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Theorem iseqcaopr3 9556
 Description: Lemma for iseqcaopr2 . (Contributed by Jim Kingdon, 16-Aug-2021.)
Hypotheses
Ref Expression
iseqcaopr3.1 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
iseqcaopr3.2 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)
iseqcaopr3.3 (𝜑𝑁 ∈ (ℤ𝑀))
iseqcaopr3.4 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) ∈ 𝑆)
iseqcaopr3.5 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐺𝑘) ∈ 𝑆)
iseqcaopr3.6 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐻𝑘) = ((𝐹𝑘)𝑄(𝐺𝑘)))
iseqcaopr3.7 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑛)) + ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1)))) = (((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄((seq𝑀( + , 𝐺, 𝑆)‘𝑛) + (𝐺‘(𝑛 + 1)))))
iseqcaopr3.s (𝜑𝑆𝑉)
Assertion
Ref Expression
iseqcaopr3 (𝜑 → (seq𝑀( + , 𝐻, 𝑆)‘𝑁) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑁)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑁)))
Distinct variable groups:   + ,𝑛,𝑥,𝑦   𝑘,𝐹,𝑛,𝑥,𝑦   𝑘,𝐺,𝑛,𝑥,𝑦   𝑘,𝐻,𝑛,𝑥,𝑦   𝑘,𝑀,𝑛,𝑥,𝑦   𝑘,𝑁,𝑛,𝑥,𝑦   𝑄,𝑘,𝑛,𝑥,𝑦   𝑆,𝑘,𝑛,𝑥,𝑦   𝜑,𝑘,𝑛,𝑥,𝑦
Allowed substitution hints:   + (𝑘)   𝑉(𝑥,𝑦,𝑘,𝑛)

Proof of Theorem iseqcaopr3
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 iseqcaopr3.3 . . 3 (𝜑𝑁 ∈ (ℤ𝑀))
2 eluzfz2 9127 . . 3 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
31, 2syl 14 . 2 (𝜑𝑁 ∈ (𝑀...𝑁))
4 fveq2 5209 . . . . 5 (𝑧 = 𝑀 → (seq𝑀( + , 𝐻, 𝑆)‘𝑧) = (seq𝑀( + , 𝐻, 𝑆)‘𝑀))
5 fveq2 5209 . . . . . 6 (𝑧 = 𝑀 → (seq𝑀( + , 𝐹, 𝑆)‘𝑧) = (seq𝑀( + , 𝐹, 𝑆)‘𝑀))
6 fveq2 5209 . . . . . 6 (𝑧 = 𝑀 → (seq𝑀( + , 𝐺, 𝑆)‘𝑧) = (seq𝑀( + , 𝐺, 𝑆)‘𝑀))
75, 6oveq12d 5561 . . . . 5 (𝑧 = 𝑀 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑧)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑧)) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑀)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑀)))
84, 7eqeq12d 2096 . . . 4 (𝑧 = 𝑀 → ((seq𝑀( + , 𝐻, 𝑆)‘𝑧) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑧)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑧)) ↔ (seq𝑀( + , 𝐻, 𝑆)‘𝑀) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑀)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑀))))
98imbi2d 228 . . 3 (𝑧 = 𝑀 → ((𝜑 → (seq𝑀( + , 𝐻, 𝑆)‘𝑧) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑧)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑧))) ↔ (𝜑 → (seq𝑀( + , 𝐻, 𝑆)‘𝑀) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑀)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑀)))))
10 fveq2 5209 . . . . 5 (𝑧 = 𝑛 → (seq𝑀( + , 𝐻, 𝑆)‘𝑧) = (seq𝑀( + , 𝐻, 𝑆)‘𝑛))
11 fveq2 5209 . . . . . 6 (𝑧 = 𝑛 → (seq𝑀( + , 𝐹, 𝑆)‘𝑧) = (seq𝑀( + , 𝐹, 𝑆)‘𝑛))
12 fveq2 5209 . . . . . 6 (𝑧 = 𝑛 → (seq𝑀( + , 𝐺, 𝑆)‘𝑧) = (seq𝑀( + , 𝐺, 𝑆)‘𝑛))
1311, 12oveq12d 5561 . . . . 5 (𝑧 = 𝑛 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑧)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑧)) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑛)))
1410, 13eqeq12d 2096 . . . 4 (𝑧 = 𝑛 → ((seq𝑀( + , 𝐻, 𝑆)‘𝑧) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑧)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑧)) ↔ (seq𝑀( + , 𝐻, 𝑆)‘𝑛) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑛))))
1514imbi2d 228 . . 3 (𝑧 = 𝑛 → ((𝜑 → (seq𝑀( + , 𝐻, 𝑆)‘𝑧) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑧)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑧))) ↔ (𝜑 → (seq𝑀( + , 𝐻, 𝑆)‘𝑛) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑛)))))
16 fveq2 5209 . . . . 5 (𝑧 = (𝑛 + 1) → (seq𝑀( + , 𝐻, 𝑆)‘𝑧) = (seq𝑀( + , 𝐻, 𝑆)‘(𝑛 + 1)))
17 fveq2 5209 . . . . . 6 (𝑧 = (𝑛 + 1) → (seq𝑀( + , 𝐹, 𝑆)‘𝑧) = (seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1)))
18 fveq2 5209 . . . . . 6 (𝑧 = (𝑛 + 1) → (seq𝑀( + , 𝐺, 𝑆)‘𝑧) = (seq𝑀( + , 𝐺, 𝑆)‘(𝑛 + 1)))
1917, 18oveq12d 5561 . . . . 5 (𝑧 = (𝑛 + 1) → ((seq𝑀( + , 𝐹, 𝑆)‘𝑧)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑧)) = ((seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1))𝑄(seq𝑀( + , 𝐺, 𝑆)‘(𝑛 + 1))))
2016, 19eqeq12d 2096 . . . 4 (𝑧 = (𝑛 + 1) → ((seq𝑀( + , 𝐻, 𝑆)‘𝑧) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑧)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑧)) ↔ (seq𝑀( + , 𝐻, 𝑆)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1))𝑄(seq𝑀( + , 𝐺, 𝑆)‘(𝑛 + 1)))))
2120imbi2d 228 . . 3 (𝑧 = (𝑛 + 1) → ((𝜑 → (seq𝑀( + , 𝐻, 𝑆)‘𝑧) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑧)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑧))) ↔ (𝜑 → (seq𝑀( + , 𝐻, 𝑆)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1))𝑄(seq𝑀( + , 𝐺, 𝑆)‘(𝑛 + 1))))))
22 fveq2 5209 . . . . 5 (𝑧 = 𝑁 → (seq𝑀( + , 𝐻, 𝑆)‘𝑧) = (seq𝑀( + , 𝐻, 𝑆)‘𝑁))
23 fveq2 5209 . . . . . 6 (𝑧 = 𝑁 → (seq𝑀( + , 𝐹, 𝑆)‘𝑧) = (seq𝑀( + , 𝐹, 𝑆)‘𝑁))
24 fveq2 5209 . . . . . 6 (𝑧 = 𝑁 → (seq𝑀( + , 𝐺, 𝑆)‘𝑧) = (seq𝑀( + , 𝐺, 𝑆)‘𝑁))
2523, 24oveq12d 5561 . . . . 5 (𝑧 = 𝑁 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑧)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑧)) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑁)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑁)))
2622, 25eqeq12d 2096 . . . 4 (𝑧 = 𝑁 → ((seq𝑀( + , 𝐻, 𝑆)‘𝑧) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑧)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑧)) ↔ (seq𝑀( + , 𝐻, 𝑆)‘𝑁) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑁)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑁))))
2726imbi2d 228 . . 3 (𝑧 = 𝑁 → ((𝜑 → (seq𝑀( + , 𝐻, 𝑆)‘𝑧) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑧)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑧))) ↔ (𝜑 → (seq𝑀( + , 𝐻, 𝑆)‘𝑁) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑁)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑁)))))
28 fveq2 5209 . . . . . . 7 (𝑘 = 𝑀 → (𝐻𝑘) = (𝐻𝑀))
29 fveq2 5209 . . . . . . . 8 (𝑘 = 𝑀 → (𝐹𝑘) = (𝐹𝑀))
30 fveq2 5209 . . . . . . . 8 (𝑘 = 𝑀 → (𝐺𝑘) = (𝐺𝑀))
3129, 30oveq12d 5561 . . . . . . 7 (𝑘 = 𝑀 → ((𝐹𝑘)𝑄(𝐺𝑘)) = ((𝐹𝑀)𝑄(𝐺𝑀)))
3228, 31eqeq12d 2096 . . . . . 6 (𝑘 = 𝑀 → ((𝐻𝑘) = ((𝐹𝑘)𝑄(𝐺𝑘)) ↔ (𝐻𝑀) = ((𝐹𝑀)𝑄(𝐺𝑀))))
33 iseqcaopr3.6 . . . . . . 7 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐻𝑘) = ((𝐹𝑘)𝑄(𝐺𝑘)))
3433ralrimiva 2435 . . . . . 6 (𝜑 → ∀𝑘 ∈ (ℤ𝑀)(𝐻𝑘) = ((𝐹𝑘)𝑄(𝐺𝑘)))
35 eluzel2 8705 . . . . . . . 8 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
361, 35syl 14 . . . . . . 7 (𝜑𝑀 ∈ ℤ)
37 uzid 8714 . . . . . . 7 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
3836, 37syl 14 . . . . . 6 (𝜑𝑀 ∈ (ℤ𝑀))
3932, 34, 38rspcdva 2708 . . . . 5 (𝜑 → (𝐻𝑀) = ((𝐹𝑀)𝑄(𝐺𝑀)))
40 iseqcaopr3.2 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)
4140ralrimivva 2444 . . . . . . . . . . 11 (𝜑 → ∀𝑥𝑆𝑦𝑆 (𝑥𝑄𝑦) ∈ 𝑆)
4241adantr 270 . . . . . . . . . 10 ((𝜑𝑘 ∈ (ℤ𝑀)) → ∀𝑥𝑆𝑦𝑆 (𝑥𝑄𝑦) ∈ 𝑆)
43 iseqcaopr3.4 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) ∈ 𝑆)
44 iseqcaopr3.5 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐺𝑘) ∈ 𝑆)
45 oveq1 5550 . . . . . . . . . . . . 13 (𝑥 = (𝐹𝑘) → (𝑥𝑄𝑦) = ((𝐹𝑘)𝑄𝑦))
4645eleq1d 2148 . . . . . . . . . . . 12 (𝑥 = (𝐹𝑘) → ((𝑥𝑄𝑦) ∈ 𝑆 ↔ ((𝐹𝑘)𝑄𝑦) ∈ 𝑆))
47 oveq2 5551 . . . . . . . . . . . . 13 (𝑦 = (𝐺𝑘) → ((𝐹𝑘)𝑄𝑦) = ((𝐹𝑘)𝑄(𝐺𝑘)))
4847eleq1d 2148 . . . . . . . . . . . 12 (𝑦 = (𝐺𝑘) → (((𝐹𝑘)𝑄𝑦) ∈ 𝑆 ↔ ((𝐹𝑘)𝑄(𝐺𝑘)) ∈ 𝑆))
4946, 48rspc2v 2714 . . . . . . . . . . 11 (((𝐹𝑘) ∈ 𝑆 ∧ (𝐺𝑘) ∈ 𝑆) → (∀𝑥𝑆𝑦𝑆 (𝑥𝑄𝑦) ∈ 𝑆 → ((𝐹𝑘)𝑄(𝐺𝑘)) ∈ 𝑆))
5043, 44, 49syl2anc 403 . . . . . . . . . 10 ((𝜑𝑘 ∈ (ℤ𝑀)) → (∀𝑥𝑆𝑦𝑆 (𝑥𝑄𝑦) ∈ 𝑆 → ((𝐹𝑘)𝑄(𝐺𝑘)) ∈ 𝑆))
5142, 50mpd 13 . . . . . . . . 9 ((𝜑𝑘 ∈ (ℤ𝑀)) → ((𝐹𝑘)𝑄(𝐺𝑘)) ∈ 𝑆)
5233, 51eqeltrd 2156 . . . . . . . 8 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐻𝑘) ∈ 𝑆)
5352ralrimiva 2435 . . . . . . 7 (𝜑 → ∀𝑘 ∈ (ℤ𝑀)(𝐻𝑘) ∈ 𝑆)
54 fveq2 5209 . . . . . . . . 9 (𝑘 = 𝑥 → (𝐻𝑘) = (𝐻𝑥))
5554eleq1d 2148 . . . . . . . 8 (𝑘 = 𝑥 → ((𝐻𝑘) ∈ 𝑆 ↔ (𝐻𝑥) ∈ 𝑆))
5655rspcv 2698 . . . . . . 7 (𝑥 ∈ (ℤ𝑀) → (∀𝑘 ∈ (ℤ𝑀)(𝐻𝑘) ∈ 𝑆 → (𝐻𝑥) ∈ 𝑆))
5753, 56mpan9 275 . . . . . 6 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐻𝑥) ∈ 𝑆)
58 iseqcaopr3.1 . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
5936, 57, 58iseq1 9533 . . . . 5 (𝜑 → (seq𝑀( + , 𝐻, 𝑆)‘𝑀) = (𝐻𝑀))
6043ralrimiva 2435 . . . . . . . 8 (𝜑 → ∀𝑘 ∈ (ℤ𝑀)(𝐹𝑘) ∈ 𝑆)
61 fveq2 5209 . . . . . . . . . 10 (𝑘 = 𝑥 → (𝐹𝑘) = (𝐹𝑥))
6261eleq1d 2148 . . . . . . . . 9 (𝑘 = 𝑥 → ((𝐹𝑘) ∈ 𝑆 ↔ (𝐹𝑥) ∈ 𝑆))
6362rspcv 2698 . . . . . . . 8 (𝑥 ∈ (ℤ𝑀) → (∀𝑘 ∈ (ℤ𝑀)(𝐹𝑘) ∈ 𝑆 → (𝐹𝑥) ∈ 𝑆))
6460, 63mpan9 275 . . . . . . 7 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
6536, 64, 58iseq1 9533 . . . . . 6 (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (𝐹𝑀))
6644ralrimiva 2435 . . . . . . . 8 (𝜑 → ∀𝑘 ∈ (ℤ𝑀)(𝐺𝑘) ∈ 𝑆)
67 fveq2 5209 . . . . . . . . . 10 (𝑘 = 𝑥 → (𝐺𝑘) = (𝐺𝑥))
6867eleq1d 2148 . . . . . . . . 9 (𝑘 = 𝑥 → ((𝐺𝑘) ∈ 𝑆 ↔ (𝐺𝑥) ∈ 𝑆))
6968rspcv 2698 . . . . . . . 8 (𝑥 ∈ (ℤ𝑀) → (∀𝑘 ∈ (ℤ𝑀)(𝐺𝑘) ∈ 𝑆 → (𝐺𝑥) ∈ 𝑆))
7066, 69mpan9 275 . . . . . . 7 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)
7136, 70, 58iseq1 9533 . . . . . 6 (𝜑 → (seq𝑀( + , 𝐺, 𝑆)‘𝑀) = (𝐺𝑀))
7265, 71oveq12d 5561 . . . . 5 (𝜑 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑀)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑀)) = ((𝐹𝑀)𝑄(𝐺𝑀)))
7339, 59, 723eqtr4d 2124 . . . 4 (𝜑 → (seq𝑀( + , 𝐻, 𝑆)‘𝑀) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑀)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑀)))
7473a1i 9 . . 3 (𝑁 ∈ (ℤ𝑀) → (𝜑 → (seq𝑀( + , 𝐻, 𝑆)‘𝑀) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑀)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑀))))
75 oveq1 5550 . . . . . 6 ((seq𝑀( + , 𝐻, 𝑆)‘𝑛) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑛)) → ((seq𝑀( + , 𝐻, 𝑆)‘𝑛) + (𝐻‘(𝑛 + 1))) = (((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑛)) + (𝐻‘(𝑛 + 1))))
76 elfzouz 9238 . . . . . . . . 9 (𝑛 ∈ (𝑀..^𝑁) → 𝑛 ∈ (ℤ𝑀))
7776adantl 271 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → 𝑛 ∈ (ℤ𝑀))
7857adantlr 461 . . . . . . . 8 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐻𝑥) ∈ 𝑆)
7958adantlr 461 . . . . . . . 8 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
8077, 78, 79iseqp1 9538 . . . . . . 7 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( + , 𝐻, 𝑆)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐻, 𝑆)‘𝑛) + (𝐻‘(𝑛 + 1))))
81 iseqcaopr3.7 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑛)) + ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1)))) = (((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄((seq𝑀( + , 𝐺, 𝑆)‘𝑛) + (𝐺‘(𝑛 + 1)))))
82 fveq2 5209 . . . . . . . . . . 11 (𝑘 = (𝑛 + 1) → (𝐻𝑘) = (𝐻‘(𝑛 + 1)))
83 fveq2 5209 . . . . . . . . . . . 12 (𝑘 = (𝑛 + 1) → (𝐹𝑘) = (𝐹‘(𝑛 + 1)))
84 fveq2 5209 . . . . . . . . . . . 12 (𝑘 = (𝑛 + 1) → (𝐺𝑘) = (𝐺‘(𝑛 + 1)))
8583, 84oveq12d 5561 . . . . . . . . . . 11 (𝑘 = (𝑛 + 1) → ((𝐹𝑘)𝑄(𝐺𝑘)) = ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1))))
8682, 85eqeq12d 2096 . . . . . . . . . 10 (𝑘 = (𝑛 + 1) → ((𝐻𝑘) = ((𝐹𝑘)𝑄(𝐺𝑘)) ↔ (𝐻‘(𝑛 + 1)) = ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1)))))
8734adantr 270 . . . . . . . . . 10 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ∀𝑘 ∈ (ℤ𝑀)(𝐻𝑘) = ((𝐹𝑘)𝑄(𝐺𝑘)))
88 fzofzp1 9313 . . . . . . . . . . . 12 (𝑛 ∈ (𝑀..^𝑁) → (𝑛 + 1) ∈ (𝑀...𝑁))
89 elfzuz 9117 . . . . . . . . . . . 12 ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝑛 + 1) ∈ (ℤ𝑀))
9088, 89syl 14 . . . . . . . . . . 11 (𝑛 ∈ (𝑀..^𝑁) → (𝑛 + 1) ∈ (ℤ𝑀))
9190adantl 271 . . . . . . . . . 10 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝑛 + 1) ∈ (ℤ𝑀))
9286, 87, 91rspcdva 2708 . . . . . . . . 9 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝐻‘(𝑛 + 1)) = ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1))))
9392oveq2d 5559 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑛)) + (𝐻‘(𝑛 + 1))) = (((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑛)) + ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1)))))
9464adantlr 461 . . . . . . . . . 10 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
9577, 94, 79iseqp1 9538 . . . . . . . . 9 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1))))
9670adantlr 461 . . . . . . . . . 10 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)
9777, 96, 79iseqp1 9538 . . . . . . . . 9 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( + , 𝐺, 𝑆)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐺, 𝑆)‘𝑛) + (𝐺‘(𝑛 + 1))))
9895, 97oveq12d 5561 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ((seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1))𝑄(seq𝑀( + , 𝐺, 𝑆)‘(𝑛 + 1))) = (((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄((seq𝑀( + , 𝐺, 𝑆)‘𝑛) + (𝐺‘(𝑛 + 1)))))
9981, 93, 983eqtr4rd 2125 . . . . . . 7 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ((seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1))𝑄(seq𝑀( + , 𝐺, 𝑆)‘(𝑛 + 1))) = (((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑛)) + (𝐻‘(𝑛 + 1))))
10080, 99eqeq12d 2096 . . . . . 6 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ((seq𝑀( + , 𝐻, 𝑆)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1))𝑄(seq𝑀( + , 𝐺, 𝑆)‘(𝑛 + 1))) ↔ ((seq𝑀( + , 𝐻, 𝑆)‘𝑛) + (𝐻‘(𝑛 + 1))) = (((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑛)) + (𝐻‘(𝑛 + 1)))))
10175, 100syl5ibr 154 . . . . 5 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ((seq𝑀( + , 𝐻, 𝑆)‘𝑛) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑛)) → (seq𝑀( + , 𝐻, 𝑆)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1))𝑄(seq𝑀( + , 𝐺, 𝑆)‘(𝑛 + 1)))))
102101expcom 114 . . . 4 (𝑛 ∈ (𝑀..^𝑁) → (𝜑 → ((seq𝑀( + , 𝐻, 𝑆)‘𝑛) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑛)) → (seq𝑀( + , 𝐻, 𝑆)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1))𝑄(seq𝑀( + , 𝐺, 𝑆)‘(𝑛 + 1))))))
103102a2d 26 . . 3 (𝑛 ∈ (𝑀..^𝑁) → ((𝜑 → (seq𝑀( + , 𝐻, 𝑆)‘𝑛) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑛))) → (𝜑 → (seq𝑀( + , 𝐻, 𝑆)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1))𝑄(seq𝑀( + , 𝐺, 𝑆)‘(𝑛 + 1))))))
1049, 15, 21, 27, 74, 103fzind2 9325 . 2 (𝑁 ∈ (𝑀...𝑁) → (𝜑 → (seq𝑀( + , 𝐻, 𝑆)‘𝑁) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑁)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑁))))
1053, 104mpcom 36 1 (𝜑 → (seq𝑀( + , 𝐻, 𝑆)‘𝑁) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑁)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑁)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 102   = wceq 1285   ∈ wcel 1434  ∀wral 2349  ‘cfv 4932  (class class class)co 5543  1c1 7044   + caddc 7046  ℤcz 8432  ℤ≥cuz 8700  ...cfz 9105  ..^cfzo 9229  seqcseq 9521 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 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-addcom 7138  ax-addass 7140  ax-distr 7142  ax-i2m1 7143  ax-0lt1 7144  ax-0id 7146  ax-rnegex 7147  ax-cnre 7149  ax-pre-ltirr 7150  ax-pre-ltwlin 7151  ax-pre-lttrn 7152  ax-pre-ltadd 7154 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 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-id 4056  df-iord 4129  df-on 4131  df-ilim 4132  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-frec 6040  df-pnf 7217  df-mnf 7218  df-xr 7219  df-ltxr 7220  df-le 7221  df-sub 7348  df-neg 7349  df-inn 8107  df-n0 8356  df-z 8433  df-uz 8701  df-fz 9106  df-fzo 9230  df-iseq 9522 This theorem is referenced by:  iseqcaopr2  9557
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