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Theorem seq3caopr3 10095
Description: Lemma for seq3caopr2 10096. (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 9653 . . 3 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
31, 2syl 14 . 2 (𝜑𝑁 ∈ (𝑀...𝑁))
4 fveq2 5353 . . . . 5 (𝑧 = 𝑀 → (seq𝑀( + , 𝐻)‘𝑧) = (seq𝑀( + , 𝐻)‘𝑀))
5 fveq2 5353 . . . . . 6 (𝑧 = 𝑀 → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝑀( + , 𝐹)‘𝑀))
6 fveq2 5353 . . . . . 6 (𝑧 = 𝑀 → (seq𝑀( + , 𝐺)‘𝑧) = (seq𝑀( + , 𝐺)‘𝑀))
75, 6oveq12d 5724 . . . . 5 (𝑧 = 𝑀 → ((seq𝑀( + , 𝐹)‘𝑧)𝑄(seq𝑀( + , 𝐺)‘𝑧)) = ((seq𝑀( + , 𝐹)‘𝑀)𝑄(seq𝑀( + , 𝐺)‘𝑀)))
84, 7eqeq12d 2114 . . . 4 (𝑧 = 𝑀 → ((seq𝑀( + , 𝐻)‘𝑧) = ((seq𝑀( + , 𝐹)‘𝑧)𝑄(seq𝑀( + , 𝐺)‘𝑧)) ↔ (seq𝑀( + , 𝐻)‘𝑀) = ((seq𝑀( + , 𝐹)‘𝑀)𝑄(seq𝑀( + , 𝐺)‘𝑀))))
98imbi2d 229 . . 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 5353 . . . . 5 (𝑧 = 𝑛 → (seq𝑀( + , 𝐻)‘𝑧) = (seq𝑀( + , 𝐻)‘𝑛))
11 fveq2 5353 . . . . . 6 (𝑧 = 𝑛 → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝑀( + , 𝐹)‘𝑛))
12 fveq2 5353 . . . . . 6 (𝑧 = 𝑛 → (seq𝑀( + , 𝐺)‘𝑧) = (seq𝑀( + , 𝐺)‘𝑛))
1311, 12oveq12d 5724 . . . . 5 (𝑧 = 𝑛 → ((seq𝑀( + , 𝐹)‘𝑧)𝑄(seq𝑀( + , 𝐺)‘𝑧)) = ((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛)))
1410, 13eqeq12d 2114 . . . 4 (𝑧 = 𝑛 → ((seq𝑀( + , 𝐻)‘𝑧) = ((seq𝑀( + , 𝐹)‘𝑧)𝑄(seq𝑀( + , 𝐺)‘𝑧)) ↔ (seq𝑀( + , 𝐻)‘𝑛) = ((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛))))
1514imbi2d 229 . . 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 5353 . . . . 5 (𝑧 = (𝑛 + 1) → (seq𝑀( + , 𝐻)‘𝑧) = (seq𝑀( + , 𝐻)‘(𝑛 + 1)))
17 fveq2 5353 . . . . . 6 (𝑧 = (𝑛 + 1) → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝑀( + , 𝐹)‘(𝑛 + 1)))
18 fveq2 5353 . . . . . 6 (𝑧 = (𝑛 + 1) → (seq𝑀( + , 𝐺)‘𝑧) = (seq𝑀( + , 𝐺)‘(𝑛 + 1)))
1917, 18oveq12d 5724 . . . . 5 (𝑧 = (𝑛 + 1) → ((seq𝑀( + , 𝐹)‘𝑧)𝑄(seq𝑀( + , 𝐺)‘𝑧)) = ((seq𝑀( + , 𝐹)‘(𝑛 + 1))𝑄(seq𝑀( + , 𝐺)‘(𝑛 + 1))))
2016, 19eqeq12d 2114 . . . 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 229 . . 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 5353 . . . . 5 (𝑧 = 𝑁 → (seq𝑀( + , 𝐻)‘𝑧) = (seq𝑀( + , 𝐻)‘𝑁))
23 fveq2 5353 . . . . . 6 (𝑧 = 𝑁 → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝑀( + , 𝐹)‘𝑁))
24 fveq2 5353 . . . . . 6 (𝑧 = 𝑁 → (seq𝑀( + , 𝐺)‘𝑧) = (seq𝑀( + , 𝐺)‘𝑁))
2523, 24oveq12d 5724 . . . . 5 (𝑧 = 𝑁 → ((seq𝑀( + , 𝐹)‘𝑧)𝑄(seq𝑀( + , 𝐺)‘𝑧)) = ((seq𝑀( + , 𝐹)‘𝑁)𝑄(seq𝑀( + , 𝐺)‘𝑁)))
2622, 25eqeq12d 2114 . . . 4 (𝑧 = 𝑁 → ((seq𝑀( + , 𝐻)‘𝑧) = ((seq𝑀( + , 𝐹)‘𝑧)𝑄(seq𝑀( + , 𝐺)‘𝑧)) ↔ (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁)𝑄(seq𝑀( + , 𝐺)‘𝑁))))
2726imbi2d 229 . . 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 5353 . . . . . . 7 (𝑘 = 𝑀 → (𝐻𝑘) = (𝐻𝑀))
29 fveq2 5353 . . . . . . . 8 (𝑘 = 𝑀 → (𝐹𝑘) = (𝐹𝑀))
30 fveq2 5353 . . . . . . . 8 (𝑘 = 𝑀 → (𝐺𝑘) = (𝐺𝑀))
3129, 30oveq12d 5724 . . . . . . 7 (𝑘 = 𝑀 → ((𝐹𝑘)𝑄(𝐺𝑘)) = ((𝐹𝑀)𝑄(𝐺𝑀)))
3228, 31eqeq12d 2114 . . . . . 6 (𝑘 = 𝑀 → ((𝐻𝑘) = ((𝐹𝑘)𝑄(𝐺𝑘)) ↔ (𝐻𝑀) = ((𝐹𝑀)𝑄(𝐺𝑀))))
33 seq3caopr3.6 . . . . . . 7 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐻𝑘) = ((𝐹𝑘)𝑄(𝐺𝑘)))
3433ralrimiva 2464 . . . . . 6 (𝜑 → ∀𝑘 ∈ (ℤ𝑀)(𝐻𝑘) = ((𝐹𝑘)𝑄(𝐺𝑘)))
35 eluzel2 9181 . . . . . . . 8 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
361, 35syl 14 . . . . . . 7 (𝜑𝑀 ∈ ℤ)
37 uzid 9190 . . . . . . 7 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
3836, 37syl 14 . . . . . 6 (𝜑𝑀 ∈ (ℤ𝑀))
3932, 34, 38rspcdva 2749 . . . . 5 (𝜑 → (𝐻𝑀) = ((𝐹𝑀)𝑄(𝐺𝑀)))
40 seqcaopr3.2 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)
4140ralrimivva 2473 . . . . . . . . . . 11 (𝜑 → ∀𝑥𝑆𝑦𝑆 (𝑥𝑄𝑦) ∈ 𝑆)
4241adantr 272 . . . . . . . . . 10 ((𝜑𝑘 ∈ (ℤ𝑀)) → ∀𝑥𝑆𝑦𝑆 (𝑥𝑄𝑦) ∈ 𝑆)
43 seq3caopr3.4 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) ∈ 𝑆)
44 seq3caopr3.5 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐺𝑘) ∈ 𝑆)
45 oveq1 5713 . . . . . . . . . . . . 13 (𝑥 = (𝐹𝑘) → (𝑥𝑄𝑦) = ((𝐹𝑘)𝑄𝑦))
4645eleq1d 2168 . . . . . . . . . . . 12 (𝑥 = (𝐹𝑘) → ((𝑥𝑄𝑦) ∈ 𝑆 ↔ ((𝐹𝑘)𝑄𝑦) ∈ 𝑆))
47 oveq2 5714 . . . . . . . . . . . . 13 (𝑦 = (𝐺𝑘) → ((𝐹𝑘)𝑄𝑦) = ((𝐹𝑘)𝑄(𝐺𝑘)))
4847eleq1d 2168 . . . . . . . . . . . 12 (𝑦 = (𝐺𝑘) → (((𝐹𝑘)𝑄𝑦) ∈ 𝑆 ↔ ((𝐹𝑘)𝑄(𝐺𝑘)) ∈ 𝑆))
4946, 48rspc2v 2756 . . . . . . . . . . 11 (((𝐹𝑘) ∈ 𝑆 ∧ (𝐺𝑘) ∈ 𝑆) → (∀𝑥𝑆𝑦𝑆 (𝑥𝑄𝑦) ∈ 𝑆 → ((𝐹𝑘)𝑄(𝐺𝑘)) ∈ 𝑆))
5043, 44, 49syl2anc 406 . . . . . . . . . 10 ((𝜑𝑘 ∈ (ℤ𝑀)) → (∀𝑥𝑆𝑦𝑆 (𝑥𝑄𝑦) ∈ 𝑆 → ((𝐹𝑘)𝑄(𝐺𝑘)) ∈ 𝑆))
5142, 50mpd 13 . . . . . . . . 9 ((𝜑𝑘 ∈ (ℤ𝑀)) → ((𝐹𝑘)𝑄(𝐺𝑘)) ∈ 𝑆)
5233, 51eqeltrd 2176 . . . . . . . 8 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐻𝑘) ∈ 𝑆)
5352ralrimiva 2464 . . . . . . 7 (𝜑 → ∀𝑘 ∈ (ℤ𝑀)(𝐻𝑘) ∈ 𝑆)
54 fveq2 5353 . . . . . . . . 9 (𝑘 = 𝑥 → (𝐻𝑘) = (𝐻𝑥))
5554eleq1d 2168 . . . . . . . 8 (𝑘 = 𝑥 → ((𝐻𝑘) ∈ 𝑆 ↔ (𝐻𝑥) ∈ 𝑆))
5655rspcv 2740 . . . . . . 7 (𝑥 ∈ (ℤ𝑀) → (∀𝑘 ∈ (ℤ𝑀)(𝐻𝑘) ∈ 𝑆 → (𝐻𝑥) ∈ 𝑆))
5753, 56mpan9 277 . . . . . 6 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐻𝑥) ∈ 𝑆)
58 seqcaopr3.1 . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
5936, 57, 58seq3-1 10074 . . . . 5 (𝜑 → (seq𝑀( + , 𝐻)‘𝑀) = (𝐻𝑀))
6043ralrimiva 2464 . . . . . . . 8 (𝜑 → ∀𝑘 ∈ (ℤ𝑀)(𝐹𝑘) ∈ 𝑆)
61 fveq2 5353 . . . . . . . . . 10 (𝑘 = 𝑥 → (𝐹𝑘) = (𝐹𝑥))
6261eleq1d 2168 . . . . . . . . 9 (𝑘 = 𝑥 → ((𝐹𝑘) ∈ 𝑆 ↔ (𝐹𝑥) ∈ 𝑆))
6362rspcv 2740 . . . . . . . 8 (𝑥 ∈ (ℤ𝑀) → (∀𝑘 ∈ (ℤ𝑀)(𝐹𝑘) ∈ 𝑆 → (𝐹𝑥) ∈ 𝑆))
6460, 63mpan9 277 . . . . . . 7 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
6536, 64, 58seq3-1 10074 . . . . . 6 (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹𝑀))
6644ralrimiva 2464 . . . . . . . 8 (𝜑 → ∀𝑘 ∈ (ℤ𝑀)(𝐺𝑘) ∈ 𝑆)
67 fveq2 5353 . . . . . . . . . 10 (𝑘 = 𝑥 → (𝐺𝑘) = (𝐺𝑥))
6867eleq1d 2168 . . . . . . . . 9 (𝑘 = 𝑥 → ((𝐺𝑘) ∈ 𝑆 ↔ (𝐺𝑥) ∈ 𝑆))
6968rspcv 2740 . . . . . . . 8 (𝑥 ∈ (ℤ𝑀) → (∀𝑘 ∈ (ℤ𝑀)(𝐺𝑘) ∈ 𝑆 → (𝐺𝑥) ∈ 𝑆))
7066, 69mpan9 277 . . . . . . 7 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)
7136, 70, 58seq3-1 10074 . . . . . 6 (𝜑 → (seq𝑀( + , 𝐺)‘𝑀) = (𝐺𝑀))
7265, 71oveq12d 5724 . . . . 5 (𝜑 → ((seq𝑀( + , 𝐹)‘𝑀)𝑄(seq𝑀( + , 𝐺)‘𝑀)) = ((𝐹𝑀)𝑄(𝐺𝑀)))
7339, 59, 723eqtr4d 2142 . . . 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 5713 . . . . . 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 9769 . . . . . . . . 9 (𝑛 ∈ (𝑀..^𝑁) → 𝑛 ∈ (ℤ𝑀))
7776adantl 273 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → 𝑛 ∈ (ℤ𝑀))
7857adantlr 464 . . . . . . . 8 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐻𝑥) ∈ 𝑆)
7958adantlr 464 . . . . . . . 8 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
8077, 78, 79seq3p1 10076 . . . . . . 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 5353 . . . . . . . . . . 11 (𝑘 = (𝑛 + 1) → (𝐻𝑘) = (𝐻‘(𝑛 + 1)))
83 fveq2 5353 . . . . . . . . . . . 12 (𝑘 = (𝑛 + 1) → (𝐹𝑘) = (𝐹‘(𝑛 + 1)))
84 fveq2 5353 . . . . . . . . . . . 12 (𝑘 = (𝑛 + 1) → (𝐺𝑘) = (𝐺‘(𝑛 + 1)))
8583, 84oveq12d 5724 . . . . . . . . . . 11 (𝑘 = (𝑛 + 1) → ((𝐹𝑘)𝑄(𝐺𝑘)) = ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1))))
8682, 85eqeq12d 2114 . . . . . . . . . 10 (𝑘 = (𝑛 + 1) → ((𝐻𝑘) = ((𝐹𝑘)𝑄(𝐺𝑘)) ↔ (𝐻‘(𝑛 + 1)) = ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1)))))
8734adantr 272 . . . . . . . . . 10 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ∀𝑘 ∈ (ℤ𝑀)(𝐻𝑘) = ((𝐹𝑘)𝑄(𝐺𝑘)))
88 fzofzp1 9845 . . . . . . . . . . . 12 (𝑛 ∈ (𝑀..^𝑁) → (𝑛 + 1) ∈ (𝑀...𝑁))
89 elfzuz 9643 . . . . . . . . . . . 12 ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝑛 + 1) ∈ (ℤ𝑀))
9088, 89syl 14 . . . . . . . . . . 11 (𝑛 ∈ (𝑀..^𝑁) → (𝑛 + 1) ∈ (ℤ𝑀))
9190adantl 273 . . . . . . . . . 10 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝑛 + 1) ∈ (ℤ𝑀))
9286, 87, 91rspcdva 2749 . . . . . . . . 9 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝐻‘(𝑛 + 1)) = ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1))))
9392oveq2d 5722 . . . . . . . 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 464 . . . . . . . . . 10 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
9577, 94, 79seq3p1 10076 . . . . . . . . 9 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
9670adantlr 464 . . . . . . . . . 10 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)
9777, 96, 79seq3p1 10076 . . . . . . . . 9 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( + , 𝐺)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1))))
9895, 97oveq12d 5724 . . . . . . . 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 2143 . . . . . . 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 2114 . . . . . 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, 100syl5ibr 155 . . . . 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 115 . . . 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 9857 . 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 103   = wceq 1299  wcel 1448  wral 2375  cfv 5059  (class class class)co 5706  1c1 7501   + caddc 7503  cz 8906  cuz 9176  ...cfz 9631  ..^cfzo 9760  seqcseq 10059
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-addcom 7595  ax-addass 7597  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-0id 7603  ax-rnegex 7604  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-ltadd 7611
This theorem depends on definitions:  df-bi 116  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-iord 4226  df-on 4228  df-ilim 4229  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-frec 6218  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-inn 8579  df-n0 8830  df-z 8907  df-uz 9177  df-fz 9632  df-fzo 9761  df-seqfrec 10060
This theorem is referenced by:  seq3caopr2  10096
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