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Theorem seq3caopr3 10499
Description: Lemma for seq3caopr2 10500. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by Jim Kingdon, 22-Apr-2023.)
Hypotheses
Ref Expression
seqcaopr3.1 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
seqcaopr3.2 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)
seqcaopr3.3 (𝜑𝑁 ∈ (ℤ𝑀))
seq3caopr3.4 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) ∈ 𝑆)
seq3caopr3.5 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐺𝑘) ∈ 𝑆)
seq3caopr3.6 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐻𝑘) = ((𝐹𝑘)𝑄(𝐺𝑘)))
seqcaopr3.7 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛)) + ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1)))) = (((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1)))))
Assertion
Ref Expression
seq3caopr3 (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁)𝑄(seq𝑀( + , 𝐺)‘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 hint:   + (𝑘)

Proof of Theorem seq3caopr3
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 seqcaopr3.3 . . 3 (𝜑𝑁 ∈ (ℤ𝑀))
2 eluzfz2 10050 . . 3 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
31, 2syl 14 . 2 (𝜑𝑁 ∈ (𝑀...𝑁))
4 fveq2 5530 . . . . 5 (𝑧 = 𝑀 → (seq𝑀( + , 𝐻)‘𝑧) = (seq𝑀( + , 𝐻)‘𝑀))
5 fveq2 5530 . . . . . 6 (𝑧 = 𝑀 → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝑀( + , 𝐹)‘𝑀))
6 fveq2 5530 . . . . . 6 (𝑧 = 𝑀 → (seq𝑀( + , 𝐺)‘𝑧) = (seq𝑀( + , 𝐺)‘𝑀))
75, 6oveq12d 5909 . . . . 5 (𝑧 = 𝑀 → ((seq𝑀( + , 𝐹)‘𝑧)𝑄(seq𝑀( + , 𝐺)‘𝑧)) = ((seq𝑀( + , 𝐹)‘𝑀)𝑄(seq𝑀( + , 𝐺)‘𝑀)))
84, 7eqeq12d 2204 . . . 4 (𝑧 = 𝑀 → ((seq𝑀( + , 𝐻)‘𝑧) = ((seq𝑀( + , 𝐹)‘𝑧)𝑄(seq𝑀( + , 𝐺)‘𝑧)) ↔ (seq𝑀( + , 𝐻)‘𝑀) = ((seq𝑀( + , 𝐹)‘𝑀)𝑄(seq𝑀( + , 𝐺)‘𝑀))))
98imbi2d 230 . . 3 (𝑧 = 𝑀 → ((𝜑 → (seq𝑀( + , 𝐻)‘𝑧) = ((seq𝑀( + , 𝐹)‘𝑧)𝑄(seq𝑀( + , 𝐺)‘𝑧))) ↔ (𝜑 → (seq𝑀( + , 𝐻)‘𝑀) = ((seq𝑀( + , 𝐹)‘𝑀)𝑄(seq𝑀( + , 𝐺)‘𝑀)))))
10 fveq2 5530 . . . . 5 (𝑧 = 𝑛 → (seq𝑀( + , 𝐻)‘𝑧) = (seq𝑀( + , 𝐻)‘𝑛))
11 fveq2 5530 . . . . . 6 (𝑧 = 𝑛 → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝑀( + , 𝐹)‘𝑛))
12 fveq2 5530 . . . . . 6 (𝑧 = 𝑛 → (seq𝑀( + , 𝐺)‘𝑧) = (seq𝑀( + , 𝐺)‘𝑛))
1311, 12oveq12d 5909 . . . . 5 (𝑧 = 𝑛 → ((seq𝑀( + , 𝐹)‘𝑧)𝑄(seq𝑀( + , 𝐺)‘𝑧)) = ((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛)))
1410, 13eqeq12d 2204 . . . 4 (𝑧 = 𝑛 → ((seq𝑀( + , 𝐻)‘𝑧) = ((seq𝑀( + , 𝐹)‘𝑧)𝑄(seq𝑀( + , 𝐺)‘𝑧)) ↔ (seq𝑀( + , 𝐻)‘𝑛) = ((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛))))
1514imbi2d 230 . . 3 (𝑧 = 𝑛 → ((𝜑 → (seq𝑀( + , 𝐻)‘𝑧) = ((seq𝑀( + , 𝐹)‘𝑧)𝑄(seq𝑀( + , 𝐺)‘𝑧))) ↔ (𝜑 → (seq𝑀( + , 𝐻)‘𝑛) = ((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛)))))
16 fveq2 5530 . . . . 5 (𝑧 = (𝑛 + 1) → (seq𝑀( + , 𝐻)‘𝑧) = (seq𝑀( + , 𝐻)‘(𝑛 + 1)))
17 fveq2 5530 . . . . . 6 (𝑧 = (𝑛 + 1) → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝑀( + , 𝐹)‘(𝑛 + 1)))
18 fveq2 5530 . . . . . 6 (𝑧 = (𝑛 + 1) → (seq𝑀( + , 𝐺)‘𝑧) = (seq𝑀( + , 𝐺)‘(𝑛 + 1)))
1917, 18oveq12d 5909 . . . . 5 (𝑧 = (𝑛 + 1) → ((seq𝑀( + , 𝐹)‘𝑧)𝑄(seq𝑀( + , 𝐺)‘𝑧)) = ((seq𝑀( + , 𝐹)‘(𝑛 + 1))𝑄(seq𝑀( + , 𝐺)‘(𝑛 + 1))))
2016, 19eqeq12d 2204 . . . 4 (𝑧 = (𝑛 + 1) → ((seq𝑀( + , 𝐻)‘𝑧) = ((seq𝑀( + , 𝐹)‘𝑧)𝑄(seq𝑀( + , 𝐺)‘𝑧)) ↔ (seq𝑀( + , 𝐻)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘(𝑛 + 1))𝑄(seq𝑀( + , 𝐺)‘(𝑛 + 1)))))
2120imbi2d 230 . . 3 (𝑧 = (𝑛 + 1) → ((𝜑 → (seq𝑀( + , 𝐻)‘𝑧) = ((seq𝑀( + , 𝐹)‘𝑧)𝑄(seq𝑀( + , 𝐺)‘𝑧))) ↔ (𝜑 → (seq𝑀( + , 𝐻)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘(𝑛 + 1))𝑄(seq𝑀( + , 𝐺)‘(𝑛 + 1))))))
22 fveq2 5530 . . . . 5 (𝑧 = 𝑁 → (seq𝑀( + , 𝐻)‘𝑧) = (seq𝑀( + , 𝐻)‘𝑁))
23 fveq2 5530 . . . . . 6 (𝑧 = 𝑁 → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝑀( + , 𝐹)‘𝑁))
24 fveq2 5530 . . . . . 6 (𝑧 = 𝑁 → (seq𝑀( + , 𝐺)‘𝑧) = (seq𝑀( + , 𝐺)‘𝑁))
2523, 24oveq12d 5909 . . . . 5 (𝑧 = 𝑁 → ((seq𝑀( + , 𝐹)‘𝑧)𝑄(seq𝑀( + , 𝐺)‘𝑧)) = ((seq𝑀( + , 𝐹)‘𝑁)𝑄(seq𝑀( + , 𝐺)‘𝑁)))
2622, 25eqeq12d 2204 . . . 4 (𝑧 = 𝑁 → ((seq𝑀( + , 𝐻)‘𝑧) = ((seq𝑀( + , 𝐹)‘𝑧)𝑄(seq𝑀( + , 𝐺)‘𝑧)) ↔ (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁)𝑄(seq𝑀( + , 𝐺)‘𝑁))))
2726imbi2d 230 . . 3 (𝑧 = 𝑁 → ((𝜑 → (seq𝑀( + , 𝐻)‘𝑧) = ((seq𝑀( + , 𝐹)‘𝑧)𝑄(seq𝑀( + , 𝐺)‘𝑧))) ↔ (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁)𝑄(seq𝑀( + , 𝐺)‘𝑁)))))
28 fveq2 5530 . . . . . . 7 (𝑘 = 𝑀 → (𝐻𝑘) = (𝐻𝑀))
29 fveq2 5530 . . . . . . . 8 (𝑘 = 𝑀 → (𝐹𝑘) = (𝐹𝑀))
30 fveq2 5530 . . . . . . . 8 (𝑘 = 𝑀 → (𝐺𝑘) = (𝐺𝑀))
3129, 30oveq12d 5909 . . . . . . 7 (𝑘 = 𝑀 → ((𝐹𝑘)𝑄(𝐺𝑘)) = ((𝐹𝑀)𝑄(𝐺𝑀)))
3228, 31eqeq12d 2204 . . . . . 6 (𝑘 = 𝑀 → ((𝐻𝑘) = ((𝐹𝑘)𝑄(𝐺𝑘)) ↔ (𝐻𝑀) = ((𝐹𝑀)𝑄(𝐺𝑀))))
33 seq3caopr3.6 . . . . . . 7 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐻𝑘) = ((𝐹𝑘)𝑄(𝐺𝑘)))
3433ralrimiva 2563 . . . . . 6 (𝜑 → ∀𝑘 ∈ (ℤ𝑀)(𝐻𝑘) = ((𝐹𝑘)𝑄(𝐺𝑘)))
35 eluzel2 9551 . . . . . . . 8 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
361, 35syl 14 . . . . . . 7 (𝜑𝑀 ∈ ℤ)
37 uzid 9560 . . . . . . 7 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
3836, 37syl 14 . . . . . 6 (𝜑𝑀 ∈ (ℤ𝑀))
3932, 34, 38rspcdva 2861 . . . . 5 (𝜑 → (𝐻𝑀) = ((𝐹𝑀)𝑄(𝐺𝑀)))
40 seqcaopr3.2 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)
4140ralrimivva 2572 . . . . . . . . . . 11 (𝜑 → ∀𝑥𝑆𝑦𝑆 (𝑥𝑄𝑦) ∈ 𝑆)
4241adantr 276 . . . . . . . . . 10 ((𝜑𝑘 ∈ (ℤ𝑀)) → ∀𝑥𝑆𝑦𝑆 (𝑥𝑄𝑦) ∈ 𝑆)
43 seq3caopr3.4 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) ∈ 𝑆)
44 seq3caopr3.5 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐺𝑘) ∈ 𝑆)
45 oveq1 5898 . . . . . . . . . . . . 13 (𝑥 = (𝐹𝑘) → (𝑥𝑄𝑦) = ((𝐹𝑘)𝑄𝑦))
4645eleq1d 2258 . . . . . . . . . . . 12 (𝑥 = (𝐹𝑘) → ((𝑥𝑄𝑦) ∈ 𝑆 ↔ ((𝐹𝑘)𝑄𝑦) ∈ 𝑆))
47 oveq2 5899 . . . . . . . . . . . . 13 (𝑦 = (𝐺𝑘) → ((𝐹𝑘)𝑄𝑦) = ((𝐹𝑘)𝑄(𝐺𝑘)))
4847eleq1d 2258 . . . . . . . . . . . 12 (𝑦 = (𝐺𝑘) → (((𝐹𝑘)𝑄𝑦) ∈ 𝑆 ↔ ((𝐹𝑘)𝑄(𝐺𝑘)) ∈ 𝑆))
4946, 48rspc2v 2869 . . . . . . . . . . 11 (((𝐹𝑘) ∈ 𝑆 ∧ (𝐺𝑘) ∈ 𝑆) → (∀𝑥𝑆𝑦𝑆 (𝑥𝑄𝑦) ∈ 𝑆 → ((𝐹𝑘)𝑄(𝐺𝑘)) ∈ 𝑆))
5043, 44, 49syl2anc 411 . . . . . . . . . 10 ((𝜑𝑘 ∈ (ℤ𝑀)) → (∀𝑥𝑆𝑦𝑆 (𝑥𝑄𝑦) ∈ 𝑆 → ((𝐹𝑘)𝑄(𝐺𝑘)) ∈ 𝑆))
5142, 50mpd 13 . . . . . . . . 9 ((𝜑𝑘 ∈ (ℤ𝑀)) → ((𝐹𝑘)𝑄(𝐺𝑘)) ∈ 𝑆)
5233, 51eqeltrd 2266 . . . . . . . 8 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐻𝑘) ∈ 𝑆)
5352ralrimiva 2563 . . . . . . 7 (𝜑 → ∀𝑘 ∈ (ℤ𝑀)(𝐻𝑘) ∈ 𝑆)
54 fveq2 5530 . . . . . . . . 9 (𝑘 = 𝑥 → (𝐻𝑘) = (𝐻𝑥))
5554eleq1d 2258 . . . . . . . 8 (𝑘 = 𝑥 → ((𝐻𝑘) ∈ 𝑆 ↔ (𝐻𝑥) ∈ 𝑆))
5655rspcv 2852 . . . . . . 7 (𝑥 ∈ (ℤ𝑀) → (∀𝑘 ∈ (ℤ𝑀)(𝐻𝑘) ∈ 𝑆 → (𝐻𝑥) ∈ 𝑆))
5753, 56mpan9 281 . . . . . 6 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐻𝑥) ∈ 𝑆)
58 seqcaopr3.1 . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
5936, 57, 58seq3-1 10478 . . . . 5 (𝜑 → (seq𝑀( + , 𝐻)‘𝑀) = (𝐻𝑀))
6043ralrimiva 2563 . . . . . . . 8 (𝜑 → ∀𝑘 ∈ (ℤ𝑀)(𝐹𝑘) ∈ 𝑆)
61 fveq2 5530 . . . . . . . . . 10 (𝑘 = 𝑥 → (𝐹𝑘) = (𝐹𝑥))
6261eleq1d 2258 . . . . . . . . 9 (𝑘 = 𝑥 → ((𝐹𝑘) ∈ 𝑆 ↔ (𝐹𝑥) ∈ 𝑆))
6362rspcv 2852 . . . . . . . 8 (𝑥 ∈ (ℤ𝑀) → (∀𝑘 ∈ (ℤ𝑀)(𝐹𝑘) ∈ 𝑆 → (𝐹𝑥) ∈ 𝑆))
6460, 63mpan9 281 . . . . . . 7 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
6536, 64, 58seq3-1 10478 . . . . . 6 (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹𝑀))
6644ralrimiva 2563 . . . . . . . 8 (𝜑 → ∀𝑘 ∈ (ℤ𝑀)(𝐺𝑘) ∈ 𝑆)
67 fveq2 5530 . . . . . . . . . 10 (𝑘 = 𝑥 → (𝐺𝑘) = (𝐺𝑥))
6867eleq1d 2258 . . . . . . . . 9 (𝑘 = 𝑥 → ((𝐺𝑘) ∈ 𝑆 ↔ (𝐺𝑥) ∈ 𝑆))
6968rspcv 2852 . . . . . . . 8 (𝑥 ∈ (ℤ𝑀) → (∀𝑘 ∈ (ℤ𝑀)(𝐺𝑘) ∈ 𝑆 → (𝐺𝑥) ∈ 𝑆))
7066, 69mpan9 281 . . . . . . 7 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)
7136, 70, 58seq3-1 10478 . . . . . 6 (𝜑 → (seq𝑀( + , 𝐺)‘𝑀) = (𝐺𝑀))
7265, 71oveq12d 5909 . . . . 5 (𝜑 → ((seq𝑀( + , 𝐹)‘𝑀)𝑄(seq𝑀( + , 𝐺)‘𝑀)) = ((𝐹𝑀)𝑄(𝐺𝑀)))
7339, 59, 723eqtr4d 2232 . . . 4 (𝜑 → (seq𝑀( + , 𝐻)‘𝑀) = ((seq𝑀( + , 𝐹)‘𝑀)𝑄(seq𝑀( + , 𝐺)‘𝑀)))
7473a1i 9 . . 3 (𝑁 ∈ (ℤ𝑀) → (𝜑 → (seq𝑀( + , 𝐻)‘𝑀) = ((seq𝑀( + , 𝐹)‘𝑀)𝑄(seq𝑀( + , 𝐺)‘𝑀))))
75 oveq1 5898 . . . . . 6 ((seq𝑀( + , 𝐻)‘𝑛) = ((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛)) → ((seq𝑀( + , 𝐻)‘𝑛) + (𝐻‘(𝑛 + 1))) = (((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛)) + (𝐻‘(𝑛 + 1))))
76 elfzouz 10169 . . . . . . . . 9 (𝑛 ∈ (𝑀..^𝑁) → 𝑛 ∈ (ℤ𝑀))
7776adantl 277 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → 𝑛 ∈ (ℤ𝑀))
7857adantlr 477 . . . . . . . 8 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐻𝑥) ∈ 𝑆)
7958adantlr 477 . . . . . . . 8 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
8077, 78, 79seq3p1 10480 . . . . . . 7 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( + , 𝐻)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐻)‘𝑛) + (𝐻‘(𝑛 + 1))))
81 seqcaopr3.7 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛)) + ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1)))) = (((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1)))))
82 fveq2 5530 . . . . . . . . . . 11 (𝑘 = (𝑛 + 1) → (𝐻𝑘) = (𝐻‘(𝑛 + 1)))
83 fveq2 5530 . . . . . . . . . . . 12 (𝑘 = (𝑛 + 1) → (𝐹𝑘) = (𝐹‘(𝑛 + 1)))
84 fveq2 5530 . . . . . . . . . . . 12 (𝑘 = (𝑛 + 1) → (𝐺𝑘) = (𝐺‘(𝑛 + 1)))
8583, 84oveq12d 5909 . . . . . . . . . . 11 (𝑘 = (𝑛 + 1) → ((𝐹𝑘)𝑄(𝐺𝑘)) = ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1))))
8682, 85eqeq12d 2204 . . . . . . . . . 10 (𝑘 = (𝑛 + 1) → ((𝐻𝑘) = ((𝐹𝑘)𝑄(𝐺𝑘)) ↔ (𝐻‘(𝑛 + 1)) = ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1)))))
8734adantr 276 . . . . . . . . . 10 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ∀𝑘 ∈ (ℤ𝑀)(𝐻𝑘) = ((𝐹𝑘)𝑄(𝐺𝑘)))
88 fzofzp1 10245 . . . . . . . . . . . 12 (𝑛 ∈ (𝑀..^𝑁) → (𝑛 + 1) ∈ (𝑀...𝑁))
89 elfzuz 10039 . . . . . . . . . . . 12 ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝑛 + 1) ∈ (ℤ𝑀))
9088, 89syl 14 . . . . . . . . . . 11 (𝑛 ∈ (𝑀..^𝑁) → (𝑛 + 1) ∈ (ℤ𝑀))
9190adantl 277 . . . . . . . . . 10 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝑛 + 1) ∈ (ℤ𝑀))
9286, 87, 91rspcdva 2861 . . . . . . . . 9 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝐻‘(𝑛 + 1)) = ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1))))
9392oveq2d 5907 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛)) + (𝐻‘(𝑛 + 1))) = (((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛)) + ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1)))))
9464adantlr 477 . . . . . . . . . 10 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
9577, 94, 79seq3p1 10480 . . . . . . . . 9 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
9670adantlr 477 . . . . . . . . . 10 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)
9777, 96, 79seq3p1 10480 . . . . . . . . 9 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( + , 𝐺)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1))))
9895, 97oveq12d 5909 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ((seq𝑀( + , 𝐹)‘(𝑛 + 1))𝑄(seq𝑀( + , 𝐺)‘(𝑛 + 1))) = (((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1)))))
9981, 93, 983eqtr4rd 2233 . . . . . . 7 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ((seq𝑀( + , 𝐹)‘(𝑛 + 1))𝑄(seq𝑀( + , 𝐺)‘(𝑛 + 1))) = (((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛)) + (𝐻‘(𝑛 + 1))))
10080, 99eqeq12d 2204 . . . . . 6 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ((seq𝑀( + , 𝐻)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘(𝑛 + 1))𝑄(seq𝑀( + , 𝐺)‘(𝑛 + 1))) ↔ ((seq𝑀( + , 𝐻)‘𝑛) + (𝐻‘(𝑛 + 1))) = (((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛)) + (𝐻‘(𝑛 + 1)))))
10175, 100imbitrrid 156 . . . . 5 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ((seq𝑀( + , 𝐻)‘𝑛) = ((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛)) → (seq𝑀( + , 𝐻)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘(𝑛 + 1))𝑄(seq𝑀( + , 𝐺)‘(𝑛 + 1)))))
102101expcom 116 . . . 4 (𝑛 ∈ (𝑀..^𝑁) → (𝜑 → ((seq𝑀( + , 𝐻)‘𝑛) = ((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛)) → (seq𝑀( + , 𝐻)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘(𝑛 + 1))𝑄(seq𝑀( + , 𝐺)‘(𝑛 + 1))))))
103102a2d 26 . . 3 (𝑛 ∈ (𝑀..^𝑁) → ((𝜑 → (seq𝑀( + , 𝐻)‘𝑛) = ((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛))) → (𝜑 → (seq𝑀( + , 𝐻)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘(𝑛 + 1))𝑄(seq𝑀( + , 𝐺)‘(𝑛 + 1))))))
1049, 15, 21, 27, 74, 103fzind2 10257 . 2 (𝑁 ∈ (𝑀...𝑁) → (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁)𝑄(seq𝑀( + , 𝐺)‘𝑁))))
1053, 104mpcom 36 1 (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁)𝑄(seq𝑀( + , 𝐺)‘𝑁)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1364  wcel 2160  wral 2468  cfv 5231  (class class class)co 5891  1c1 7830   + caddc 7832  cz 9271  cuz 9546  ...cfz 10026  ..^cfzo 10160  seqcseq 10463
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602  ax-cnex 7920  ax-resscn 7921  ax-1cn 7922  ax-1re 7923  ax-icn 7924  ax-addcl 7925  ax-addrcl 7926  ax-mulcl 7927  ax-addcom 7929  ax-addass 7931  ax-distr 7933  ax-i2m1 7934  ax-0lt1 7935  ax-0id 7937  ax-rnegex 7938  ax-cnre 7940  ax-pre-ltirr 7941  ax-pre-ltwlin 7942  ax-pre-lttrn 7943  ax-pre-ltadd 7945
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-iord 4381  df-on 4383  df-ilim 4384  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-recs 6324  df-frec 6410  df-pnf 8012  df-mnf 8013  df-xr 8014  df-ltxr 8015  df-le 8016  df-sub 8148  df-neg 8149  df-inn 8938  df-n0 9195  df-z 9272  df-uz 9547  df-fz 10027  df-fzo 10161  df-seqfrec 10464
This theorem is referenced by:  seq3caopr2  10500
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