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Theorem iseqcaopr3 9214
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 8894 . . 3 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
31, 2syl 14 . 2 (𝜑𝑁 ∈ (𝑀...𝑁))
4 fveq2 5178 . . . . 5 (𝑧 = 𝑀 → (seq𝑀( + , 𝐻, 𝑆)‘𝑧) = (seq𝑀( + , 𝐻, 𝑆)‘𝑀))
5 fveq2 5178 . . . . . 6 (𝑧 = 𝑀 → (seq𝑀( + , 𝐹, 𝑆)‘𝑧) = (seq𝑀( + , 𝐹, 𝑆)‘𝑀))
6 fveq2 5178 . . . . . 6 (𝑧 = 𝑀 → (seq𝑀( + , 𝐺, 𝑆)‘𝑧) = (seq𝑀( + , 𝐺, 𝑆)‘𝑀))
75, 6oveq12d 5530 . . . . 5 (𝑧 = 𝑀 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑧)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑧)) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑀)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑀)))
84, 7eqeq12d 2054 . . . 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 219 . . 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 5178 . . . . 5 (𝑧 = 𝑛 → (seq𝑀( + , 𝐻, 𝑆)‘𝑧) = (seq𝑀( + , 𝐻, 𝑆)‘𝑛))
11 fveq2 5178 . . . . . 6 (𝑧 = 𝑛 → (seq𝑀( + , 𝐹, 𝑆)‘𝑧) = (seq𝑀( + , 𝐹, 𝑆)‘𝑛))
12 fveq2 5178 . . . . . 6 (𝑧 = 𝑛 → (seq𝑀( + , 𝐺, 𝑆)‘𝑧) = (seq𝑀( + , 𝐺, 𝑆)‘𝑛))
1311, 12oveq12d 5530 . . . . 5 (𝑧 = 𝑛 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑧)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑧)) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑛)))
1410, 13eqeq12d 2054 . . . 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 219 . . 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 5178 . . . . 5 (𝑧 = (𝑛 + 1) → (seq𝑀( + , 𝐻, 𝑆)‘𝑧) = (seq𝑀( + , 𝐻, 𝑆)‘(𝑛 + 1)))
17 fveq2 5178 . . . . . 6 (𝑧 = (𝑛 + 1) → (seq𝑀( + , 𝐹, 𝑆)‘𝑧) = (seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1)))
18 fveq2 5178 . . . . . 6 (𝑧 = (𝑛 + 1) → (seq𝑀( + , 𝐺, 𝑆)‘𝑧) = (seq𝑀( + , 𝐺, 𝑆)‘(𝑛 + 1)))
1917, 18oveq12d 5530 . . . . 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 2054 . . . 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 219 . . 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 5178 . . . . 5 (𝑧 = 𝑁 → (seq𝑀( + , 𝐻, 𝑆)‘𝑧) = (seq𝑀( + , 𝐻, 𝑆)‘𝑁))
23 fveq2 5178 . . . . . 6 (𝑧 = 𝑁 → (seq𝑀( + , 𝐹, 𝑆)‘𝑧) = (seq𝑀( + , 𝐹, 𝑆)‘𝑁))
24 fveq2 5178 . . . . . 6 (𝑧 = 𝑁 → (seq𝑀( + , 𝐺, 𝑆)‘𝑧) = (seq𝑀( + , 𝐺, 𝑆)‘𝑁))
2523, 24oveq12d 5530 . . . . 5 (𝑧 = 𝑁 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑧)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑧)) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑁)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑁)))
2622, 25eqeq12d 2054 . . . 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 219 . . 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 eluzel2 8476 . . . . . . . 8 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
291, 28syl 14 . . . . . . 7 (𝜑𝑀 ∈ ℤ)
30 uzid 8485 . . . . . . 7 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
3129, 30syl 14 . . . . . 6 (𝜑𝑀 ∈ (ℤ𝑀))
32 iseqcaopr3.6 . . . . . . 7 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐻𝑘) = ((𝐹𝑘)𝑄(𝐺𝑘)))
3332ralrimiva 2392 . . . . . 6 (𝜑 → ∀𝑘 ∈ (ℤ𝑀)(𝐻𝑘) = ((𝐹𝑘)𝑄(𝐺𝑘)))
34 fveq2 5178 . . . . . . . 8 (𝑘 = 𝑀 → (𝐻𝑘) = (𝐻𝑀))
35 fveq2 5178 . . . . . . . . 9 (𝑘 = 𝑀 → (𝐹𝑘) = (𝐹𝑀))
36 fveq2 5178 . . . . . . . . 9 (𝑘 = 𝑀 → (𝐺𝑘) = (𝐺𝑀))
3735, 36oveq12d 5530 . . . . . . . 8 (𝑘 = 𝑀 → ((𝐹𝑘)𝑄(𝐺𝑘)) = ((𝐹𝑀)𝑄(𝐺𝑀)))
3834, 37eqeq12d 2054 . . . . . . 7 (𝑘 = 𝑀 → ((𝐻𝑘) = ((𝐹𝑘)𝑄(𝐺𝑘)) ↔ (𝐻𝑀) = ((𝐹𝑀)𝑄(𝐺𝑀))))
3938rspcv 2652 . . . . . 6 (𝑀 ∈ (ℤ𝑀) → (∀𝑘 ∈ (ℤ𝑀)(𝐻𝑘) = ((𝐹𝑘)𝑄(𝐺𝑘)) → (𝐻𝑀) = ((𝐹𝑀)𝑄(𝐺𝑀))))
4031, 33, 39sylc 56 . . . . 5 (𝜑 → (𝐻𝑀) = ((𝐹𝑀)𝑄(𝐺𝑀)))
41 iseqcaopr3.s . . . . . 6 (𝜑𝑆𝑉)
42 iseqcaopr3.2 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)
4342ralrimivva 2401 . . . . . . . . . . 11 (𝜑 → ∀𝑥𝑆𝑦𝑆 (𝑥𝑄𝑦) ∈ 𝑆)
4443adantr 261 . . . . . . . . . 10 ((𝜑𝑘 ∈ (ℤ𝑀)) → ∀𝑥𝑆𝑦𝑆 (𝑥𝑄𝑦) ∈ 𝑆)
45 iseqcaopr3.4 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) ∈ 𝑆)
46 iseqcaopr3.5 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐺𝑘) ∈ 𝑆)
47 oveq1 5519 . . . . . . . . . . . . 13 (𝑥 = (𝐹𝑘) → (𝑥𝑄𝑦) = ((𝐹𝑘)𝑄𝑦))
4847eleq1d 2106 . . . . . . . . . . . 12 (𝑥 = (𝐹𝑘) → ((𝑥𝑄𝑦) ∈ 𝑆 ↔ ((𝐹𝑘)𝑄𝑦) ∈ 𝑆))
49 oveq2 5520 . . . . . . . . . . . . 13 (𝑦 = (𝐺𝑘) → ((𝐹𝑘)𝑄𝑦) = ((𝐹𝑘)𝑄(𝐺𝑘)))
5049eleq1d 2106 . . . . . . . . . . . 12 (𝑦 = (𝐺𝑘) → (((𝐹𝑘)𝑄𝑦) ∈ 𝑆 ↔ ((𝐹𝑘)𝑄(𝐺𝑘)) ∈ 𝑆))
5148, 50rspc2v 2662 . . . . . . . . . . 11 (((𝐹𝑘) ∈ 𝑆 ∧ (𝐺𝑘) ∈ 𝑆) → (∀𝑥𝑆𝑦𝑆 (𝑥𝑄𝑦) ∈ 𝑆 → ((𝐹𝑘)𝑄(𝐺𝑘)) ∈ 𝑆))
5245, 46, 51syl2anc 391 . . . . . . . . . 10 ((𝜑𝑘 ∈ (ℤ𝑀)) → (∀𝑥𝑆𝑦𝑆 (𝑥𝑄𝑦) ∈ 𝑆 → ((𝐹𝑘)𝑄(𝐺𝑘)) ∈ 𝑆))
5344, 52mpd 13 . . . . . . . . 9 ((𝜑𝑘 ∈ (ℤ𝑀)) → ((𝐹𝑘)𝑄(𝐺𝑘)) ∈ 𝑆)
5432, 53eqeltrd 2114 . . . . . . . 8 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐻𝑘) ∈ 𝑆)
5554ralrimiva 2392 . . . . . . 7 (𝜑 → ∀𝑘 ∈ (ℤ𝑀)(𝐻𝑘) ∈ 𝑆)
56 fveq2 5178 . . . . . . . . 9 (𝑘 = 𝑥 → (𝐻𝑘) = (𝐻𝑥))
5756eleq1d 2106 . . . . . . . 8 (𝑘 = 𝑥 → ((𝐻𝑘) ∈ 𝑆 ↔ (𝐻𝑥) ∈ 𝑆))
5857rspcv 2652 . . . . . . 7 (𝑥 ∈ (ℤ𝑀) → (∀𝑘 ∈ (ℤ𝑀)(𝐻𝑘) ∈ 𝑆 → (𝐻𝑥) ∈ 𝑆))
5955, 58mpan9 265 . . . . . 6 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐻𝑥) ∈ 𝑆)
60 iseqcaopr3.1 . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
6129, 41, 59, 60iseq1 9196 . . . . 5 (𝜑 → (seq𝑀( + , 𝐻, 𝑆)‘𝑀) = (𝐻𝑀))
6245ralrimiva 2392 . . . . . . . 8 (𝜑 → ∀𝑘 ∈ (ℤ𝑀)(𝐹𝑘) ∈ 𝑆)
63 fveq2 5178 . . . . . . . . . 10 (𝑘 = 𝑥 → (𝐹𝑘) = (𝐹𝑥))
6463eleq1d 2106 . . . . . . . . 9 (𝑘 = 𝑥 → ((𝐹𝑘) ∈ 𝑆 ↔ (𝐹𝑥) ∈ 𝑆))
6564rspcv 2652 . . . . . . . 8 (𝑥 ∈ (ℤ𝑀) → (∀𝑘 ∈ (ℤ𝑀)(𝐹𝑘) ∈ 𝑆 → (𝐹𝑥) ∈ 𝑆))
6662, 65mpan9 265 . . . . . . 7 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
6729, 41, 66, 60iseq1 9196 . . . . . 6 (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (𝐹𝑀))
6846ralrimiva 2392 . . . . . . . 8 (𝜑 → ∀𝑘 ∈ (ℤ𝑀)(𝐺𝑘) ∈ 𝑆)
69 fveq2 5178 . . . . . . . . . 10 (𝑘 = 𝑥 → (𝐺𝑘) = (𝐺𝑥))
7069eleq1d 2106 . . . . . . . . 9 (𝑘 = 𝑥 → ((𝐺𝑘) ∈ 𝑆 ↔ (𝐺𝑥) ∈ 𝑆))
7170rspcv 2652 . . . . . . . 8 (𝑥 ∈ (ℤ𝑀) → (∀𝑘 ∈ (ℤ𝑀)(𝐺𝑘) ∈ 𝑆 → (𝐺𝑥) ∈ 𝑆))
7268, 71mpan9 265 . . . . . . 7 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)
7329, 41, 72, 60iseq1 9196 . . . . . 6 (𝜑 → (seq𝑀( + , 𝐺, 𝑆)‘𝑀) = (𝐺𝑀))
7467, 73oveq12d 5530 . . . . 5 (𝜑 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑀)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑀)) = ((𝐹𝑀)𝑄(𝐺𝑀)))
7540, 61, 743eqtr4d 2082 . . . 4 (𝜑 → (seq𝑀( + , 𝐻, 𝑆)‘𝑀) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑀)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑀)))
7675a1i 9 . . 3 (𝑁 ∈ (ℤ𝑀) → (𝜑 → (seq𝑀( + , 𝐻, 𝑆)‘𝑀) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑀)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑀))))
77 oveq1 5519 . . . . . 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))))
78 elfzouz 9006 . . . . . . . . 9 (𝑛 ∈ (𝑀..^𝑁) → 𝑛 ∈ (ℤ𝑀))
7978adantl 262 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → 𝑛 ∈ (ℤ𝑀))
8041adantr 261 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → 𝑆𝑉)
8159adantlr 446 . . . . . . . 8 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐻𝑥) ∈ 𝑆)
8260adantlr 446 . . . . . . . 8 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
8379, 80, 81, 82iseqp1 9199 . . . . . . 7 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( + , 𝐻, 𝑆)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐻, 𝑆)‘𝑛) + (𝐻‘(𝑛 + 1))))
84 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)))))
85 fzofzp1 9081 . . . . . . . . . . . 12 (𝑛 ∈ (𝑀..^𝑁) → (𝑛 + 1) ∈ (𝑀...𝑁))
86 elfzuz 8884 . . . . . . . . . . . 12 ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝑛 + 1) ∈ (ℤ𝑀))
8785, 86syl 14 . . . . . . . . . . 11 (𝑛 ∈ (𝑀..^𝑁) → (𝑛 + 1) ∈ (ℤ𝑀))
8887adantl 262 . . . . . . . . . 10 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝑛 + 1) ∈ (ℤ𝑀))
8933adantr 261 . . . . . . . . . 10 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ∀𝑘 ∈ (ℤ𝑀)(𝐻𝑘) = ((𝐹𝑘)𝑄(𝐺𝑘)))
90 fveq2 5178 . . . . . . . . . . . 12 (𝑘 = (𝑛 + 1) → (𝐻𝑘) = (𝐻‘(𝑛 + 1)))
91 fveq2 5178 . . . . . . . . . . . . 13 (𝑘 = (𝑛 + 1) → (𝐹𝑘) = (𝐹‘(𝑛 + 1)))
92 fveq2 5178 . . . . . . . . . . . . 13 (𝑘 = (𝑛 + 1) → (𝐺𝑘) = (𝐺‘(𝑛 + 1)))
9391, 92oveq12d 5530 . . . . . . . . . . . 12 (𝑘 = (𝑛 + 1) → ((𝐹𝑘)𝑄(𝐺𝑘)) = ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1))))
9490, 93eqeq12d 2054 . . . . . . . . . . 11 (𝑘 = (𝑛 + 1) → ((𝐻𝑘) = ((𝐹𝑘)𝑄(𝐺𝑘)) ↔ (𝐻‘(𝑛 + 1)) = ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1)))))
9594rspcv 2652 . . . . . . . . . 10 ((𝑛 + 1) ∈ (ℤ𝑀) → (∀𝑘 ∈ (ℤ𝑀)(𝐻𝑘) = ((𝐹𝑘)𝑄(𝐺𝑘)) → (𝐻‘(𝑛 + 1)) = ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1)))))
9688, 89, 95sylc 56 . . . . . . . . 9 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝐻‘(𝑛 + 1)) = ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1))))
9796oveq2d 5528 . . . . . . . 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)))))
9866adantlr 446 . . . . . . . . . 10 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
9979, 80, 98, 82iseqp1 9199 . . . . . . . . 9 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1))))
10072adantlr 446 . . . . . . . . . 10 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)
10179, 80, 100, 82iseqp1 9199 . . . . . . . . 9 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( + , 𝐺, 𝑆)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐺, 𝑆)‘𝑛) + (𝐺‘(𝑛 + 1))))
10299, 101oveq12d 5530 . . . . . . . 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)))))
10384, 97, 1023eqtr4rd 2083 . . . . . . 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))))
10483, 103eqeq12d 2054 . . . . . 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)))))
10577, 104syl5ibr 145 . . . . 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)))))
106105expcom 109 . . . 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))))))
107106a2d 23 . . 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))))))
1089, 15, 21, 27, 76, 107fzind2 9093 . 2 (𝑁 ∈ (𝑀...𝑁) → (𝜑 → (seq𝑀( + , 𝐻, 𝑆)‘𝑁) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑁)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑁))))
1093, 108mpcom 32 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 97   = wceq 1243  wcel 1393  wral 2306  cfv 4902  (class class class)co 5512  1c1 6888   + caddc 6890  cz 8243  cuz 8471  ...cfz 8872  ..^cfzo 8997  seqcseq 9185
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 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311  ax-cnex 6973  ax-resscn 6974  ax-1cn 6975  ax-1re 6976  ax-icn 6977  ax-addcl 6978  ax-addrcl 6979  ax-mulcl 6980  ax-addcom 6982  ax-addass 6984  ax-distr 6986  ax-i2m1 6987  ax-0id 6990  ax-rnegex 6991  ax-cnre 6993  ax-pre-ltirr 6994  ax-pre-ltwlin 6995  ax-pre-lttrn 6996  ax-pre-ltadd 6998
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 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-frec 5978  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6400  df-pli 6401  df-mi 6402  df-lti 6403  df-plpq 6440  df-mpq 6441  df-enq 6443  df-nqqs 6444  df-plqqs 6445  df-mqqs 6446  df-1nqqs 6447  df-rq 6448  df-ltnqqs 6449  df-enq0 6520  df-nq0 6521  df-0nq0 6522  df-plq0 6523  df-mq0 6524  df-inp 6562  df-i1p 6563  df-iplp 6564  df-iltp 6566  df-enr 6809  df-nr 6810  df-ltr 6813  df-0r 6814  df-1r 6815  df-0 6894  df-1 6895  df-r 6897  df-lt 6900  df-pnf 7060  df-mnf 7061  df-xr 7062  df-ltxr 7063  df-le 7064  df-sub 7182  df-neg 7183  df-inn 7913  df-n0 8180  df-z 8244  df-uz 8472  df-fz 8873  df-fzo 8998  df-iseq 9186
This theorem is referenced by:  iseqcaopr2  9215
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