Theorem seqf1olem2a 14072

Description: Lemma for seqf1o 14075. (Contributed by Mario Carneiro, 24-Apr-2016.)

Hypotheses

Ref	Expression
seqf1o.1	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
seqf1o.2	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
seqf1o.3	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
seqf1o.4	⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
seqf1o.5	⊢ (𝜑 → 𝐶 ⊆ 𝑆)
seqf1olem2a.1	⊢ (𝜑 → 𝐺:𝐴⟶𝐶)
seqf1olem2a.3	⊢ (𝜑 → 𝐾 ∈ 𝐴)
seqf1olem2a.4	⊢ (𝜑 → (𝑀...𝑁) ⊆ 𝐴)

Assertion

Ref	Expression
seqf1olem2a	⊢ (𝜑 → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑁)) = ((seq𝑀( + , 𝐺)‘𝑁) + (𝐺‘𝐾)))

Distinct variable groups:   𝑥,𝑦,𝑧,𝐺   𝑥,𝑀,𝑦,𝑧   𝑥, + ,𝑦,𝑧   𝑥,𝑁,𝑦,𝑧   𝑥,𝐾,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧

Allowed substitution hints:   𝐴(𝑥,𝑦,𝑧)

Proof of Theorem seqf1olem2a

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

Step	Hyp	Ref	Expression
1		seqf1o.4	. . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
2		eluzfz2 13555	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
3	1, 2	syl 18	. 2 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
4		fveq2 6881	. . . . . 6 ⊢ (𝑚 = 𝑀 → (seq𝑀( + , 𝐺)‘𝑚) = (seq𝑀( + , 𝐺)‘𝑀))
5	4	oveq2d 7426	. . . . 5 ⊢ (𝑚 = 𝑀 → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑀)))
6	4	oveq1d 7425	. . . . 5 ⊢ (𝑚 = 𝑀 → ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺‘𝐾)) = ((seq𝑀( + , 𝐺)‘𝑀) + (𝐺‘𝐾)))
7	5, 6	eqeq12d 2779	. . . 4 ⊢ (𝑚 = 𝑀 → (((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺‘𝐾)) ↔ ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑀)) = ((seq𝑀( + , 𝐺)‘𝑀) + (𝐺‘𝐾))))
8	7	imbi2d 343	. . 3 ⊢ (𝑚 = 𝑀 → ((𝜑 → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺‘𝐾))) ↔ (𝜑 → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑀)) = ((seq𝑀( + , 𝐺)‘𝑀) + (𝐺‘𝐾)))))
9		fveq2 6881	. . . . . 6 ⊢ (𝑚 = 𝑛 → (seq𝑀( + , 𝐺)‘𝑚) = (seq𝑀( + , 𝐺)‘𝑛))
10	9	oveq2d 7426	. . . . 5 ⊢ (𝑚 = 𝑛 → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑛)))
11	9	oveq1d 7425	. . . . 5 ⊢ (𝑚 = 𝑛 → ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺‘𝐾)) = ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘𝐾)))
12	10, 11	eqeq12d 2779	. . . 4 ⊢ (𝑚 = 𝑛 → (((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺‘𝐾)) ↔ ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑛)) = ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘𝐾))))
13	12	imbi2d 343	. . 3 ⊢ (𝑚 = 𝑛 → ((𝜑 → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺‘𝐾))) ↔ (𝜑 → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑛)) = ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘𝐾)))))
14		fveq2 6881	. . . . . 6 ⊢ (𝑚 = (𝑛 + 1) → (seq𝑀( + , 𝐺)‘𝑚) = (seq𝑀( + , 𝐺)‘(𝑛 + 1)))
15	14	oveq2d 7426	. . . . 5 ⊢ (𝑚 = (𝑛 + 1) → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘(𝑛 + 1))))
16	14	oveq1d 7425	. . . . 5 ⊢ (𝑚 = (𝑛 + 1) → ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺‘𝐾)) = ((seq𝑀( + , 𝐺)‘(𝑛 + 1)) + (𝐺‘𝐾)))
17	15, 16	eqeq12d 2779	. . . 4 ⊢ (𝑚 = (𝑛 + 1) → (((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺‘𝐾)) ↔ ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘(𝑛 + 1))) = ((seq𝑀( + , 𝐺)‘(𝑛 + 1)) + (𝐺‘𝐾))))
18	17	imbi2d 343	. . 3 ⊢ (𝑚 = (𝑛 + 1) → ((𝜑 → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺‘𝐾))) ↔ (𝜑 → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘(𝑛 + 1))) = ((seq𝑀( + , 𝐺)‘(𝑛 + 1)) + (𝐺‘𝐾)))))
19		fveq2 6881	. . . . . 6 ⊢ (𝑚 = 𝑁 → (seq𝑀( + , 𝐺)‘𝑚) = (seq𝑀( + , 𝐺)‘𝑁))
20	19	oveq2d 7426	. . . . 5 ⊢ (𝑚 = 𝑁 → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑁)))
21	19	oveq1d 7425	. . . . 5 ⊢ (𝑚 = 𝑁 → ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺‘𝐾)) = ((seq𝑀( + , 𝐺)‘𝑁) + (𝐺‘𝐾)))
22	20, 21	eqeq12d 2779	. . . 4 ⊢ (𝑚 = 𝑁 → (((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺‘𝐾)) ↔ ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑁)) = ((seq𝑀( + , 𝐺)‘𝑁) + (𝐺‘𝐾))))
23	22	imbi2d 343	. . 3 ⊢ (𝑚 = 𝑁 → ((𝜑 → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺‘𝐾))) ↔ (𝜑 → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑁)) = ((seq𝑀( + , 𝐺)‘𝑁) + (𝐺‘𝐾)))))
24		seqf1o.2	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
25		seqf1olem2a.1	. . . . . 6 ⊢ (𝜑 → 𝐺:𝐴⟶𝐶)
26		seqf1olem2a.3	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ 𝐴)
27	25, 26	ffvelcdmd 7080	. . . . 5 ⊢ (𝜑 → (𝐺‘𝐾) ∈ 𝐶)
28		eluzel2 12862	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
29		seq1 14046	. . . . . . 7 ⊢ (𝑀 ∈ ℤ → (seq𝑀( + , 𝐺)‘𝑀) = (𝐺‘𝑀))
30	1, 28, 29	3syl 19	. . . . . 6 ⊢ (𝜑 → (seq𝑀( + , 𝐺)‘𝑀) = (𝐺‘𝑀))
31		seqf1olem2a.4	. . . . . . . 8 ⊢ (𝜑 → (𝑀...𝑁) ⊆ 𝐴)
32		eluzfz1 13554	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
33	1, 32	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁))
34	31, 33	sseldd 3938	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ 𝐴)
35	25, 34	ffvelcdmd 7080	. . . . . 6 ⊢ (𝜑 → (𝐺‘𝑀) ∈ 𝐶)
36	30, 35	eqeltrd 2863	. . . . 5 ⊢ (𝜑 → (seq𝑀( + , 𝐺)‘𝑀) ∈ 𝐶)
37	24, 27, 36	caovcomd 7606	. . . 4 ⊢ (𝜑 → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑀)) = ((seq𝑀( + , 𝐺)‘𝑀) + (𝐺‘𝐾)))
38	37	a1i 11	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝜑 → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑀)) = ((seq𝑀( + , 𝐺)‘𝑀) + (𝐺‘𝐾))))
39		oveq1 7417	. . . . . 6 ⊢ (((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑛)) = ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘𝐾)) → (((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑛)) + (𝐺‘(𝑛 + 1))) = (((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘𝐾)) + (𝐺‘(𝑛 + 1))))
40		elfzouz 13688	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (𝑀..^𝑁) → 𝑛 ∈ (ℤ≥‘𝑀))
41	40	adantl 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → 𝑛 ∈ (ℤ≥‘𝑀))
42		seqp1 14048	. . . . . . . . . 10 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → (seq𝑀( + , 𝐺)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1))))
43	41, 42	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( + , 𝐺)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1))))
44	43	oveq2d 7426	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘(𝑛 + 1))) = ((𝐺‘𝐾) + ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1)))))
45		seqf1o.3	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
46	45	adantlr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
47		seqf1o.5	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ⊆ 𝑆)
48	47, 27	sseldd 3938	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺‘𝐾) ∈ 𝑆)
49	48	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝐺‘𝐾) ∈ 𝑆)
50	47	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → 𝐶 ⊆ 𝑆)
51	50	adantr 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑛)) → 𝐶 ⊆ 𝑆)
52	25	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → 𝐺:𝐴⟶𝐶)
53	52	adantr 485	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑛)) → 𝐺:𝐴⟶𝐶)
54		elfzouz2 13699	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (𝑀..^𝑁) → 𝑁 ∈ (ℤ≥‘𝑛))
55	54	adantl 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → 𝑁 ∈ (ℤ≥‘𝑛))
56		fzss2 13588	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (ℤ≥‘𝑛) → (𝑀...𝑛) ⊆ (𝑀...𝑁))
57	55, 56	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝑀...𝑛) ⊆ (𝑀...𝑁))
58	31	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝑀...𝑁) ⊆ 𝐴)
59	57, 58	sstrd 3947	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝑀...𝑛) ⊆ 𝐴)
60	59	sselda 3937	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑛)) → 𝑥 ∈ 𝐴)
61	53, 60	ffvelcdmd 7080	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑛)) → (𝐺‘𝑥) ∈ 𝐶)
62	51, 61	sseldd 3938	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑛)) → (𝐺‘𝑥) ∈ 𝑆)
63		seqf1o.1	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
64	63	adantlr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
65	41, 62, 64	seqcl 14054	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( + , 𝐺)‘𝑛) ∈ 𝑆)
66		fzofzp1 13789	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (𝑀..^𝑁) → (𝑛 + 1) ∈ (𝑀...𝑁))
67	66	adantl 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝑛 + 1) ∈ (𝑀...𝑁))
68	58, 67	sseldd 3938	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝑛 + 1) ∈ 𝐴)
69	52, 68	ffvelcdmd 7080	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝐺‘(𝑛 + 1)) ∈ 𝐶)
70	50, 69	sseldd 3938	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝐺‘(𝑛 + 1)) ∈ 𝑆)
71	46, 49, 65, 70	caovassd 7609	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑛)) + (𝐺‘(𝑛 + 1))) = ((𝐺‘𝐾) + ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1)))))
72	44, 71	eqtr4d 2801	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘(𝑛 + 1))) = (((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑛)) + (𝐺‘(𝑛 + 1))))
73	46, 65, 70, 49	caovassd 7609	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1))) + (𝐺‘𝐾)) = ((seq𝑀( + , 𝐺)‘𝑛) + ((𝐺‘(𝑛 + 1)) + (𝐺‘𝐾))))
74	43	oveq1d 7425	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → ((seq𝑀( + , 𝐺)‘(𝑛 + 1)) + (𝐺‘𝐾)) = (((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1))) + (𝐺‘𝐾)))
75	46, 65, 49, 70	caovassd 7609	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘𝐾)) + (𝐺‘(𝑛 + 1))) = ((seq𝑀( + , 𝐺)‘𝑛) + ((𝐺‘𝐾) + (𝐺‘(𝑛 + 1)))))
76	24	adantlr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
77	27	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝐺‘𝐾) ∈ 𝐶)
78	76, 69, 77	caovcomd 7606	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → ((𝐺‘(𝑛 + 1)) + (𝐺‘𝐾)) = ((𝐺‘𝐾) + (𝐺‘(𝑛 + 1))))
79	78	oveq2d 7426	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → ((seq𝑀( + , 𝐺)‘𝑛) + ((𝐺‘(𝑛 + 1)) + (𝐺‘𝐾))) = ((seq𝑀( + , 𝐺)‘𝑛) + ((𝐺‘𝐾) + (𝐺‘(𝑛 + 1)))))
80	75, 79	eqtr4d 2801	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘𝐾)) + (𝐺‘(𝑛 + 1))) = ((seq𝑀( + , 𝐺)‘𝑛) + ((𝐺‘(𝑛 + 1)) + (𝐺‘𝐾))))
81	73, 74, 80	3eqtr4d 2808	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → ((seq𝑀( + , 𝐺)‘(𝑛 + 1)) + (𝐺‘𝐾)) = (((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘𝐾)) + (𝐺‘(𝑛 + 1))))
82	72, 81	eqeq12d 2779	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘(𝑛 + 1))) = ((seq𝑀( + , 𝐺)‘(𝑛 + 1)) + (𝐺‘𝐾)) ↔ (((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑛)) + (𝐺‘(𝑛 + 1))) = (((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘𝐾)) + (𝐺‘(𝑛 + 1)))))
83	39, 82	imbitrrid 249	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑛)) = ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘𝐾)) → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘(𝑛 + 1))) = ((seq𝑀( + , 𝐺)‘(𝑛 + 1)) + (𝐺‘𝐾))))
84	83	expcom 418	. . . 4 ⊢ (𝑛 ∈ (𝑀..^𝑁) → (𝜑 → (((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑛)) = ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘𝐾)) → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘(𝑛 + 1))) = ((seq𝑀( + , 𝐺)‘(𝑛 + 1)) + (𝐺‘𝐾)))))
85	84	a2d 30	. . 3 ⊢ (𝑛 ∈ (𝑀..^𝑁) → ((𝜑 → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑛)) = ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘𝐾))) → (𝜑 → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘(𝑛 + 1))) = ((seq𝑀( + , 𝐺)‘(𝑛 + 1)) + (𝐺‘𝐾)))))
86	8, 13, 18, 23, 38, 85	fzind2 13813	. 2 ⊢ (𝑁 ∈ (𝑀...𝑁) → (𝜑 → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑁)) = ((seq𝑀( + , 𝐺)‘𝑁) + (𝐺‘𝐾))))
87	3, 86	mpcom 39	1 ⊢ (𝜑 → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑁)) = ((seq𝑀( + , 𝐺)‘𝑁) + (𝐺‘𝐾)))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ⊆ wss 3905  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  1c1 11096   + caddc 11098  ℤcz 12586  ℤ_≥cuz 12857  ...cfz 13530  ..^cfzo 13678  seqcseq 14033

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-er 8690  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-nn 12229  df-n0 12500  df-z 12587  df-uz 12858  df-fz 13531  df-fzo 13679  df-seq 14034

This theorem is referenced by:  seqf1olem2  14074