Theorem seqf1oglem2a 10933

Description: Lemma for seqf1og 10936. (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)
seqf1og.p	⊢ (𝜑 → + ∈ 𝑉)
seqf1olem2a.1	⊢ (𝜑 → 𝐺:𝐴⟶𝐶)
seqf1olem2a.3	⊢ (𝜑 → 𝐾 ∈ 𝐴)
seqf1olem2a.4	⊢ (𝜑 → (𝑀...𝑁) ⊆ 𝐴)
seqf1oglem2a.a	⊢ (𝜑 → 𝐴 ∈ 𝑊)

Assertion

Ref	Expression
seqf1oglem2a	⊢ (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)   𝑉(𝑥,𝑦,𝑧)   𝑊(𝑥,𝑦,𝑧)

Proof of Theorem seqf1oglem2a

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 10415	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
3	1, 2	syl 14	. 2 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
4		fveq2 5690	. . . . . 6 ⊢ (𝑚 = 𝑀 → (seq𝑀( + , 𝐺)‘𝑚) = (seq𝑀( + , 𝐺)‘𝑀))
5	4	oveq2d 6091	. . . . 5 ⊢ (𝑚 = 𝑀 → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑀)))
6	4	oveq1d 6090	. . . . 5 ⊢ (𝑚 = 𝑀 → ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺‘𝐾)) = ((seq𝑀( + , 𝐺)‘𝑀) + (𝐺‘𝐾)))
7	5, 6	eqeq12d 2253	. . . 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 230	. . 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 5690	. . . . . 6 ⊢ (𝑚 = 𝑛 → (seq𝑀( + , 𝐺)‘𝑚) = (seq𝑀( + , 𝐺)‘𝑛))
10	9	oveq2d 6091	. . . . 5 ⊢ (𝑚 = 𝑛 → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑛)))
11	9	oveq1d 6090	. . . . 5 ⊢ (𝑚 = 𝑛 → ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺‘𝐾)) = ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘𝐾)))
12	10, 11	eqeq12d 2253	. . . 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 230	. . 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 5690	. . . . . 6 ⊢ (𝑚 = (𝑛 + 1) → (seq𝑀( + , 𝐺)‘𝑚) = (seq𝑀( + , 𝐺)‘(𝑛 + 1)))
15	14	oveq2d 6091	. . . . 5 ⊢ (𝑚 = (𝑛 + 1) → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘(𝑛 + 1))))
16	14	oveq1d 6090	. . . . 5 ⊢ (𝑚 = (𝑛 + 1) → ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺‘𝐾)) = ((seq𝑀( + , 𝐺)‘(𝑛 + 1)) + (𝐺‘𝐾)))
17	15, 16	eqeq12d 2253	. . . 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 230	. . 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 5690	. . . . . 6 ⊢ (𝑚 = 𝑁 → (seq𝑀( + , 𝐺)‘𝑚) = (seq𝑀( + , 𝐺)‘𝑁))
20	19	oveq2d 6091	. . . . 5 ⊢ (𝑚 = 𝑁 → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑁)))
21	19	oveq1d 6090	. . . . 5 ⊢ (𝑚 = 𝑁 → ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺‘𝐾)) = ((seq𝑀( + , 𝐺)‘𝑁) + (𝐺‘𝐾)))
22	20, 21	eqeq12d 2253	. . . 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 230	. . 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 5835	. . . . 5 ⊢ (𝜑 → (𝐺‘𝐾) ∈ 𝐶)
28		eluzel2 9905	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
29	1, 28	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℤ)
30		seqf1oglem2a.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝑊)
31	25, 30	fexd 5938	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ V)
32		seqf1og.p	. . . . . . 7 ⊢ (𝜑 → + ∈ 𝑉)
33		seq1g 10878	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝐺 ∈ V ∧ + ∈ 𝑉) → (seq𝑀( + , 𝐺)‘𝑀) = (𝐺‘𝑀))
34	29, 31, 32, 33	syl3anc 1278	. . . . . 6 ⊢ (𝜑 → (seq𝑀( + , 𝐺)‘𝑀) = (𝐺‘𝑀))
35		seqf1olem2a.4	. . . . . . . 8 ⊢ (𝜑 → (𝑀...𝑁) ⊆ 𝐴)
36		eluzfz1 10414	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
37	1, 36	syl 14	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁))
38	35, 37	sseldd 3249	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ 𝐴)
39	25, 38	ffvelcdmd 5835	. . . . . 6 ⊢ (𝜑 → (𝐺‘𝑀) ∈ 𝐶)
40	34, 39	eqeltrd 2315	. . . . 5 ⊢ (𝜑 → (seq𝑀( + , 𝐺)‘𝑀) ∈ 𝐶)
41	24, 27, 40	caovcomd 6236	. . . 4 ⊢ (𝜑 → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑀)) = ((seq𝑀( + , 𝐺)‘𝑀) + (𝐺‘𝐾)))
42	41	a1i 9	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝜑 → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑀)) = ((seq𝑀( + , 𝐺)‘𝑀) + (𝐺‘𝐾))))
43		oveq1 6082	. . . . . 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))))
44		elfzouz 10536	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (𝑀..^𝑁) → 𝑛 ∈ (ℤ≥‘𝑀))
45	44	adantl 277	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → 𝑛 ∈ (ℤ≥‘𝑀))
46	31	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → 𝐺 ∈ V)
47	32	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → + ∈ 𝑉)
48		seqp1g 10881	. . . . . . . . . 10 ⊢ ((𝑛 ∈ (ℤ≥‘𝑀) ∧ 𝐺 ∈ V ∧ + ∈ 𝑉) → (seq𝑀( + , 𝐺)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1))))
49	45, 46, 47, 48	syl3anc 1278	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( + , 𝐺)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1))))
50	49	oveq2d 6091	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘(𝑛 + 1))) = ((𝐺‘𝐾) + ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1)))))
51		seqf1o.3	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
52	51	adantlr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
53		seqf1o.5	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ⊆ 𝑆)
54	53, 27	sseldd 3249	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺‘𝐾) ∈ 𝑆)
55	54	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝐺‘𝐾) ∈ 𝑆)
56	53	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → 𝐶 ⊆ 𝑆)
57	56	adantr 276	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑛)) → 𝐶 ⊆ 𝑆)
58	25	adantr 276	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → 𝐺:𝐴⟶𝐶)
59	58	adantr 276	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑛)) → 𝐺:𝐴⟶𝐶)
60		elfzouz2 10547	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (𝑀..^𝑁) → 𝑁 ∈ (ℤ≥‘𝑛))
61	60	adantl 277	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → 𝑁 ∈ (ℤ≥‘𝑛))
62		fzss2 10448	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (ℤ≥‘𝑛) → (𝑀...𝑛) ⊆ (𝑀...𝑁))
63	61, 62	syl 14	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝑀...𝑛) ⊆ (𝑀...𝑁))
64	35	adantr 276	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝑀...𝑁) ⊆ 𝐴)
65	63, 64	sstrd 3258	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝑀...𝑛) ⊆ 𝐴)
66	65	sselda 3248	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑛)) → 𝑥 ∈ 𝐴)
67	59, 66	ffvelcdmd 5835	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑛)) → (𝐺‘𝑥) ∈ 𝐶)
68	57, 67	sseldd 3249	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑛)) → (𝐺‘𝑥) ∈ 𝑆)
69		seqf1o.1	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
70	69	adantlr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
71	45, 68, 70, 46, 47	seqclg 10887	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( + , 𝐺)‘𝑛) ∈ 𝑆)
72		fzofzp1 10623	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (𝑀..^𝑁) → (𝑛 + 1) ∈ (𝑀...𝑁))
73	72	adantl 277	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝑛 + 1) ∈ (𝑀...𝑁))
74	64, 73	sseldd 3249	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝑛 + 1) ∈ 𝐴)
75	58, 74	ffvelcdmd 5835	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝐺‘(𝑛 + 1)) ∈ 𝐶)
76	56, 75	sseldd 3249	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝐺‘(𝑛 + 1)) ∈ 𝑆)
77	52, 55, 71, 76	caovassd 6239	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑛)) + (𝐺‘(𝑛 + 1))) = ((𝐺‘𝐾) + ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1)))))
78	50, 77	eqtr4d 2274	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘(𝑛 + 1))) = (((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑛)) + (𝐺‘(𝑛 + 1))))
79	52, 71, 76, 55	caovassd 6239	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1))) + (𝐺‘𝐾)) = ((seq𝑀( + , 𝐺)‘𝑛) + ((𝐺‘(𝑛 + 1)) + (𝐺‘𝐾))))
80	49	oveq1d 6090	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → ((seq𝑀( + , 𝐺)‘(𝑛 + 1)) + (𝐺‘𝐾)) = (((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1))) + (𝐺‘𝐾)))
81	52, 71, 55, 76	caovassd 6239	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘𝐾)) + (𝐺‘(𝑛 + 1))) = ((seq𝑀( + , 𝐺)‘𝑛) + ((𝐺‘𝐾) + (𝐺‘(𝑛 + 1)))))
82	24	adantlr 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
83	27	adantr 276	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝐺‘𝐾) ∈ 𝐶)
84	82, 75, 83	caovcomd 6236	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → ((𝐺‘(𝑛 + 1)) + (𝐺‘𝐾)) = ((𝐺‘𝐾) + (𝐺‘(𝑛 + 1))))
85	84	oveq2d 6091	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → ((seq𝑀( + , 𝐺)‘𝑛) + ((𝐺‘(𝑛 + 1)) + (𝐺‘𝐾))) = ((seq𝑀( + , 𝐺)‘𝑛) + ((𝐺‘𝐾) + (𝐺‘(𝑛 + 1)))))
86	81, 85	eqtr4d 2274	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘𝐾)) + (𝐺‘(𝑛 + 1))) = ((seq𝑀( + , 𝐺)‘𝑛) + ((𝐺‘(𝑛 + 1)) + (𝐺‘𝐾))))
87	79, 80, 86	3eqtr4d 2281	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → ((seq𝑀( + , 𝐺)‘(𝑛 + 1)) + (𝐺‘𝐾)) = (((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘𝐾)) + (𝐺‘(𝑛 + 1))))
88	78, 87	eqeq12d 2253	. . . . . 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)))))
89	43, 88	imbitrrid 156	. . . . 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))))
90	89	expcom 116	. . . 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)))))
91	90	a2d 26	. . 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)))))
92	8, 13, 18, 23, 42, 91	fzind2 10636	. 2 ⊢ (𝑁 ∈ (𝑀...𝑁) → (𝜑 → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑁)) = ((seq𝑀( + , 𝐺)‘𝑁) + (𝐺‘𝐾))))
93	3, 92	mpcom 36	1 ⊢ (𝜑 → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑁)) = ((seq𝑀( + , 𝐺)‘𝑁) + (𝐺‘𝐾)))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  Vcvv 2821   ⊆ wss 3220  ⟶wf 5368  ‘cfv 5372  (class class class)co 6075  1c1 8170   + caddc 8172  ℤcz 9623  ℤ_≥cuz 9900  ...cfz 10390  ..^cfzo 10527  seqcseq 10862

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-n0 9543  df-z 9624  df-uz 9901  df-fz 10391  df-fzo 10528  df-seqfrec 10863

This theorem is referenced by:  seqf1oglem2  10935