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Theorem iseqhomo 9565
Description: Apply a homomorphism to a sequence. (Contributed by Jim Kingdon, 21-Aug-2021.)
Hypotheses
Ref Expression
iseqhomo.1 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
iseqhomo.2 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
iseqhomo.s (𝜑𝑆𝑉)
iseqhomo.3 (𝜑𝑁 ∈ (ℤ𝑀))
iseqhomo.4 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝐻‘(𝑥 + 𝑦)) = ((𝐻𝑥)𝑄(𝐻𝑦)))
iseqhomo.5 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐻‘(𝐹𝑥)) = (𝐺𝑥))
iseqhomo.g ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)
iseqhomo.qcl ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)
Assertion
Ref Expression
iseqhomo (𝜑 → (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑁)) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑁))
Distinct variable groups:   𝑥,𝑦,𝐹   𝑥,𝐻,𝑦   𝑥,𝑀,𝑦   𝑥,𝑁,𝑦   𝜑,𝑥,𝑦   𝑥,𝐺   𝑥, + ,𝑦   𝑥,𝑄,𝑦   𝑥,𝑆,𝑦   𝑦,𝐺
Allowed substitution hints:   𝑉(𝑥,𝑦)

Proof of Theorem iseqhomo
Dummy variables 𝑛 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iseqhomo.3 . 2 (𝜑𝑁 ∈ (ℤ𝑀))
2 fveq2 5209 . . . . . 6 (𝑤 = 𝑀 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘𝑀))
32fveq2d 5213 . . . . 5 (𝑤 = 𝑀 → (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑤)) = (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑀)))
4 fveq2 5209 . . . . 5 (𝑤 = 𝑀 → (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑤) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑀))
53, 4eqeq12d 2096 . . . 4 (𝑤 = 𝑀 → ((𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑤)) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑤) ↔ (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑀)) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑀)))
65imbi2d 228 . . 3 (𝑤 = 𝑀 → ((𝜑 → (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑤)) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑤)) ↔ (𝜑 → (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑀)) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑀))))
7 fveq2 5209 . . . . . 6 (𝑤 = 𝑛 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘𝑛))
87fveq2d 5213 . . . . 5 (𝑤 = 𝑛 → (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑤)) = (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑛)))
9 fveq2 5209 . . . . 5 (𝑤 = 𝑛 → (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑤) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑛))
108, 9eqeq12d 2096 . . . 4 (𝑤 = 𝑛 → ((𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑤)) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑤) ↔ (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑛)) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑛)))
1110imbi2d 228 . . 3 (𝑤 = 𝑛 → ((𝜑 → (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑤)) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑤)) ↔ (𝜑 → (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑛)) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑛))))
12 fveq2 5209 . . . . . 6 (𝑤 = (𝑛 + 1) → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1)))
1312fveq2d 5213 . . . . 5 (𝑤 = (𝑛 + 1) → (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑤)) = (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1))))
14 fveq2 5209 . . . . 5 (𝑤 = (𝑛 + 1) → (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑤) = (seq𝑀(𝑄, 𝐺, 𝑆)‘(𝑛 + 1)))
1513, 14eqeq12d 2096 . . . 4 (𝑤 = (𝑛 + 1) → ((𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑤)) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑤) ↔ (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1))) = (seq𝑀(𝑄, 𝐺, 𝑆)‘(𝑛 + 1))))
1615imbi2d 228 . . 3 (𝑤 = (𝑛 + 1) → ((𝜑 → (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑤)) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑤)) ↔ (𝜑 → (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1))) = (seq𝑀(𝑄, 𝐺, 𝑆)‘(𝑛 + 1)))))
17 fveq2 5209 . . . . . 6 (𝑤 = 𝑁 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘𝑁))
1817fveq2d 5213 . . . . 5 (𝑤 = 𝑁 → (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑤)) = (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑁)))
19 fveq2 5209 . . . . 5 (𝑤 = 𝑁 → (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑤) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑁))
2018, 19eqeq12d 2096 . . . 4 (𝑤 = 𝑁 → ((𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑤)) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑤) ↔ (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑁)) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑁)))
2120imbi2d 228 . . 3 (𝑤 = 𝑁 → ((𝜑 → (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑤)) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑤)) ↔ (𝜑 → (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑁)) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑁))))
22 fveq2 5209 . . . . . . . 8 (𝑥 = 𝑀 → (𝐹𝑥) = (𝐹𝑀))
2322fveq2d 5213 . . . . . . 7 (𝑥 = 𝑀 → (𝐻‘(𝐹𝑥)) = (𝐻‘(𝐹𝑀)))
24 fveq2 5209 . . . . . . 7 (𝑥 = 𝑀 → (𝐺𝑥) = (𝐺𝑀))
2523, 24eqeq12d 2096 . . . . . 6 (𝑥 = 𝑀 → ((𝐻‘(𝐹𝑥)) = (𝐺𝑥) ↔ (𝐻‘(𝐹𝑀)) = (𝐺𝑀)))
26 iseqhomo.5 . . . . . . 7 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐻‘(𝐹𝑥)) = (𝐺𝑥))
2726ralrimiva 2435 . . . . . 6 (𝜑 → ∀𝑥 ∈ (ℤ𝑀)(𝐻‘(𝐹𝑥)) = (𝐺𝑥))
28 eluzel2 8705 . . . . . . . 8 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
291, 28syl 14 . . . . . . 7 (𝜑𝑀 ∈ ℤ)
30 uzid 8714 . . . . . . 7 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
3129, 30syl 14 . . . . . 6 (𝜑𝑀 ∈ (ℤ𝑀))
3225, 27, 31rspcdva 2708 . . . . 5 (𝜑 → (𝐻‘(𝐹𝑀)) = (𝐺𝑀))
33 iseqhomo.2 . . . . . . 7 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
34 iseqhomo.1 . . . . . . 7 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
3529, 33, 34iseq1 9533 . . . . . 6 (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (𝐹𝑀))
3635fveq2d 5213 . . . . 5 (𝜑 → (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑀)) = (𝐻‘(𝐹𝑀)))
37 iseqhomo.g . . . . . 6 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)
38 iseqhomo.qcl . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)
3929, 37, 38iseq1 9533 . . . . 5 (𝜑 → (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑀) = (𝐺𝑀))
4032, 36, 393eqtr4d 2124 . . . 4 (𝜑 → (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑀)) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑀))
4140a1i 9 . . 3 (𝑀 ∈ ℤ → (𝜑 → (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑀)) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑀)))
42 oveq1 5550 . . . . . 6 ((𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑛)) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑛) → ((𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑛))𝑄(𝐺‘(𝑛 + 1))) = ((seq𝑀(𝑄, 𝐺, 𝑆)‘𝑛)𝑄(𝐺‘(𝑛 + 1))))
43 simpr 108 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → 𝑛 ∈ (ℤ𝑀))
4433adantlr 461 . . . . . . . . . 10 (((𝜑𝑛 ∈ (ℤ𝑀)) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
4534adantlr 461 . . . . . . . . . 10 (((𝜑𝑛 ∈ (ℤ𝑀)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
4643, 44, 45iseqp1 9538 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℤ𝑀)) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1))))
4746fveq2d 5213 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1))) = (𝐻‘((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1)))))
48 iseqhomo.4 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝐻‘(𝑥 + 𝑦)) = ((𝐻𝑥)𝑄(𝐻𝑦)))
4948ralrimivva 2444 . . . . . . . . . 10 (𝜑 → ∀𝑥𝑆𝑦𝑆 (𝐻‘(𝑥 + 𝑦)) = ((𝐻𝑥)𝑄(𝐻𝑦)))
5049adantr 270 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℤ𝑀)) → ∀𝑥𝑆𝑦𝑆 (𝐻‘(𝑥 + 𝑦)) = ((𝐻𝑥)𝑄(𝐻𝑦)))
5143, 44, 45iseqcl 9537 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → (seq𝑀( + , 𝐹, 𝑆)‘𝑛) ∈ 𝑆)
52 fveq2 5209 . . . . . . . . . . . 12 (𝑥 = (𝑛 + 1) → (𝐹𝑥) = (𝐹‘(𝑛 + 1)))
5352eleq1d 2148 . . . . . . . . . . 11 (𝑥 = (𝑛 + 1) → ((𝐹𝑥) ∈ 𝑆 ↔ (𝐹‘(𝑛 + 1)) ∈ 𝑆))
5433ralrimiva 2435 . . . . . . . . . . . 12 (𝜑 → ∀𝑥 ∈ (ℤ𝑀)(𝐹𝑥) ∈ 𝑆)
5554adantr 270 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (ℤ𝑀)) → ∀𝑥 ∈ (ℤ𝑀)(𝐹𝑥) ∈ 𝑆)
56 peano2uz 8752 . . . . . . . . . . . 12 (𝑛 ∈ (ℤ𝑀) → (𝑛 + 1) ∈ (ℤ𝑀))
5743, 56syl 14 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝑛 + 1) ∈ (ℤ𝑀))
5853, 55, 57rspcdva 2708 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝐹‘(𝑛 + 1)) ∈ 𝑆)
59 oveq1 5550 . . . . . . . . . . . . 13 (𝑥 = (seq𝑀( + , 𝐹, 𝑆)‘𝑛) → (𝑥 + 𝑦) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + 𝑦))
6059fveq2d 5213 . . . . . . . . . . . 12 (𝑥 = (seq𝑀( + , 𝐹, 𝑆)‘𝑛) → (𝐻‘(𝑥 + 𝑦)) = (𝐻‘((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + 𝑦)))
61 fveq2 5209 . . . . . . . . . . . . 13 (𝑥 = (seq𝑀( + , 𝐹, 𝑆)‘𝑛) → (𝐻𝑥) = (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑛)))
6261oveq1d 5558 . . . . . . . . . . . 12 (𝑥 = (seq𝑀( + , 𝐹, 𝑆)‘𝑛) → ((𝐻𝑥)𝑄(𝐻𝑦)) = ((𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑛))𝑄(𝐻𝑦)))
6360, 62eqeq12d 2096 . . . . . . . . . . 11 (𝑥 = (seq𝑀( + , 𝐹, 𝑆)‘𝑛) → ((𝐻‘(𝑥 + 𝑦)) = ((𝐻𝑥)𝑄(𝐻𝑦)) ↔ (𝐻‘((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + 𝑦)) = ((𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑛))𝑄(𝐻𝑦))))
64 oveq2 5551 . . . . . . . . . . . . 13 (𝑦 = (𝐹‘(𝑛 + 1)) → ((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + 𝑦) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1))))
6564fveq2d 5213 . . . . . . . . . . . 12 (𝑦 = (𝐹‘(𝑛 + 1)) → (𝐻‘((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + 𝑦)) = (𝐻‘((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1)))))
66 fveq2 5209 . . . . . . . . . . . . 13 (𝑦 = (𝐹‘(𝑛 + 1)) → (𝐻𝑦) = (𝐻‘(𝐹‘(𝑛 + 1))))
6766oveq2d 5559 . . . . . . . . . . . 12 (𝑦 = (𝐹‘(𝑛 + 1)) → ((𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑛))𝑄(𝐻𝑦)) = ((𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑛))𝑄(𝐻‘(𝐹‘(𝑛 + 1)))))
6865, 67eqeq12d 2096 . . . . . . . . . . 11 (𝑦 = (𝐹‘(𝑛 + 1)) → ((𝐻‘((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + 𝑦)) = ((𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑛))𝑄(𝐻𝑦)) ↔ (𝐻‘((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1)))) = ((𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑛))𝑄(𝐻‘(𝐹‘(𝑛 + 1))))))
6963, 68rspc2v 2714 . . . . . . . . . 10 (((seq𝑀( + , 𝐹, 𝑆)‘𝑛) ∈ 𝑆 ∧ (𝐹‘(𝑛 + 1)) ∈ 𝑆) → (∀𝑥𝑆𝑦𝑆 (𝐻‘(𝑥 + 𝑦)) = ((𝐻𝑥)𝑄(𝐻𝑦)) → (𝐻‘((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1)))) = ((𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑛))𝑄(𝐻‘(𝐹‘(𝑛 + 1))))))
7051, 58, 69syl2anc 403 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℤ𝑀)) → (∀𝑥𝑆𝑦𝑆 (𝐻‘(𝑥 + 𝑦)) = ((𝐻𝑥)𝑄(𝐻𝑦)) → (𝐻‘((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1)))) = ((𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑛))𝑄(𝐻‘(𝐹‘(𝑛 + 1))))))
7150, 70mpd 13 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝐻‘((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1)))) = ((𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑛))𝑄(𝐻‘(𝐹‘(𝑛 + 1)))))
7252fveq2d 5213 . . . . . . . . . . 11 (𝑥 = (𝑛 + 1) → (𝐻‘(𝐹𝑥)) = (𝐻‘(𝐹‘(𝑛 + 1))))
73 fveq2 5209 . . . . . . . . . . 11 (𝑥 = (𝑛 + 1) → (𝐺𝑥) = (𝐺‘(𝑛 + 1)))
7472, 73eqeq12d 2096 . . . . . . . . . 10 (𝑥 = (𝑛 + 1) → ((𝐻‘(𝐹𝑥)) = (𝐺𝑥) ↔ (𝐻‘(𝐹‘(𝑛 + 1))) = (𝐺‘(𝑛 + 1))))
7527adantr 270 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → ∀𝑥 ∈ (ℤ𝑀)(𝐻‘(𝐹𝑥)) = (𝐺𝑥))
7674, 75, 57rspcdva 2708 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝐻‘(𝐹‘(𝑛 + 1))) = (𝐺‘(𝑛 + 1)))
7776oveq2d 5559 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → ((𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑛))𝑄(𝐻‘(𝐹‘(𝑛 + 1)))) = ((𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑛))𝑄(𝐺‘(𝑛 + 1))))
7847, 71, 773eqtrd 2118 . . . . . . 7 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1))) = ((𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑛))𝑄(𝐺‘(𝑛 + 1))))
7937adantlr 461 . . . . . . . 8 (((𝜑𝑛 ∈ (ℤ𝑀)) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)
8038adantlr 461 . . . . . . . 8 (((𝜑𝑛 ∈ (ℤ𝑀)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)
8143, 79, 80iseqp1 9538 . . . . . . 7 ((𝜑𝑛 ∈ (ℤ𝑀)) → (seq𝑀(𝑄, 𝐺, 𝑆)‘(𝑛 + 1)) = ((seq𝑀(𝑄, 𝐺, 𝑆)‘𝑛)𝑄(𝐺‘(𝑛 + 1))))
8278, 81eqeq12d 2096 . . . . . 6 ((𝜑𝑛 ∈ (ℤ𝑀)) → ((𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1))) = (seq𝑀(𝑄, 𝐺, 𝑆)‘(𝑛 + 1)) ↔ ((𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑛))𝑄(𝐺‘(𝑛 + 1))) = ((seq𝑀(𝑄, 𝐺, 𝑆)‘𝑛)𝑄(𝐺‘(𝑛 + 1)))))
8342, 82syl5ibr 154 . . . . 5 ((𝜑𝑛 ∈ (ℤ𝑀)) → ((𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑛)) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑛) → (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1))) = (seq𝑀(𝑄, 𝐺, 𝑆)‘(𝑛 + 1))))
8483expcom 114 . . . 4 (𝑛 ∈ (ℤ𝑀) → (𝜑 → ((𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑛)) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑛) → (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1))) = (seq𝑀(𝑄, 𝐺, 𝑆)‘(𝑛 + 1)))))
8584a2d 26 . . 3 (𝑛 ∈ (ℤ𝑀) → ((𝜑 → (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑛)) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑛)) → (𝜑 → (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1))) = (seq𝑀(𝑄, 𝐺, 𝑆)‘(𝑛 + 1)))))
866, 11, 16, 21, 41, 85uzind4 8757 . 2 (𝑁 ∈ (ℤ𝑀) → (𝜑 → (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑁)) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑁)))
871, 86mpcom 36 1 (𝜑 → (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑁)) = (seq𝑀(𝑄, 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  cz 8432  cuz 8700  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-iseq 9522
This theorem is referenced by:  iseqdistr  9567
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