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Theorem seqhomo 13990

Description: Apply a homomorphism to a sequence. (Contributed by Mario Carneiro, 28-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)

**Hypotheses**
Ref	Expression
seqhomo.1	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
seqhomo.2	⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ 𝑆)
seqhomo.3	⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
seqhomo.4	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝐻‘(𝑥 + 𝑦)) = ((𝐻‘𝑥)𝑄(𝐻‘𝑦)))
seqhomo.5	⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐻‘(𝐹‘𝑥)) = (𝐺‘𝑥))

**Assertion**
Ref	Expression
seqhomo	⊢ (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 Allowed substitution hint: 𝐺(𝑦)

Proof of Theorem seqhomo Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		seqhomo.3	. . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
2		eluzfz2 13469	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
3	1, 2	syl 17	. 2 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
4		eleq1 2816	. . . . . 6 ⊢ (𝑥 = 𝑀 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑀 ∈ (𝑀...𝑁)))
5		2fveq3 6845	. . . . . . 7 ⊢ (𝑥 = 𝑀 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑥)) = (𝐻‘(seq𝑀( + , 𝐹)‘𝑀)))
6		fveq2 6840	. . . . . . 7 ⊢ (𝑥 = 𝑀 → (seq𝑀(𝑄, 𝐺)‘𝑥) = (seq𝑀(𝑄, 𝐺)‘𝑀))
7	5, 6	eqeq12d 2745	. . . . . 6 ⊢ (𝑥 = 𝑀 → ((𝐻‘(seq𝑀( + , 𝐹)‘𝑥)) = (seq𝑀(𝑄, 𝐺)‘𝑥) ↔ (𝐻‘(seq𝑀( + , 𝐹)‘𝑀)) = (seq𝑀(𝑄, 𝐺)‘𝑀)))
8	4, 7	imbi12d 344	. . . . 5 ⊢ (𝑥 = 𝑀 → ((𝑥 ∈ (𝑀...𝑁) → (𝐻‘(seq𝑀( + , 𝐹)‘𝑥)) = (seq𝑀(𝑄, 𝐺)‘𝑥)) ↔ (𝑀 ∈ (𝑀...𝑁) → (𝐻‘(seq𝑀( + , 𝐹)‘𝑀)) = (seq𝑀(𝑄, 𝐺)‘𝑀))))
9	8	imbi2d 340	. . . 4 ⊢ (𝑥 = 𝑀 → ((𝜑 → (𝑥 ∈ (𝑀...𝑁) → (𝐻‘(seq𝑀( + , 𝐹)‘𝑥)) = (seq𝑀(𝑄, 𝐺)‘𝑥))) ↔ (𝜑 → (𝑀 ∈ (𝑀...𝑁) → (𝐻‘(seq𝑀( + , 𝐹)‘𝑀)) = (seq𝑀(𝑄, 𝐺)‘𝑀)))))
10		eleq1 2816	. . . . . 6 ⊢ (𝑥 = 𝑛 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑛 ∈ (𝑀...𝑁)))
11		2fveq3 6845	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑥)) = (𝐻‘(seq𝑀( + , 𝐹)‘𝑛)))
12		fveq2 6840	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (seq𝑀(𝑄, 𝐺)‘𝑥) = (seq𝑀(𝑄, 𝐺)‘𝑛))
13	11, 12	eqeq12d 2745	. . . . . 6 ⊢ (𝑥 = 𝑛 → ((𝐻‘(seq𝑀( + , 𝐹)‘𝑥)) = (seq𝑀(𝑄, 𝐺)‘𝑥) ↔ (𝐻‘(seq𝑀( + , 𝐹)‘𝑛)) = (seq𝑀(𝑄, 𝐺)‘𝑛)))
14	10, 13	imbi12d 344	. . . . 5 ⊢ (𝑥 = 𝑛 → ((𝑥 ∈ (𝑀...𝑁) → (𝐻‘(seq𝑀( + , 𝐹)‘𝑥)) = (seq𝑀(𝑄, 𝐺)‘𝑥)) ↔ (𝑛 ∈ (𝑀...𝑁) → (𝐻‘(seq𝑀( + , 𝐹)‘𝑛)) = (seq𝑀(𝑄, 𝐺)‘𝑛))))
15	14	imbi2d 340	. . . 4 ⊢ (𝑥 = 𝑛 → ((𝜑 → (𝑥 ∈ (𝑀...𝑁) → (𝐻‘(seq𝑀( + , 𝐹)‘𝑥)) = (seq𝑀(𝑄, 𝐺)‘𝑥))) ↔ (𝜑 → (𝑛 ∈ (𝑀...𝑁) → (𝐻‘(seq𝑀( + , 𝐹)‘𝑛)) = (seq𝑀(𝑄, 𝐺)‘𝑛)))))
16		eleq1 2816	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → (𝑥 ∈ (𝑀...𝑁) ↔ (𝑛 + 1) ∈ (𝑀...𝑁)))
17		2fveq3 6845	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → (𝐻‘(seq𝑀( + , 𝐹)‘𝑥)) = (𝐻‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))))
18		fveq2 6840	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → (seq𝑀(𝑄, 𝐺)‘𝑥) = (seq𝑀(𝑄, 𝐺)‘(𝑛 + 1)))
19	17, 18	eqeq12d 2745	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → ((𝐻‘(seq𝑀( + , 𝐹)‘𝑥)) = (seq𝑀(𝑄, 𝐺)‘𝑥) ↔ (𝐻‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))) = (seq𝑀(𝑄, 𝐺)‘(𝑛 + 1))))
20	16, 19	imbi12d 344	. . . . 5 ⊢ (𝑥 = (𝑛 + 1) → ((𝑥 ∈ (𝑀...𝑁) → (𝐻‘(seq𝑀( + , 𝐹)‘𝑥)) = (seq𝑀(𝑄, 𝐺)‘𝑥)) ↔ ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝐻‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))) = (seq𝑀(𝑄, 𝐺)‘(𝑛 + 1)))))
21	20	imbi2d 340	. . . 4 ⊢ (𝑥 = (𝑛 + 1) → ((𝜑 → (𝑥 ∈ (𝑀...𝑁) → (𝐻‘(seq𝑀( + , 𝐹)‘𝑥)) = (seq𝑀(𝑄, 𝐺)‘𝑥))) ↔ (𝜑 → ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝐻‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))) = (seq𝑀(𝑄, 𝐺)‘(𝑛 + 1))))))
22		eleq1 2816	. . . . . 6 ⊢ (𝑥 = 𝑁 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑁 ∈ (𝑀...𝑁)))
23		2fveq3 6845	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑥)) = (𝐻‘(seq𝑀( + , 𝐹)‘𝑁)))
24		fveq2 6840	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (seq𝑀(𝑄, 𝐺)‘𝑥) = (seq𝑀(𝑄, 𝐺)‘𝑁))
25	23, 24	eqeq12d 2745	. . . . . 6 ⊢ (𝑥 = 𝑁 → ((𝐻‘(seq𝑀( + , 𝐹)‘𝑥)) = (seq𝑀(𝑄, 𝐺)‘𝑥) ↔ (𝐻‘(seq𝑀( + , 𝐹)‘𝑁)) = (seq𝑀(𝑄, 𝐺)‘𝑁)))
26	22, 25	imbi12d 344	. . . . 5 ⊢ (𝑥 = 𝑁 → ((𝑥 ∈ (𝑀...𝑁) → (𝐻‘(seq𝑀( + , 𝐹)‘𝑥)) = (seq𝑀(𝑄, 𝐺)‘𝑥)) ↔ (𝑁 ∈ (𝑀...𝑁) → (𝐻‘(seq𝑀( + , 𝐹)‘𝑁)) = (seq𝑀(𝑄, 𝐺)‘𝑁))))
27	26	imbi2d 340	. . . 4 ⊢ (𝑥 = 𝑁 → ((𝜑 → (𝑥 ∈ (𝑀...𝑁) → (𝐻‘(seq𝑀( + , 𝐹)‘𝑥)) = (seq𝑀(𝑄, 𝐺)‘𝑥))) ↔ (𝜑 → (𝑁 ∈ (𝑀...𝑁) → (𝐻‘(seq𝑀( + , 𝐹)‘𝑁)) = (seq𝑀(𝑄, 𝐺)‘𝑁)))))
28		2fveq3 6845	. . . . . . . 8 ⊢ (𝑥 = 𝑀 → (𝐻‘(𝐹‘𝑥)) = (𝐻‘(𝐹‘𝑀)))
29		fveq2 6840	. . . . . . . 8 ⊢ (𝑥 = 𝑀 → (𝐺‘𝑥) = (𝐺‘𝑀))
30	28, 29	eqeq12d 2745	. . . . . . 7 ⊢ (𝑥 = 𝑀 → ((𝐻‘(𝐹‘𝑥)) = (𝐺‘𝑥) ↔ (𝐻‘(𝐹‘𝑀)) = (𝐺‘𝑀)))
31		seqhomo.5	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐻‘(𝐹‘𝑥)) = (𝐺‘𝑥))
32	31	ralrimiva 3125	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ (𝑀...𝑁)(𝐻‘(𝐹‘𝑥)) = (𝐺‘𝑥))
33		eluzfz1 13468	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
34	1, 33	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁))
35	30, 32, 34	rspcdva 3586	. . . . . 6 ⊢ (𝜑 → (𝐻‘(𝐹‘𝑀)) = (𝐺‘𝑀))
36		eluzel2 12774	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ ℤ)
37		seq1 13955	. . . . . . . 8 ⊢ (𝑀 ∈ ℤ → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀))
38	1, 36, 37	3syl 18	. . . . . . 7 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀))
39	38	fveq2d 6844	. . . . . 6 ⊢ (𝜑 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑀)) = (𝐻‘(𝐹‘𝑀)))
40		seq1 13955	. . . . . . 7 ⊢ (𝑀 ∈ ℤ → (seq𝑀(𝑄, 𝐺)‘𝑀) = (𝐺‘𝑀))
41	1, 36, 40	3syl 18	. . . . . 6 ⊢ (𝜑 → (seq𝑀(𝑄, 𝐺)‘𝑀) = (𝐺‘𝑀))
42	35, 39, 41	3eqtr4d 2774	. . . . 5 ⊢ (𝜑 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑀)) = (seq𝑀(𝑄, 𝐺)‘𝑀))
43	42	a1d 25	. . . 4 ⊢ (𝜑 → (𝑀 ∈ (𝑀...𝑁) → (𝐻‘(seq𝑀( + , 𝐹)‘𝑀)) = (seq𝑀(𝑄, 𝐺)‘𝑀)))
44		peano2fzr 13474	. . . . . . . 8 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → 𝑛 ∈ (𝑀...𝑁))
45	44	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ (𝑀...𝑁))
46	45	expr 456	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → ((𝑛 + 1) ∈ (𝑀...𝑁) → 𝑛 ∈ (𝑀...𝑁)))
47	46	imim1d 82	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → ((𝑛 ∈ (𝑀...𝑁) → (𝐻‘(seq𝑀( + , 𝐹)‘𝑛)) = (seq𝑀(𝑄, 𝐺)‘𝑛)) → ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝐻‘(seq𝑀( + , 𝐹)‘𝑛)) = (seq𝑀(𝑄, 𝐺)‘𝑛))))
48		oveq1 7376	. . . . . 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))))
49		seqp1 13957	. . . . . . . . . 10 ⊢ (𝑛 ∈ (ℤ_≥‘𝑀) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
50	49	ad2antrl 728	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
51	50	fveq2d 6844	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐻‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))) = (𝐻‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))))
52		seqhomo.4	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝐻‘(𝑥 + 𝑦)) = ((𝐻‘𝑥)𝑄(𝐻‘𝑦)))
53	52	ralrimivva 3178	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝐻‘(𝑥 + 𝑦)) = ((𝐻‘𝑥)𝑄(𝐻‘𝑦)))
54	53	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝐻‘(𝑥 + 𝑦)) = ((𝐻‘𝑥)𝑄(𝐻‘𝑦)))
55		simprl 770	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ (ℤ_≥‘𝑀))
56		elfzuz3 13458	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ_≥‘𝑛))
57		fzss2 13501	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (ℤ_≥‘𝑛) → (𝑀...𝑛) ⊆ (𝑀...𝑁))
58	45, 56, 57	3syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝑀...𝑛) ⊆ (𝑀...𝑁))
59	58	sselda 3943	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) ∧ 𝑥 ∈ (𝑀...𝑛)) → 𝑥 ∈ (𝑀...𝑁))
60		seqhomo.2	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ 𝑆)
61	60	adantlr 715	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ 𝑆)
62	59, 61	syldan 591	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) ∧ 𝑥 ∈ (𝑀...𝑛)) → (𝐹‘𝑥) ∈ 𝑆)
63		seqhomo.1	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
64	63	adantlr 715	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
65	55, 62, 64	seqcl 13963	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (seq𝑀( + , 𝐹)‘𝑛) ∈ 𝑆)
66		fveq2 6840	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑛 + 1) → (𝐹‘𝑥) = (𝐹‘(𝑛 + 1)))
67	66	eleq1d 2813	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑛 + 1) → ((𝐹‘𝑥) ∈ 𝑆 ↔ (𝐹‘(𝑛 + 1)) ∈ 𝑆))
68	60	ralrimiva 3125	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑥 ∈ (𝑀...𝑁)(𝐹‘𝑥) ∈ 𝑆)
69	68	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ∀𝑥 ∈ (𝑀...𝑁)(𝐹‘𝑥) ∈ 𝑆)
70		simprr 772	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝑛 + 1) ∈ (𝑀...𝑁))
71	67, 69, 70	rspcdva 3586	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐹‘(𝑛 + 1)) ∈ 𝑆)
72		fvoveq1 7392	. . . . . . . . . . . 12 ⊢ (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → (𝐻‘(𝑥 + 𝑦)) = (𝐻‘((seq𝑀( + , 𝐹)‘𝑛) + 𝑦)))
73		fveq2 6840	. . . . . . . . . . . . 13 ⊢ (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → (𝐻‘𝑥) = (𝐻‘(seq𝑀( + , 𝐹)‘𝑛)))
74	73	oveq1d 7384	. . . . . . . . . . . 12 ⊢ (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → ((𝐻‘𝑥)𝑄(𝐻‘𝑦)) = ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐻‘𝑦)))
75	72, 74	eqeq12d 2745	. . . . . . . . . . 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))))
76		oveq2 7377	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝐹‘(𝑛 + 1)) → ((seq𝑀( + , 𝐹)‘𝑛) + 𝑦) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
77	76	fveq2d 6844	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐹‘(𝑛 + 1)) → (𝐻‘((seq𝑀( + , 𝐹)‘𝑛) + 𝑦)) = (𝐻‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))))
78		fveq2 6840	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝐹‘(𝑛 + 1)) → (𝐻‘𝑦) = (𝐻‘(𝐹‘(𝑛 + 1))))
79	78	oveq2d 7385	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐹‘(𝑛 + 1)) → ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐻‘𝑦)) = ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐻‘(𝐹‘(𝑛 + 1)))))
80	77, 79	eqeq12d 2745	. . . . . . . . . . 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))))))
81	75, 80	rspc2v 3596	. . . . . . . . . 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))))))
82	65, 71, 81	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (∀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))))))
83	54, 82	mpd 15	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐻‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))) = ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐻‘(𝐹‘(𝑛 + 1)))))
84		2fveq3 6845	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑛 + 1) → (𝐻‘(𝐹‘𝑥)) = (𝐻‘(𝐹‘(𝑛 + 1))))
85		fveq2 6840	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑛 + 1) → (𝐺‘𝑥) = (𝐺‘(𝑛 + 1)))
86	84, 85	eqeq12d 2745	. . . . . . . . . 10 ⊢ (𝑥 = (𝑛 + 1) → ((𝐻‘(𝐹‘𝑥)) = (𝐺‘𝑥) ↔ (𝐻‘(𝐹‘(𝑛 + 1))) = (𝐺‘(𝑛 + 1))))
87	32	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ∀𝑥 ∈ (𝑀...𝑁)(𝐻‘(𝐹‘𝑥)) = (𝐺‘𝑥))
88	86, 87, 70	rspcdva 3586	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐻‘(𝐹‘(𝑛 + 1))) = (𝐺‘(𝑛 + 1)))
89	88	oveq2d 7385	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐻‘(𝐹‘(𝑛 + 1)))) = ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐺‘(𝑛 + 1))))
90	51, 83, 89	3eqtrd 2768	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐻‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))) = ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛))𝑄(𝐺‘(𝑛 + 1))))
91		seqp1 13957	. . . . . . . 8 ⊢ (𝑛 ∈ (ℤ_≥‘𝑀) → (seq𝑀(𝑄, 𝐺)‘(𝑛 + 1)) = ((seq𝑀(𝑄, 𝐺)‘𝑛)𝑄(𝐺‘(𝑛 + 1))))
92	91	ad2antrl 728	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (seq𝑀(𝑄, 𝐺)‘(𝑛 + 1)) = ((seq𝑀(𝑄, 𝐺)‘𝑛)𝑄(𝐺‘(𝑛 + 1))))
93	90, 92	eqeq12d 2745	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ((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)))))
94	48, 93	imbitrrid 246	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ((𝐻‘(seq𝑀( + , 𝐹)‘𝑛)) = (seq𝑀(𝑄, 𝐺)‘𝑛) → (𝐻‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))) = (seq𝑀(𝑄, 𝐺)‘(𝑛 + 1))))
95	47, 94	animpimp2impd 846	. . . 4 ⊢ (𝑛 ∈ (ℤ_≥‘𝑀) → ((𝜑 → (𝑛 ∈ (𝑀...𝑁) → (𝐻‘(seq𝑀( + , 𝐹)‘𝑛)) = (seq𝑀(𝑄, 𝐺)‘𝑛))) → (𝜑 → ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝐻‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))) = (seq𝑀(𝑄, 𝐺)‘(𝑛 + 1))))))
96	9, 15, 21, 27, 43, 95	uzind4i 12845	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (𝜑 → (𝑁 ∈ (𝑀...𝑁) → (𝐻‘(seq𝑀( + , 𝐹)‘𝑁)) = (seq𝑀(𝑄, 𝐺)‘𝑁))))
97	1, 96	mpcom 38	. 2 ⊢ (𝜑 → (𝑁 ∈ (𝑀...𝑁) → (𝐻‘(seq𝑀( + , 𝐹)‘𝑁)) = (seq𝑀(𝑄, 𝐺)‘𝑁)))
98	3, 97	mpd 15	1 ⊢ (𝜑 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑁)) = (seq𝑀(𝑄, 𝐺)‘𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∀wral 3044 ⊆ wss 3911 ‘cfv 6499 (class class class)co 7369 1c1 11045 + caddc 11047 ℤcz 12505 ℤ_≥cuz 12769 ...cfz 13444 seqcseq 13942

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-1st 7947 df-2nd 7948 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-er 8648 df-en 8896 df-dom 8897 df-sdom 8898 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-nn 12163 df-n0 12419 df-z 12506 df-uz 12770 df-fz 13445 df-seq 13943

This theorem is referenced by: seqfeq4 13992 seqdistr 13994 seqof 14000 fsumrelem 15749 efcj 16034 gsumwmhm 18748 gsumzmhm 19843 elqaalem2 26204 logfac 26486 gamcvg2lem 26945 prmorcht 27064 pclogsum 27102

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