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Theorem iseqhomo 9222
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 5178 . . . . . 6 (𝑤 = 𝑀 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘𝑀))
32fveq2d 5182 . . . . 5 (𝑤 = 𝑀 → (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑤)) = (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑀)))
4 fveq2 5178 . . . . 5 (𝑤 = 𝑀 → (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑤) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑀))
53, 4eqeq12d 2054 . . . 4 (𝑤 = 𝑀 → ((𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑤)) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑤) ↔ (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑀)) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑀)))
65imbi2d 219 . . 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 5178 . . . . . 6 (𝑤 = 𝑛 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘𝑛))
87fveq2d 5182 . . . . 5 (𝑤 = 𝑛 → (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑤)) = (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑛)))
9 fveq2 5178 . . . . 5 (𝑤 = 𝑛 → (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑤) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑛))
108, 9eqeq12d 2054 . . . 4 (𝑤 = 𝑛 → ((𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑤)) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑤) ↔ (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑛)) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑛)))
1110imbi2d 219 . . 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 5178 . . . . . 6 (𝑤 = (𝑛 + 1) → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1)))
1312fveq2d 5182 . . . . 5 (𝑤 = (𝑛 + 1) → (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑤)) = (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1))))
14 fveq2 5178 . . . . 5 (𝑤 = (𝑛 + 1) → (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑤) = (seq𝑀(𝑄, 𝐺, 𝑆)‘(𝑛 + 1)))
1513, 14eqeq12d 2054 . . . 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 219 . . 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 5178 . . . . . 6 (𝑤 = 𝑁 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘𝑁))
1817fveq2d 5182 . . . . 5 (𝑤 = 𝑁 → (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑤)) = (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑁)))
19 fveq2 5178 . . . . 5 (𝑤 = 𝑁 → (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑤) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑁))
2018, 19eqeq12d 2054 . . . 4 (𝑤 = 𝑁 → ((𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑤)) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑤) ↔ (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑁)) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑁)))
2120imbi2d 219 . . 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 eluzel2 8476 . . . . . . . 8 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
231, 22syl 14 . . . . . . 7 (𝜑𝑀 ∈ ℤ)
24 uzid 8485 . . . . . . 7 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
2523, 24syl 14 . . . . . 6 (𝜑𝑀 ∈ (ℤ𝑀))
26 iseqhomo.5 . . . . . . 7 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐻‘(𝐹𝑥)) = (𝐺𝑥))
2726ralrimiva 2392 . . . . . 6 (𝜑 → ∀𝑥 ∈ (ℤ𝑀)(𝐻‘(𝐹𝑥)) = (𝐺𝑥))
28 fveq2 5178 . . . . . . . . 9 (𝑥 = 𝑀 → (𝐹𝑥) = (𝐹𝑀))
2928fveq2d 5182 . . . . . . . 8 (𝑥 = 𝑀 → (𝐻‘(𝐹𝑥)) = (𝐻‘(𝐹𝑀)))
30 fveq2 5178 . . . . . . . 8 (𝑥 = 𝑀 → (𝐺𝑥) = (𝐺𝑀))
3129, 30eqeq12d 2054 . . . . . . 7 (𝑥 = 𝑀 → ((𝐻‘(𝐹𝑥)) = (𝐺𝑥) ↔ (𝐻‘(𝐹𝑀)) = (𝐺𝑀)))
3231rspcv 2652 . . . . . 6 (𝑀 ∈ (ℤ𝑀) → (∀𝑥 ∈ (ℤ𝑀)(𝐻‘(𝐹𝑥)) = (𝐺𝑥) → (𝐻‘(𝐹𝑀)) = (𝐺𝑀)))
3325, 27, 32sylc 56 . . . . 5 (𝜑 → (𝐻‘(𝐹𝑀)) = (𝐺𝑀))
34 iseqhomo.s . . . . . . 7 (𝜑𝑆𝑉)
35 iseqhomo.2 . . . . . . 7 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
36 iseqhomo.1 . . . . . . 7 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
3723, 34, 35, 36iseq1 9196 . . . . . 6 (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (𝐹𝑀))
3837fveq2d 5182 . . . . 5 (𝜑 → (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑀)) = (𝐻‘(𝐹𝑀)))
39 iseqhomo.g . . . . . 6 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)
40 iseqhomo.qcl . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)
4123, 34, 39, 40iseq1 9196 . . . . 5 (𝜑 → (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑀) = (𝐺𝑀))
4233, 38, 413eqtr4d 2082 . . . 4 (𝜑 → (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑀)) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑀))
4342a1i 9 . . 3 (𝑀 ∈ ℤ → (𝜑 → (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑀)) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑀)))
44 oveq1 5519 . . . . . 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))))
45 simpr 103 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → 𝑛 ∈ (ℤ𝑀))
4634adantr 261 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → 𝑆𝑉)
4735adantlr 446 . . . . . . . . . 10 (((𝜑𝑛 ∈ (ℤ𝑀)) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
4836adantlr 446 . . . . . . . . . 10 (((𝜑𝑛 ∈ (ℤ𝑀)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
4945, 46, 47, 48iseqp1 9199 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℤ𝑀)) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1))))
5049fveq2d 5182 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1))) = (𝐻‘((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1)))))
51 iseqhomo.4 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝐻‘(𝑥 + 𝑦)) = ((𝐻𝑥)𝑄(𝐻𝑦)))
5251ralrimivva 2401 . . . . . . . . . 10 (𝜑 → ∀𝑥𝑆𝑦𝑆 (𝐻‘(𝑥 + 𝑦)) = ((𝐻𝑥)𝑄(𝐻𝑦)))
5352adantr 261 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℤ𝑀)) → ∀𝑥𝑆𝑦𝑆 (𝐻‘(𝑥 + 𝑦)) = ((𝐻𝑥)𝑄(𝐻𝑦)))
5445, 46, 47, 48iseqcl 9197 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → (seq𝑀( + , 𝐹, 𝑆)‘𝑛) ∈ 𝑆)
55 peano2uz 8524 . . . . . . . . . . . 12 (𝑛 ∈ (ℤ𝑀) → (𝑛 + 1) ∈ (ℤ𝑀))
5645, 55syl 14 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝑛 + 1) ∈ (ℤ𝑀))
5735ralrimiva 2392 . . . . . . . . . . . 12 (𝜑 → ∀𝑥 ∈ (ℤ𝑀)(𝐹𝑥) ∈ 𝑆)
5857adantr 261 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (ℤ𝑀)) → ∀𝑥 ∈ (ℤ𝑀)(𝐹𝑥) ∈ 𝑆)
59 fveq2 5178 . . . . . . . . . . . . 13 (𝑥 = (𝑛 + 1) → (𝐹𝑥) = (𝐹‘(𝑛 + 1)))
6059eleq1d 2106 . . . . . . . . . . . 12 (𝑥 = (𝑛 + 1) → ((𝐹𝑥) ∈ 𝑆 ↔ (𝐹‘(𝑛 + 1)) ∈ 𝑆))
6160rspcv 2652 . . . . . . . . . . 11 ((𝑛 + 1) ∈ (ℤ𝑀) → (∀𝑥 ∈ (ℤ𝑀)(𝐹𝑥) ∈ 𝑆 → (𝐹‘(𝑛 + 1)) ∈ 𝑆))
6256, 58, 61sylc 56 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝐹‘(𝑛 + 1)) ∈ 𝑆)
63 oveq1 5519 . . . . . . . . . . . . 13 (𝑥 = (seq𝑀( + , 𝐹, 𝑆)‘𝑛) → (𝑥 + 𝑦) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + 𝑦))
6463fveq2d 5182 . . . . . . . . . . . 12 (𝑥 = (seq𝑀( + , 𝐹, 𝑆)‘𝑛) → (𝐻‘(𝑥 + 𝑦)) = (𝐻‘((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + 𝑦)))
65 fveq2 5178 . . . . . . . . . . . . 13 (𝑥 = (seq𝑀( + , 𝐹, 𝑆)‘𝑛) → (𝐻𝑥) = (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑛)))
6665oveq1d 5527 . . . . . . . . . . . 12 (𝑥 = (seq𝑀( + , 𝐹, 𝑆)‘𝑛) → ((𝐻𝑥)𝑄(𝐻𝑦)) = ((𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑛))𝑄(𝐻𝑦)))
6764, 66eqeq12d 2054 . . . . . . . . . . 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))))
68 oveq2 5520 . . . . . . . . . . . . 13 (𝑦 = (𝐹‘(𝑛 + 1)) → ((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + 𝑦) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1))))
6968fveq2d 5182 . . . . . . . . . . . 12 (𝑦 = (𝐹‘(𝑛 + 1)) → (𝐻‘((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + 𝑦)) = (𝐻‘((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1)))))
70 fveq2 5178 . . . . . . . . . . . . 13 (𝑦 = (𝐹‘(𝑛 + 1)) → (𝐻𝑦) = (𝐻‘(𝐹‘(𝑛 + 1))))
7170oveq2d 5528 . . . . . . . . . . . 12 (𝑦 = (𝐹‘(𝑛 + 1)) → ((𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑛))𝑄(𝐻𝑦)) = ((𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑛))𝑄(𝐻‘(𝐹‘(𝑛 + 1)))))
7269, 71eqeq12d 2054 . . . . . . . . . . 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))))))
7367, 72rspc2v 2662 . . . . . . . . . 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))))))
7454, 62, 73syl2anc 391 . . . . . . . . 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))))))
7553, 74mpd 13 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝐻‘((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1)))) = ((𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑛))𝑄(𝐻‘(𝐹‘(𝑛 + 1)))))
7627adantr 261 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → ∀𝑥 ∈ (ℤ𝑀)(𝐻‘(𝐹𝑥)) = (𝐺𝑥))
7759fveq2d 5182 . . . . . . . . . . . 12 (𝑥 = (𝑛 + 1) → (𝐻‘(𝐹𝑥)) = (𝐻‘(𝐹‘(𝑛 + 1))))
78 fveq2 5178 . . . . . . . . . . . 12 (𝑥 = (𝑛 + 1) → (𝐺𝑥) = (𝐺‘(𝑛 + 1)))
7977, 78eqeq12d 2054 . . . . . . . . . . 11 (𝑥 = (𝑛 + 1) → ((𝐻‘(𝐹𝑥)) = (𝐺𝑥) ↔ (𝐻‘(𝐹‘(𝑛 + 1))) = (𝐺‘(𝑛 + 1))))
8079rspcv 2652 . . . . . . . . . 10 ((𝑛 + 1) ∈ (ℤ𝑀) → (∀𝑥 ∈ (ℤ𝑀)(𝐻‘(𝐹𝑥)) = (𝐺𝑥) → (𝐻‘(𝐹‘(𝑛 + 1))) = (𝐺‘(𝑛 + 1))))
8156, 76, 80sylc 56 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝐻‘(𝐹‘(𝑛 + 1))) = (𝐺‘(𝑛 + 1)))
8281oveq2d 5528 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → ((𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑛))𝑄(𝐻‘(𝐹‘(𝑛 + 1)))) = ((𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑛))𝑄(𝐺‘(𝑛 + 1))))
8350, 75, 823eqtrd 2076 . . . . . . 7 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1))) = ((𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑛))𝑄(𝐺‘(𝑛 + 1))))
8439adantlr 446 . . . . . . . 8 (((𝜑𝑛 ∈ (ℤ𝑀)) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)
8540adantlr 446 . . . . . . . 8 (((𝜑𝑛 ∈ (ℤ𝑀)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)
8645, 46, 84, 85iseqp1 9199 . . . . . . 7 ((𝜑𝑛 ∈ (ℤ𝑀)) → (seq𝑀(𝑄, 𝐺, 𝑆)‘(𝑛 + 1)) = ((seq𝑀(𝑄, 𝐺, 𝑆)‘𝑛)𝑄(𝐺‘(𝑛 + 1))))
8783, 86eqeq12d 2054 . . . . . 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)))))
8844, 87syl5ibr 145 . . . . 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))))
8988expcom 109 . . . 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)))))
9089a2d 23 . . 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)))))
916, 11, 16, 21, 43, 90uzind4 8529 . 2 (𝑁 ∈ (ℤ𝑀) → (𝜑 → (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑁)) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑁)))
921, 91mpcom 32 1 (𝜑 → (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑁)) = (seq𝑀(𝑄, 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  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-iseq 9186
This theorem is referenced by:  iseqdistr  9223
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