀 + 𝐾)))
5943adantlr 468 . . . . . . . . 9 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) ∧ 𝑥 ∈ (ℤ‘(𝑀 + 𝐾))) → (𝐺𝑥) ∈ 𝑆)
6058, 59, 54seq3p1 10228 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (seq(𝑀 + 𝐾)( + , 𝐺)‘((𝑛 + 𝐾) + 1)) = ((seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑛 + 𝐾)) + (𝐺‘((𝑛 + 𝐾) + 1))))
61 eluzelz 9328 . . . . . . . . . . . 12 (𝑛 ∈ (ℤ𝑀) → 𝑛 ∈ ℤ)
6252, 61syl 14 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ ℤ)
6362zcnd 9167 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ ℂ)
64 1cnd 7775 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 1 ∈ ℂ)
6556zcnd 9167 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝐾 ∈ ℂ)
6663, 64, 65add32d 7923 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ((𝑛 + 1) + 𝐾) = ((𝑛 + 𝐾) + 1))
6766fveq2d 5418 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (seq(𝑀 + 𝐾)( + , 𝐺)‘((𝑛 + 1) + 𝐾)) = (seq(𝑀 + 𝐾)( + , 𝐺)‘((𝑛 + 𝐾) + 1)))
68 fveq2 5414 . . . . . . . . . . . 12 (𝑘 = (𝑛 + 1) → (𝐹𝑘) = (𝐹‘(𝑛 + 1)))
69 fvoveq1 5790 . . . . . . . . . . . 12 (𝑘 = (𝑛 + 1) → (𝐺‘(𝑘 + 𝐾)) = (𝐺‘((𝑛 + 1) + 𝐾)))
7068, 69eqeq12d 2152 . . . . . . . . . . 11 (𝑘 = (𝑛 + 1) → ((𝐹𝑘) = (𝐺‘(𝑘 + 𝐾)) ↔ (𝐹‘(𝑛 + 1)) = (𝐺‘((𝑛 + 1) + 𝐾))))
7132adantr 274 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ∀𝑘 ∈ (𝑀...𝑁)(𝐹𝑘) = (𝐺‘(𝑘 + 𝐾)))
72 simprr 521 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝑛 + 1) ∈ (𝑀...𝑁))
7370, 71, 72rspcdva 2789 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐹‘(𝑛 + 1)) = (𝐺‘((𝑛 + 1) + 𝐾)))
7466fveq2d 5418 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐺‘((𝑛 + 1) + 𝐾)) = (𝐺‘((𝑛 + 𝐾) + 1)))
7573, 74eqtrd 2170 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐹‘(𝑛 + 1)) = (𝐺‘((𝑛 + 𝐾) + 1)))
7675oveq2d 5783 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ((seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑛 + 𝐾)) + (𝐹‘(𝑛 + 1))) = ((seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑛 + 𝐾)) + (𝐺‘((𝑛 + 𝐾) + 1))))
7760, 67, 763eqtr4d 2180 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (seq(𝑀 + 𝐾)( + , 𝐺)‘((𝑛 + 1) + 𝐾)) = ((seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑛 + 𝐾)) + (𝐹‘(𝑛 + 1))))
7855, 77eqeq12d 2152 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ((seq𝑀( + , 𝐹)‘(𝑛 + 1)) = (seq(𝑀 + 𝐾)( + , 𝐺)‘((𝑛 + 1) + 𝐾)) ↔ ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))) = ((seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑛 + 𝐾)) + (𝐹‘(𝑛 + 1)))))
7951, 78syl5ibr 155 . . . . 5 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ((seq𝑀( + , 𝐹)‘𝑛) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑛 + 𝐾)) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = (seq(𝑀 + 𝐾)( + , 𝐺)‘((𝑛 + 1) + 𝐾))))
8050, 79animpimp2impd 548 . . . 4 (𝑛 ∈ (ℤ𝑀) → ((𝜑 → (𝑛 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑛) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑛 + 𝐾)))) → (𝜑 → ((𝑛 + 1) ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = (seq(𝑀 + 𝐾)( + , 𝐺)‘((𝑛 + 1) + 𝐾))))))
819, 15, 21, 27, 46, 80uzind4 9376 . . 3 (𝑁 ∈ (ℤ𝑀) → (𝜑 → (𝑁 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑁) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑁 + 𝐾)))))
821, 81mpcom 36 . 2 (𝜑 → (𝑁 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑁) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑁 + 𝐾))))
833, 82mpd 13 1 (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑁 + 𝐾)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1331   ∈ wcel 1480  ∀wral 2414  ‘cfv 5118  (class class class)co 5767  1c1 7614   + caddc 7616  ℤcz 9047  ℤ≥cuz 9319  ...cfz 9783  seqcseq 10211 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-addcom 7713  ax-addass 7715  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-0id 7721  ax-rnegex 7722  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-ltadd 7729 This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-iord 4283  df-on 4285  df-ilim 4286  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-frec 6281  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-inn 8714  df-n0 8971  df-z 9048  df-uz 9320  df-fz 9784  df-seqfrec 10212 This theorem is referenced by:  seq3f1olemqsumkj  10264  seq3shft  10603
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