Theorem iseqhomo 9468

Description: Apply a homomorphism to a sequence. (Contributed by Jim Kingdon, 21-Aug-2021.)

Hypotheses

Ref	Expression
iseqhomo.1	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
iseqhomo.2	⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
iseqhomo.s	⊢ (𝜑 → 𝑆 ∈ 𝑉)
iseqhomo.3	⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
iseqhomo.4	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝐻‘(𝑥 + 𝑦)) = ((𝐻‘𝑥)𝑄(𝐻‘𝑦)))
iseqhomo.5	⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐻‘(𝐹‘𝑥)) = (𝐺‘𝑥))
iseqhomo.g	⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆)
iseqhomo.qcl	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (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.

Step	Hyp	Ref	Expression
1		iseqhomo.3	. 2 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
2		fveq2 5198	. . . . . 6 ⊢ (𝑤 = 𝑀 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘𝑀))
3	2	fveq2d 5202	. . . . 5 ⊢ (𝑤 = 𝑀 → (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑤)) = (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑀)))
4		fveq2 5198	. . . . 5 ⊢ (𝑤 = 𝑀 → (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑤) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑀))
5	3, 4	eqeq12d 2095	. . . 4 ⊢ (𝑤 = 𝑀 → ((𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑤)) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑤) ↔ (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑀)) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑀)))
6	5	imbi2d 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 5198	. . . . . 6 ⊢ (𝑤 = 𝑛 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘𝑛))
8	7	fveq2d 5202	. . . . 5 ⊢ (𝑤 = 𝑛 → (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑤)) = (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑛)))
9		fveq2 5198	. . . . 5 ⊢ (𝑤 = 𝑛 → (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑤) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑛))
10	8, 9	eqeq12d 2095	. . . 4 ⊢ (𝑤 = 𝑛 → ((𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑤)) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑤) ↔ (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑛)) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑛)))
11	10	imbi2d 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 5198	. . . . . 6 ⊢ (𝑤 = (𝑛 + 1) → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1)))
13	12	fveq2d 5202	. . . . 5 ⊢ (𝑤 = (𝑛 + 1) → (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑤)) = (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1))))
14		fveq2 5198	. . . . 5 ⊢ (𝑤 = (𝑛 + 1) → (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑤) = (seq𝑀(𝑄, 𝐺, 𝑆)‘(𝑛 + 1)))
15	13, 14	eqeq12d 2095	. . . 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))))
16	15	imbi2d 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 5198	. . . . . 6 ⊢ (𝑤 = 𝑁 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘𝑁))
18	17	fveq2d 5202	. . . . 5 ⊢ (𝑤 = 𝑁 → (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑤)) = (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑁)))
19		fveq2 5198	. . . . 5 ⊢ (𝑤 = 𝑁 → (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑤) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑁))
20	18, 19	eqeq12d 2095	. . . 4 ⊢ (𝑤 = 𝑁 → ((𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑤)) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑤) ↔ (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑁)) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑁)))
21	20	imbi2d 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		eluzel2 8624	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
23	1, 22	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℤ)
24		uzid 8633	. . . . . . 7 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀))
25	23, 24	syl 14	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀))
26		iseqhomo.5	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐻‘(𝐹‘𝑥)) = (𝐺‘𝑥))
27	26	ralrimiva 2434	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ (ℤ≥‘𝑀)(𝐻‘(𝐹‘𝑥)) = (𝐺‘𝑥))
28		fveq2 5198	. . . . . . . . 9 ⊢ (𝑥 = 𝑀 → (𝐹‘𝑥) = (𝐹‘𝑀))
29	28	fveq2d 5202	. . . . . . . 8 ⊢ (𝑥 = 𝑀 → (𝐻‘(𝐹‘𝑥)) = (𝐻‘(𝐹‘𝑀)))
30		fveq2 5198	. . . . . . . 8 ⊢ (𝑥 = 𝑀 → (𝐺‘𝑥) = (𝐺‘𝑀))
31	29, 30	eqeq12d 2095	. . . . . . 7 ⊢ (𝑥 = 𝑀 → ((𝐻‘(𝐹‘𝑥)) = (𝐺‘𝑥) ↔ (𝐻‘(𝐹‘𝑀)) = (𝐺‘𝑀)))
32	31	rspcv 2697	. . . . . 6 ⊢ (𝑀 ∈ (ℤ≥‘𝑀) → (∀𝑥 ∈ (ℤ≥‘𝑀)(𝐻‘(𝐹‘𝑥)) = (𝐺‘𝑥) → (𝐻‘(𝐹‘𝑀)) = (𝐺‘𝑀)))
33	25, 27, 32	sylc 61	. . . . 5 ⊢ (𝜑 → (𝐻‘(𝐹‘𝑀)) = (𝐺‘𝑀))
34		iseqhomo.s	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
35		iseqhomo.2	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
36		iseqhomo.1	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
37	23, 34, 35, 36	iseq1 9442	. . . . . 6 ⊢ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (𝐹‘𝑀))
38	37	fveq2d 5202	. . . . 5 ⊢ (𝜑 → (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑀)) = (𝐻‘(𝐹‘𝑀)))
39		iseqhomo.g	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆)
40		iseqhomo.qcl	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)
41	23, 34, 39, 40	iseq1 9442	. . . . 5 ⊢ (𝜑 → (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑀) = (𝐺‘𝑀))
42	33, 38, 41	3eqtr4d 2123	. . . 4 ⊢ (𝜑 → (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑀)) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑀))
43	42	a1i 9	. . 3 ⊢ (𝑀 ∈ ℤ → (𝜑 → (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑀)) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑀)))
44		oveq1 5539	. . . . . 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 108	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → 𝑛 ∈ (ℤ≥‘𝑀))
46	34	adantr 270	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → 𝑆 ∈ 𝑉)
47	35	adantlr 460	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
48	36	adantlr 460	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
49	45, 46, 47, 48	iseqp1 9445	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1))))
50	49	fveq2d 5202	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1))) = (𝐻‘((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1)))))
51		iseqhomo.4	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝐻‘(𝑥 + 𝑦)) = ((𝐻‘𝑥)𝑄(𝐻‘𝑦)))
52	51	ralrimivva 2443	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝐻‘(𝑥 + 𝑦)) = ((𝐻‘𝑥)𝑄(𝐻‘𝑦)))
53	52	adantr 270	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝐻‘(𝑥 + 𝑦)) = ((𝐻‘𝑥)𝑄(𝐻‘𝑦)))
54	45, 46, 47, 48	iseqcl 9443	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (seq𝑀( + , 𝐹, 𝑆)‘𝑛) ∈ 𝑆)
55		peano2uz 8671	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → (𝑛 + 1) ∈ (ℤ≥‘𝑀))
56	45, 55	syl 14	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (𝑛 + 1) ∈ (ℤ≥‘𝑀))
57	35	ralrimiva 2434	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑥 ∈ (ℤ≥‘𝑀)(𝐹‘𝑥) ∈ 𝑆)
58	57	adantr 270	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ∀𝑥 ∈ (ℤ≥‘𝑀)(𝐹‘𝑥) ∈ 𝑆)
59		fveq2 5198	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑛 + 1) → (𝐹‘𝑥) = (𝐹‘(𝑛 + 1)))
60	59	eleq1d 2147	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑛 + 1) → ((𝐹‘𝑥) ∈ 𝑆 ↔ (𝐹‘(𝑛 + 1)) ∈ 𝑆))
61	60	rspcv 2697	. . . . . . . . . . 11 ⊢ ((𝑛 + 1) ∈ (ℤ≥‘𝑀) → (∀𝑥 ∈ (ℤ≥‘𝑀)(𝐹‘𝑥) ∈ 𝑆 → (𝐹‘(𝑛 + 1)) ∈ 𝑆))
62	56, 58, 61	sylc 61	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (𝐹‘(𝑛 + 1)) ∈ 𝑆)
63		oveq1 5539	. . . . . . . . . . . . 13 ⊢ (𝑥 = (seq𝑀( + , 𝐹, 𝑆)‘𝑛) → (𝑥 + 𝑦) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + 𝑦))
64	63	fveq2d 5202	. . . . . . . . . . . 12 ⊢ (𝑥 = (seq𝑀( + , 𝐹, 𝑆)‘𝑛) → (𝐻‘(𝑥 + 𝑦)) = (𝐻‘((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + 𝑦)))
65		fveq2 5198	. . . . . . . . . . . . 13 ⊢ (𝑥 = (seq𝑀( + , 𝐹, 𝑆)‘𝑛) → (𝐻‘𝑥) = (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑛)))
66	65	oveq1d 5547	. . . . . . . . . . . 12 ⊢ (𝑥 = (seq𝑀( + , 𝐹, 𝑆)‘𝑛) → ((𝐻‘𝑥)𝑄(𝐻‘𝑦)) = ((𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑛))𝑄(𝐻‘𝑦)))
67	64, 66	eqeq12d 2095	. . . . . . . . . . 11 ⊢ (𝑥 = (seq𝑀( + , 𝐹, 𝑆)‘𝑛) → ((𝐻‘(𝑥 + 𝑦)) = ((𝐻‘𝑥)𝑄(𝐻‘𝑦)) ↔ (𝐻‘((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + 𝑦)) = ((𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑛))𝑄(𝐻‘𝑦))))
68		oveq2 5540	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝐹‘(𝑛 + 1)) → ((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + 𝑦) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1))))
69	68	fveq2d 5202	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐹‘(𝑛 + 1)) → (𝐻‘((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + 𝑦)) = (𝐻‘((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1)))))
70		fveq2 5198	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝐹‘(𝑛 + 1)) → (𝐻‘𝑦) = (𝐻‘(𝐹‘(𝑛 + 1))))
71	70	oveq2d 5548	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐹‘(𝑛 + 1)) → ((𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑛))𝑄(𝐻‘𝑦)) = ((𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑛))𝑄(𝐻‘(𝐹‘(𝑛 + 1)))))
72	69, 71	eqeq12d 2095	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐹‘(𝑛 + 1)) → ((𝐻‘((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + 𝑦)) = ((𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑛))𝑄(𝐻‘𝑦)) ↔ (𝐻‘((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1)))) = ((𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑛))𝑄(𝐻‘(𝐹‘(𝑛 + 1))))))
73	67, 72	rspc2v 2713	. . . . . . . . . 10 ⊢ (((seq𝑀( + , 𝐹, 𝑆)‘𝑛) ∈ 𝑆 ∧ (𝐹‘(𝑛 + 1)) ∈ 𝑆) → (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝐻‘(𝑥 + 𝑦)) = ((𝐻‘𝑥)𝑄(𝐻‘𝑦)) → (𝐻‘((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1)))) = ((𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑛))𝑄(𝐻‘(𝐹‘(𝑛 + 1))))))
74	54, 62, 73	syl2anc 403	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝐻‘(𝑥 + 𝑦)) = ((𝐻‘𝑥)𝑄(𝐻‘𝑦)) → (𝐻‘((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1)))) = ((𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑛))𝑄(𝐻‘(𝐹‘(𝑛 + 1))))))
75	53, 74	mpd 13	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (𝐻‘((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1)))) = ((𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑛))𝑄(𝐻‘(𝐹‘(𝑛 + 1)))))
76	27	adantr 270	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ∀𝑥 ∈ (ℤ≥‘𝑀)(𝐻‘(𝐹‘𝑥)) = (𝐺‘𝑥))
77	59	fveq2d 5202	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑛 + 1) → (𝐻‘(𝐹‘𝑥)) = (𝐻‘(𝐹‘(𝑛 + 1))))
78		fveq2 5198	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑛 + 1) → (𝐺‘𝑥) = (𝐺‘(𝑛 + 1)))
79	77, 78	eqeq12d 2095	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑛 + 1) → ((𝐻‘(𝐹‘𝑥)) = (𝐺‘𝑥) ↔ (𝐻‘(𝐹‘(𝑛 + 1))) = (𝐺‘(𝑛 + 1))))
80	79	rspcv 2697	. . . . . . . . . 10 ⊢ ((𝑛 + 1) ∈ (ℤ≥‘𝑀) → (∀𝑥 ∈ (ℤ≥‘𝑀)(𝐻‘(𝐹‘𝑥)) = (𝐺‘𝑥) → (𝐻‘(𝐹‘(𝑛 + 1))) = (𝐺‘(𝑛 + 1))))
81	56, 76, 80	sylc 61	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (𝐻‘(𝐹‘(𝑛 + 1))) = (𝐺‘(𝑛 + 1)))
82	81	oveq2d 5548	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ((𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑛))𝑄(𝐻‘(𝐹‘(𝑛 + 1)))) = ((𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑛))𝑄(𝐺‘(𝑛 + 1))))
83	50, 75, 82	3eqtrd 2117	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1))) = ((𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑛))𝑄(𝐺‘(𝑛 + 1))))
84	39	adantlr 460	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆)
85	40	adantlr 460	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)
86	45, 46, 84, 85	iseqp1 9445	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (seq𝑀(𝑄, 𝐺, 𝑆)‘(𝑛 + 1)) = ((seq𝑀(𝑄, 𝐺, 𝑆)‘𝑛)𝑄(𝐺‘(𝑛 + 1))))
87	83, 86	eqeq12d 2095	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ((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)))))
88	44, 87	syl5ibr 154	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ((𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑛)) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑛) → (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1))) = (seq𝑀(𝑄, 𝐺, 𝑆)‘(𝑛 + 1))))
89	88	expcom 114	. . . 4 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → (𝜑 → ((𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑛)) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑛) → (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1))) = (seq𝑀(𝑄, 𝐺, 𝑆)‘(𝑛 + 1)))))
90	89	a2d 26	. . 3 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → ((𝜑 → (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑛)) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑛)) → (𝜑 → (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1))) = (seq𝑀(𝑄, 𝐺, 𝑆)‘(𝑛 + 1)))))
91	6, 11, 16, 21, 43, 90	uzind4 8676	. 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝜑 → (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑁)) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑁)))
92	1, 91	mpcom 36	1 ⊢ (𝜑 → (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑁)) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑁))

Syntax hints:   → wi 4   ∧ wa 102   = wceq 1284   ∈ wcel 1433  ∀wral 2348  ‘cfv 4922  (class class class)co 5532  1c1 6982   + caddc 6984  ℤcz 8351  ℤ_≥cuz 8619  seqcseq 9431

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 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-addcom 7076  ax-addass 7078  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-0id 7084  ax-rnegex 7085  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-ltadd 7092

This theorem depends on definitions:  df-bi 115  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-frec 6001  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-inn 8040  df-n0 8289  df-z 8352  df-uz 8620  df-iseq 9432

This theorem is referenced by:  iseqdistr  9470