Theorem seqf1oglem2a 10612

Description: Lemma for seqf1og 10615. (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 10109	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
3	1, 2	syl 14	. 2 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
4		fveq2 5559	. . . . . 6 ⊢ (𝑚 = 𝑀 → (seq𝑀( + , 𝐺)‘𝑚) = (seq𝑀( + , 𝐺)‘𝑀))
5	4	oveq2d 5939	. . . . 5 ⊢ (𝑚 = 𝑀 → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑀)))
6	4	oveq1d 5938	. . . . 5 ⊢ (𝑚 = 𝑀 → ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺‘𝐾)) = ((seq𝑀( + , 𝐺)‘𝑀) + (𝐺‘𝐾)))
7	5, 6	eqeq12d 2211	. . . 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 5559	. . . . . 6 ⊢ (𝑚 = 𝑛 → (seq𝑀( + , 𝐺)‘𝑚) = (seq𝑀( + , 𝐺)‘𝑛))
10	9	oveq2d 5939	. . . . 5 ⊢ (𝑚 = 𝑛 → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑛)))
11	9	oveq1d 5938	. . . . 5 ⊢ (𝑚 = 𝑛 → ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺‘𝐾)) = ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘𝐾)))
12	10, 11	eqeq12d 2211	. . . 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 5559	. . . . . 6 ⊢ (𝑚 = (𝑛 + 1) → (seq𝑀( + , 𝐺)‘𝑚) = (seq𝑀( + , 𝐺)‘(𝑛 + 1)))
15	14	oveq2d 5939	. . . . 5 ⊢ (𝑚 = (𝑛 + 1) → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘(𝑛 + 1))))
16	14	oveq1d 5938	. . . . 5 ⊢ (𝑚 = (𝑛 + 1) → ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺‘𝐾)) = ((seq𝑀( + , 𝐺)‘(𝑛 + 1)) + (𝐺‘𝐾)))
17	15, 16	eqeq12d 2211	. . . 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 5559	. . . . . 6 ⊢ (𝑚 = 𝑁 → (seq𝑀( + , 𝐺)‘𝑚) = (seq𝑀( + , 𝐺)‘𝑁))
20	19	oveq2d 5939	. . . . 5 ⊢ (𝑚 = 𝑁 → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑚)) = ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑁)))
21	19	oveq1d 5938	. . . . 5 ⊢ (𝑚 = 𝑁 → ((seq𝑀( + , 𝐺)‘𝑚) + (𝐺‘𝐾)) = ((seq𝑀( + , 𝐺)‘𝑁) + (𝐺‘𝐾)))
22	20, 21	eqeq12d 2211	. . . 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 5699	. . . . 5 ⊢ (𝜑 → (𝐺‘𝐾) ∈ 𝐶)
28		eluzel2 9608	. . . . . . . 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 5793	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ V)
32		seqf1og.p	. . . . . . 7 ⊢ (𝜑 → + ∈ 𝑉)
33		seq1g 10557	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝐺 ∈ V ∧ + ∈ 𝑉) → (seq𝑀( + , 𝐺)‘𝑀) = (𝐺‘𝑀))
34	29, 31, 32, 33	syl3anc 1249	. . . . . 6 ⊢ (𝜑 → (seq𝑀( + , 𝐺)‘𝑀) = (𝐺‘𝑀))
35		seqf1olem2a.4	. . . . . . . 8 ⊢ (𝜑 → (𝑀...𝑁) ⊆ 𝐴)
36		eluzfz1 10108	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
37	1, 36	syl 14	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁))
38	35, 37	sseldd 3185	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ 𝐴)
39	25, 38	ffvelcdmd 5699	. . . . . 6 ⊢ (𝜑 → (𝐺‘𝑀) ∈ 𝐶)
40	34, 39	eqeltrd 2273	. . . . 5 ⊢ (𝜑 → (seq𝑀( + , 𝐺)‘𝑀) ∈ 𝐶)
41	24, 27, 40	caovcomd 6081	. . . 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 5930	. . . . . 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 10228	. . . . . . . . . . 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 10560	. . . . . . . . . 10 ⊢ ((𝑛 ∈ (ℤ≥‘𝑀) ∧ 𝐺 ∈ V ∧ + ∈ 𝑉) → (seq𝑀( + , 𝐺)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1))))
49	45, 46, 47, 48	syl3anc 1249	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( + , 𝐺)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1))))
50	49	oveq2d 5939	. . . . . . . 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 477	. . . . . . . . 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 3185	. . . . . . . . . 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 10239	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (𝑀..^𝑁) → 𝑁 ∈ (ℤ≥‘𝑛))
61	60	adantl 277	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → 𝑁 ∈ (ℤ≥‘𝑛))
62		fzss2 10141	. . . . . . . . . . . . . . 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 3194	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝑀...𝑛) ⊆ 𝐴)
66	65	sselda 3184	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑛)) → 𝑥 ∈ 𝐴)
67	59, 66	ffvelcdmd 5699	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑛)) → (𝐺‘𝑥) ∈ 𝐶)
68	57, 67	sseldd 3185	. . . . . . . . . 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 477	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
71	45, 68, 70, 46, 47	seqclg 10566	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( + , 𝐺)‘𝑛) ∈ 𝑆)
72		fzofzp1 10305	. . . . . . . . . . . . 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 3185	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝑛 + 1) ∈ 𝐴)
75	58, 74	ffvelcdmd 5699	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝐺‘(𝑛 + 1)) ∈ 𝐶)
76	56, 75	sseldd 3185	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝐺‘(𝑛 + 1)) ∈ 𝑆)
77	52, 55, 71, 76	caovassd 6084	. . . . . . . 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 2232	. . . . . . 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 6084	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1))) + (𝐺‘𝐾)) = ((seq𝑀( + , 𝐺)‘𝑛) + ((𝐺‘(𝑛 + 1)) + (𝐺‘𝐾))))
80	49	oveq1d 5938	. . . . . . . 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 6084	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘𝐾)) + (𝐺‘(𝑛 + 1))) = ((seq𝑀( + , 𝐺)‘𝑛) + ((𝐺‘𝐾) + (𝐺‘(𝑛 + 1)))))
82	24	adantlr 477	. . . . . . . . . . 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 6081	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → ((𝐺‘(𝑛 + 1)) + (𝐺‘𝐾)) = ((𝐺‘𝐾) + (𝐺‘(𝑛 + 1))))
85	84	oveq2d 5939	. . . . . . . . 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 2232	. . . . . . . 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 2239	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → ((seq𝑀( + , 𝐺)‘(𝑛 + 1)) + (𝐺‘𝐾)) = (((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘𝐾)) + (𝐺‘(𝑛 + 1))))
88	78, 87	eqeq12d 2211	. . . . . 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 10317	. 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 980   = wceq 1364   ∈ wcel 2167  Vcvv 2763   ⊆ wss 3157  ⟶wf 5255  ‘cfv 5259  (class class class)co 5923  1c1 7882   + caddc 7884  ℤcz 9328  ℤ_≥cuz 9603  ...cfz 10085  ..^cfzo 10219  seqcseq 10541

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7972  ax-resscn 7973  ax-1cn 7974  ax-1re 7975  ax-icn 7976  ax-addcl 7977  ax-addrcl 7978  ax-mulcl 7979  ax-addcom 7981  ax-addass 7983  ax-distr 7985  ax-i2m1 7986  ax-0lt1 7987  ax-0id 7989  ax-rnegex 7990  ax-cnre 7992  ax-pre-ltirr 7993  ax-pre-ltwlin 7994  ax-pre-lttrn 7995  ax-pre-ltadd 7997

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5878  df-ov 5926  df-oprab 5927  df-mpo 5928  df-1st 6199  df-2nd 6200  df-recs 6364  df-frec 6450  df-pnf 8065  df-mnf 8066  df-xr 8067  df-ltxr 8068  df-le 8069  df-sub 8201  df-neg 8202  df-inn 8993  df-n0 9252  df-z 9329  df-uz 9604  df-fz 10086  df-fzo 10220  df-seqfrec 10542

This theorem is referenced by:  seqf1oglem2  10614