Theorem seq3homo 10694

Description: Apply a homomorphism to a sequence. (Contributed by Jim Kingdon, 10-Oct-2022.)

Hypotheses

Ref	Expression
seq3homo.1	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
seq3homo.2	⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
seq3homo.3	⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
seq3homo.4	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝐻‘(𝑥 + 𝑦)) = ((𝐻‘𝑥)𝑄(𝐻‘𝑦)))
seq3homo.5	⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐻‘(𝐹‘𝑥)) = (𝐺‘𝑥))
seq3homo.g	⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆)
seq3homo.qcl	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)

Assertion

Ref	Expression
seq3homo	⊢ (𝜑 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑁)) = (seq𝑀(𝑄, 𝐺)‘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

Proof of Theorem seq3homo

Dummy variables 𝑛 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		seq3homo.3	. 2 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
2		2fveq3 5594	. . . . 5 ⊢ (𝑤 = 𝑀 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑤)) = (𝐻‘(seq𝑀( + , 𝐹)‘𝑀)))
3		fveq2 5589	. . . . 5 ⊢ (𝑤 = 𝑀 → (seq𝑀(𝑄, 𝐺)‘𝑤) = (seq𝑀(𝑄, 𝐺)‘𝑀))
4	2, 3	eqeq12d 2221	. . . 4 ⊢ (𝑤 = 𝑀 → ((𝐻‘(seq𝑀( + , 𝐹)‘𝑤)) = (seq𝑀(𝑄, 𝐺)‘𝑤) ↔ (𝐻‘(seq𝑀( + , 𝐹)‘𝑀)) = (seq𝑀(𝑄, 𝐺)‘𝑀)))
5	4	imbi2d 230	. . 3 ⊢ (𝑤 = 𝑀 → ((𝜑 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑤)) = (seq𝑀(𝑄, 𝐺)‘𝑤)) ↔ (𝜑 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑀)) = (seq𝑀(𝑄, 𝐺)‘𝑀))))
6		2fveq3 5594	. . . . 5 ⊢ (𝑤 = 𝑛 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑤)) = (𝐻‘(seq𝑀( + , 𝐹)‘𝑛)))
7		fveq2 5589	. . . . 5 ⊢ (𝑤 = 𝑛 → (seq𝑀(𝑄, 𝐺)‘𝑤) = (seq𝑀(𝑄, 𝐺)‘𝑛))
8	6, 7	eqeq12d 2221	. . . 4 ⊢ (𝑤 = 𝑛 → ((𝐻‘(seq𝑀( + , 𝐹)‘𝑤)) = (seq𝑀(𝑄, 𝐺)‘𝑤) ↔ (𝐻‘(seq𝑀( + , 𝐹)‘𝑛)) = (seq𝑀(𝑄, 𝐺)‘𝑛)))
9	8	imbi2d 230	. . 3 ⊢ (𝑤 = 𝑛 → ((𝜑 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑤)) = (seq𝑀(𝑄, 𝐺)‘𝑤)) ↔ (𝜑 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑛)) = (seq𝑀(𝑄, 𝐺)‘𝑛))))
10		2fveq3 5594	. . . . 5 ⊢ (𝑤 = (𝑛 + 1) → (𝐻‘(seq𝑀( + , 𝐹)‘𝑤)) = (𝐻‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))))
11		fveq2 5589	. . . . 5 ⊢ (𝑤 = (𝑛 + 1) → (seq𝑀(𝑄, 𝐺)‘𝑤) = (seq𝑀(𝑄, 𝐺)‘(𝑛 + 1)))
12	10, 11	eqeq12d 2221	. . . 4 ⊢ (𝑤 = (𝑛 + 1) → ((𝐻‘(seq𝑀( + , 𝐹)‘𝑤)) = (seq𝑀(𝑄, 𝐺)‘𝑤) ↔ (𝐻‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))) = (seq𝑀(𝑄, 𝐺)‘(𝑛 + 1))))
13	12	imbi2d 230	. . 3 ⊢ (𝑤 = (𝑛 + 1) → ((𝜑 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑤)) = (seq𝑀(𝑄, 𝐺)‘𝑤)) ↔ (𝜑 → (𝐻‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))) = (seq𝑀(𝑄, 𝐺)‘(𝑛 + 1)))))
14		2fveq3 5594	. . . . 5 ⊢ (𝑤 = 𝑁 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑤)) = (𝐻‘(seq𝑀( + , 𝐹)‘𝑁)))
15		fveq2 5589	. . . . 5 ⊢ (𝑤 = 𝑁 → (seq𝑀(𝑄, 𝐺)‘𝑤) = (seq𝑀(𝑄, 𝐺)‘𝑁))
16	14, 15	eqeq12d 2221	. . . 4 ⊢ (𝑤 = 𝑁 → ((𝐻‘(seq𝑀( + , 𝐹)‘𝑤)) = (seq𝑀(𝑄, 𝐺)‘𝑤) ↔ (𝐻‘(seq𝑀( + , 𝐹)‘𝑁)) = (seq𝑀(𝑄, 𝐺)‘𝑁)))
17	16	imbi2d 230	. . 3 ⊢ (𝑤 = 𝑁 → ((𝜑 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑤)) = (seq𝑀(𝑄, 𝐺)‘𝑤)) ↔ (𝜑 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑁)) = (seq𝑀(𝑄, 𝐺)‘𝑁))))
18		2fveq3 5594	. . . . . . 7 ⊢ (𝑥 = 𝑀 → (𝐻‘(𝐹‘𝑥)) = (𝐻‘(𝐹‘𝑀)))
19		fveq2 5589	. . . . . . 7 ⊢ (𝑥 = 𝑀 → (𝐺‘𝑥) = (𝐺‘𝑀))
20	18, 19	eqeq12d 2221	. . . . . 6 ⊢ (𝑥 = 𝑀 → ((𝐻‘(𝐹‘𝑥)) = (𝐺‘𝑥) ↔ (𝐻‘(𝐹‘𝑀)) = (𝐺‘𝑀)))
21		seq3homo.5	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐻‘(𝐹‘𝑥)) = (𝐺‘𝑥))
22	21	ralrimiva 2580	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ (ℤ≥‘𝑀)(𝐻‘(𝐹‘𝑥)) = (𝐺‘𝑥))
23		eluzel2 9673	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
24	1, 23	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℤ)
25		uzid 9682	. . . . . . 7 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀))
26	24, 25	syl 14	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀))
27	20, 22, 26	rspcdva 2886	. . . . 5 ⊢ (𝜑 → (𝐻‘(𝐹‘𝑀)) = (𝐺‘𝑀))
28		seq3homo.2	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
29		seq3homo.1	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
30	24, 28, 29	seq3-1 10629	. . . . . 6 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀))
31	30	fveq2d 5593	. . . . 5 ⊢ (𝜑 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑀)) = (𝐻‘(𝐹‘𝑀)))
32		seq3homo.g	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆)
33		seq3homo.qcl	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)
34	24, 32, 33	seq3-1 10629	. . . . 5 ⊢ (𝜑 → (seq𝑀(𝑄, 𝐺)‘𝑀) = (𝐺‘𝑀))
35	27, 31, 34	3eqtr4d 2249	. . . 4 ⊢ (𝜑 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑀)) = (seq𝑀(𝑄, 𝐺)‘𝑀))
36	35	a1i 9	. . 3 ⊢ (𝑀 ∈ ℤ → (𝜑 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑀)) = (seq𝑀(𝑄, 𝐺)‘𝑀)))
37		oveq1 5964	. . . . . 6 ⊢ ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛)) = (seq𝑀(𝑄, 𝐺)‘𝑛) → ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐺‘(𝑛 + 1))) = ((seq𝑀(𝑄, 𝐺)‘𝑛)𝑄(𝐺‘(𝑛 + 1))))
38		simpr 110	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → 𝑛 ∈ (ℤ≥‘𝑀))
39	28	adantlr 477	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
40	29	adantlr 477	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
41	38, 39, 40	seq3p1 10632	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
42	41	fveq2d 5593	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (𝐻‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))) = (𝐻‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))))
43		seq3homo.4	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝐻‘(𝑥 + 𝑦)) = ((𝐻‘𝑥)𝑄(𝐻‘𝑦)))
44	43	ralrimivva 2589	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝐻‘(𝑥 + 𝑦)) = ((𝐻‘𝑥)𝑄(𝐻‘𝑦)))
45	44	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝐻‘(𝑥 + 𝑦)) = ((𝐻‘𝑥)𝑄(𝐻‘𝑦)))
46		eqid 2206	. . . . . . . . . . . 12 ⊢ (ℤ≥‘𝑀) = (ℤ≥‘𝑀)
47	46, 24, 28, 29	seqf 10631	. . . . . . . . . . 11 ⊢ (𝜑 → seq𝑀( + , 𝐹):(ℤ≥‘𝑀)⟶𝑆)
48	47	ffvelcdmda 5728	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (seq𝑀( + , 𝐹)‘𝑛) ∈ 𝑆)
49		fveq2 5589	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑛 + 1) → (𝐹‘𝑥) = (𝐹‘(𝑛 + 1)))
50	49	eleq1d 2275	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑛 + 1) → ((𝐹‘𝑥) ∈ 𝑆 ↔ (𝐹‘(𝑛 + 1)) ∈ 𝑆))
51	28	ralrimiva 2580	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑥 ∈ (ℤ≥‘𝑀)(𝐹‘𝑥) ∈ 𝑆)
52	51	adantr 276	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ∀𝑥 ∈ (ℤ≥‘𝑀)(𝐹‘𝑥) ∈ 𝑆)
53		peano2uz 9724	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → (𝑛 + 1) ∈ (ℤ≥‘𝑀))
54	38, 53	syl 14	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (𝑛 + 1) ∈ (ℤ≥‘𝑀))
55	50, 52, 54	rspcdva 2886	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (𝐹‘(𝑛 + 1)) ∈ 𝑆)
56		oveq1 5964	. . . . . . . . . . . . 13 ⊢ (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → (𝑥 + 𝑦) = ((seq𝑀( + , 𝐹)‘𝑛) + 𝑦))
57	56	fveq2d 5593	. . . . . . . . . . . 12 ⊢ (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → (𝐻‘(𝑥 + 𝑦)) = (𝐻‘((seq𝑀( + , 𝐹)‘𝑛) + 𝑦)))
58		fveq2 5589	. . . . . . . . . . . . 13 ⊢ (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → (𝐻‘𝑥) = (𝐻‘(seq𝑀( + , 𝐹)‘𝑛)))
59	58	oveq1d 5972	. . . . . . . . . . . 12 ⊢ (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → ((𝐻‘𝑥)𝑄(𝐻‘𝑦)) = ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐻‘𝑦)))
60	57, 59	eqeq12d 2221	. . . . . . . . . . 11 ⊢ (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → ((𝐻‘(𝑥 + 𝑦)) = ((𝐻‘𝑥)𝑄(𝐻‘𝑦)) ↔ (𝐻‘((seq𝑀( + , 𝐹)‘𝑛) + 𝑦)) = ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐻‘𝑦))))
61		oveq2 5965	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝐹‘(𝑛 + 1)) → ((seq𝑀( + , 𝐹)‘𝑛) + 𝑦) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
62	61	fveq2d 5593	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐹‘(𝑛 + 1)) → (𝐻‘((seq𝑀( + , 𝐹)‘𝑛) + 𝑦)) = (𝐻‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))))
63		fveq2 5589	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝐹‘(𝑛 + 1)) → (𝐻‘𝑦) = (𝐻‘(𝐹‘(𝑛 + 1))))
64	63	oveq2d 5973	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐹‘(𝑛 + 1)) → ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐻‘𝑦)) = ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐻‘(𝐹‘(𝑛 + 1)))))
65	62, 64	eqeq12d 2221	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐹‘(𝑛 + 1)) → ((𝐻‘((seq𝑀( + , 𝐹)‘𝑛) + 𝑦)) = ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐻‘𝑦)) ↔ (𝐻‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))) = ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐻‘(𝐹‘(𝑛 + 1))))))
66	60, 65	rspc2v 2894	. . . . . . . . . 10 ⊢ (((seq𝑀( + , 𝐹)‘𝑛) ∈ 𝑆 ∧ (𝐹‘(𝑛 + 1)) ∈ 𝑆) → (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝐻‘(𝑥 + 𝑦)) = ((𝐻‘𝑥)𝑄(𝐻‘𝑦)) → (𝐻‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))) = ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐻‘(𝐹‘(𝑛 + 1))))))
67	48, 55, 66	syl2anc 411	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝐻‘(𝑥 + 𝑦)) = ((𝐻‘𝑥)𝑄(𝐻‘𝑦)) → (𝐻‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))) = ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐻‘(𝐹‘(𝑛 + 1))))))
68	45, 67	mpd 13	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (𝐻‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))) = ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐻‘(𝐹‘(𝑛 + 1)))))
69		2fveq3 5594	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑛 + 1) → (𝐻‘(𝐹‘𝑥)) = (𝐻‘(𝐹‘(𝑛 + 1))))
70		fveq2 5589	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑛 + 1) → (𝐺‘𝑥) = (𝐺‘(𝑛 + 1)))
71	69, 70	eqeq12d 2221	. . . . . . . . . 10 ⊢ (𝑥 = (𝑛 + 1) → ((𝐻‘(𝐹‘𝑥)) = (𝐺‘𝑥) ↔ (𝐻‘(𝐹‘(𝑛 + 1))) = (𝐺‘(𝑛 + 1))))
72	22	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ∀𝑥 ∈ (ℤ≥‘𝑀)(𝐻‘(𝐹‘𝑥)) = (𝐺‘𝑥))
73	71, 72, 54	rspcdva 2886	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (𝐻‘(𝐹‘(𝑛 + 1))) = (𝐺‘(𝑛 + 1)))
74	73	oveq2d 5973	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐻‘(𝐹‘(𝑛 + 1)))) = ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐺‘(𝑛 + 1))))
75	42, 68, 74	3eqtrd 2243	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (𝐻‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))) = ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐺‘(𝑛 + 1))))
76	32	adantlr 477	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆)
77	33	adantlr 477	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)
78	38, 76, 77	seq3p1 10632	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (seq𝑀(𝑄, 𝐺)‘(𝑛 + 1)) = ((seq𝑀(𝑄, 𝐺)‘𝑛)𝑄(𝐺‘(𝑛 + 1))))
79	75, 78	eqeq12d 2221	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ((𝐻‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))) = (seq𝑀(𝑄, 𝐺)‘(𝑛 + 1)) ↔ ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐺‘(𝑛 + 1))) = ((seq𝑀(𝑄, 𝐺)‘𝑛)𝑄(𝐺‘(𝑛 + 1)))))
80	37, 79	imbitrrid 156	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛)) = (seq𝑀(𝑄, 𝐺)‘𝑛) → (𝐻‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))) = (seq𝑀(𝑄, 𝐺)‘(𝑛 + 1))))
81	80	expcom 116	. . . 4 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → (𝜑 → ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛)) = (seq𝑀(𝑄, 𝐺)‘𝑛) → (𝐻‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))) = (seq𝑀(𝑄, 𝐺)‘(𝑛 + 1)))))
82	81	a2d 26	. . 3 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → ((𝜑 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑛)) = (seq𝑀(𝑄, 𝐺)‘𝑛)) → (𝜑 → (𝐻‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))) = (seq𝑀(𝑄, 𝐺)‘(𝑛 + 1)))))
83	5, 9, 13, 17, 36, 82	uzind4 9729	. 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝜑 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑁)) = (seq𝑀(𝑄, 𝐺)‘𝑁)))
84	1, 83	mpcom 36	1 ⊢ (𝜑 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑁)) = (seq𝑀(𝑄, 𝐺)‘𝑁))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1373   ∈ wcel 2177  ∀wral 2485  ‘cfv 5280  (class class class)co 5957  1c1 7946   + caddc 7948  ℤcz 9392  ℤ_≥cuz 9668  seqcseq 10614

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-iinf 4644  ax-cnex 8036  ax-resscn 8037  ax-1cn 8038  ax-1re 8039  ax-icn 8040  ax-addcl 8041  ax-addrcl 8042  ax-mulcl 8043  ax-addcom 8045  ax-addass 8047  ax-distr 8049  ax-i2m1 8050  ax-0lt1 8051  ax-0id 8053  ax-rnegex 8054  ax-cnre 8056  ax-pre-ltirr 8057  ax-pre-ltwlin 8058  ax-pre-lttrn 8059  ax-pre-ltadd 8061

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-tr 4151  df-id 4348  df-iord 4421  df-on 4423  df-ilim 4424  df-suc 4426  df-iom 4647  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-riota 5912  df-ov 5960  df-oprab 5961  df-mpo 5962  df-1st 6239  df-2nd 6240  df-recs 6404  df-frec 6490  df-pnf 8129  df-mnf 8130  df-xr 8131  df-ltxr 8132  df-le 8133  df-sub 8265  df-neg 8266  df-inn 9057  df-n0 9316  df-z 9393  df-uz 9669  df-seqfrec 10615

This theorem is referenced by:  seqfeq3  10696  seq3distr  10699  efcj  12059