Theorem seqfeq4g 10623

Description: Equality of series under different addition operations which agree on an additively closed subset. (Contributed by Mario Carneiro, 25-Apr-2016.)

Hypotheses

Ref	Expression
seqfeq4.m	⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
seqfeq4.f	⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ 𝑆)
seqfeq4g.f	⊢ (𝜑 → 𝐹 ∈ 𝑉)
seqfeq4g.p	⊢ (𝜑 → + ∈ 𝑊)
seqfeq4g.q	⊢ (𝜑 → 𝑄 ∈ 𝑋)
seqfeq4.cl	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
seqfeq4.id	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑥𝑄𝑦))

Assertion

Ref	Expression
seqfeq4g	⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq𝑀(𝑄, 𝐹)‘𝑁))

Distinct variable groups:   𝑥,𝑦, +   𝑥,𝐹,𝑦   𝑥,𝑀,𝑦   𝑥,𝑁,𝑦   𝜑,𝑥,𝑦   𝑥,𝑄,𝑦   𝑥,𝑆,𝑦

Allowed substitution hints:   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)   𝑋(𝑥,𝑦)

Proof of Theorem seqfeq4g

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

Step	Hyp	Ref	Expression
1		seqfeq4.m	. . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
2		eluzfz2 10107	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
3	1, 2	syl 14	. 2 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
4		fveq2 5558	. . . . 5 ⊢ (𝑤 = 𝑀 → (seq𝑀( + , 𝐹)‘𝑤) = (seq𝑀( + , 𝐹)‘𝑀))
5		fveq2 5558	. . . . 5 ⊢ (𝑤 = 𝑀 → (seq𝑀(𝑄, 𝐹)‘𝑤) = (seq𝑀(𝑄, 𝐹)‘𝑀))
6	4, 5	eqeq12d 2211	. . . 4 ⊢ (𝑤 = 𝑀 → ((seq𝑀( + , 𝐹)‘𝑤) = (seq𝑀(𝑄, 𝐹)‘𝑤) ↔ (seq𝑀( + , 𝐹)‘𝑀) = (seq𝑀(𝑄, 𝐹)‘𝑀)))
7	6	imbi2d 230	. . 3 ⊢ (𝑤 = 𝑀 → ((𝜑 → (seq𝑀( + , 𝐹)‘𝑤) = (seq𝑀(𝑄, 𝐹)‘𝑤)) ↔ (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (seq𝑀(𝑄, 𝐹)‘𝑀))))
8		fveq2 5558	. . . . 5 ⊢ (𝑤 = 𝑘 → (seq𝑀( + , 𝐹)‘𝑤) = (seq𝑀( + , 𝐹)‘𝑘))
9		fveq2 5558	. . . . 5 ⊢ (𝑤 = 𝑘 → (seq𝑀(𝑄, 𝐹)‘𝑤) = (seq𝑀(𝑄, 𝐹)‘𝑘))
10	8, 9	eqeq12d 2211	. . . 4 ⊢ (𝑤 = 𝑘 → ((seq𝑀( + , 𝐹)‘𝑤) = (seq𝑀(𝑄, 𝐹)‘𝑤) ↔ (seq𝑀( + , 𝐹)‘𝑘) = (seq𝑀(𝑄, 𝐹)‘𝑘)))
11	10	imbi2d 230	. . 3 ⊢ (𝑤 = 𝑘 → ((𝜑 → (seq𝑀( + , 𝐹)‘𝑤) = (seq𝑀(𝑄, 𝐹)‘𝑤)) ↔ (𝜑 → (seq𝑀( + , 𝐹)‘𝑘) = (seq𝑀(𝑄, 𝐹)‘𝑘))))
12		fveq2 5558	. . . . 5 ⊢ (𝑤 = (𝑘 + 1) → (seq𝑀( + , 𝐹)‘𝑤) = (seq𝑀( + , 𝐹)‘(𝑘 + 1)))
13		fveq2 5558	. . . . 5 ⊢ (𝑤 = (𝑘 + 1) → (seq𝑀(𝑄, 𝐹)‘𝑤) = (seq𝑀(𝑄, 𝐹)‘(𝑘 + 1)))
14	12, 13	eqeq12d 2211	. . . 4 ⊢ (𝑤 = (𝑘 + 1) → ((seq𝑀( + , 𝐹)‘𝑤) = (seq𝑀(𝑄, 𝐹)‘𝑤) ↔ (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = (seq𝑀(𝑄, 𝐹)‘(𝑘 + 1))))
15	14	imbi2d 230	. . 3 ⊢ (𝑤 = (𝑘 + 1) → ((𝜑 → (seq𝑀( + , 𝐹)‘𝑤) = (seq𝑀(𝑄, 𝐹)‘𝑤)) ↔ (𝜑 → (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = (seq𝑀(𝑄, 𝐹)‘(𝑘 + 1)))))
16		fveq2 5558	. . . . 5 ⊢ (𝑤 = 𝑁 → (seq𝑀( + , 𝐹)‘𝑤) = (seq𝑀( + , 𝐹)‘𝑁))
17		fveq2 5558	. . . . 5 ⊢ (𝑤 = 𝑁 → (seq𝑀(𝑄, 𝐹)‘𝑤) = (seq𝑀(𝑄, 𝐹)‘𝑁))
18	16, 17	eqeq12d 2211	. . . 4 ⊢ (𝑤 = 𝑁 → ((seq𝑀( + , 𝐹)‘𝑤) = (seq𝑀(𝑄, 𝐹)‘𝑤) ↔ (seq𝑀( + , 𝐹)‘𝑁) = (seq𝑀(𝑄, 𝐹)‘𝑁)))
19	18	imbi2d 230	. . 3 ⊢ (𝑤 = 𝑁 → ((𝜑 → (seq𝑀( + , 𝐹)‘𝑤) = (seq𝑀(𝑄, 𝐹)‘𝑤)) ↔ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq𝑀(𝑄, 𝐹)‘𝑁))))
20		eluzel2 9606	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
21	1, 20	syl 14	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ)
22		seqfeq4g.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
23	22	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → 𝐹 ∈ 𝑉)
24		vex 2766	. . . . . . 7 ⊢ 𝑥 ∈ V
25		fvexg 5577	. . . . . . 7 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑥 ∈ V) → (𝐹‘𝑥) ∈ V)
26	23, 24, 25	sylancl 413	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ V)
27		seqfeq4g.p	. . . . . . 7 ⊢ (𝜑 → + ∈ 𝑊)
28		simprr 531	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → 𝑦 ∈ V)
29		ovexg 5956	. . . . . . 7 ⊢ ((𝑥 ∈ V ∧ + ∈ 𝑊 ∧ 𝑦 ∈ V) → (𝑥 + 𝑦) ∈ V)
30	24, 27, 28, 29	mp3an2ani 1355	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → (𝑥 + 𝑦) ∈ V)
31	21, 26, 30	seq3-1 10554	. . . . 5 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀))
32		seqfeq4g.q	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ 𝑋)
33		ovexg 5956	. . . . . . 7 ⊢ ((𝑥 ∈ V ∧ 𝑄 ∈ 𝑋 ∧ 𝑦 ∈ V) → (𝑥𝑄𝑦) ∈ V)
34	24, 32, 28, 33	mp3an2ani 1355	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → (𝑥𝑄𝑦) ∈ V)
35	21, 26, 34	seq3-1 10554	. . . . 5 ⊢ (𝜑 → (seq𝑀(𝑄, 𝐹)‘𝑀) = (𝐹‘𝑀))
36	31, 35	eqtr4d 2232	. . . 4 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (seq𝑀(𝑄, 𝐹)‘𝑀))
37	36	a1i 9	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (seq𝑀(𝑄, 𝐹)‘𝑀)))
38		simpr 110	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ (seq𝑀( + , 𝐹)‘𝑘) = (seq𝑀(𝑄, 𝐹)‘𝑘)) → (seq𝑀( + , 𝐹)‘𝑘) = (seq𝑀(𝑄, 𝐹)‘𝑘))
39	38	oveq1d 5937	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ (seq𝑀( + , 𝐹)‘𝑘) = (seq𝑀(𝑄, 𝐹)‘𝑘)) → ((seq𝑀( + , 𝐹)‘𝑘) + (𝐹‘(𝑘 + 1))) = ((seq𝑀(𝑄, 𝐹)‘𝑘) + (𝐹‘(𝑘 + 1))))
40		oveq2 5930	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐹‘(𝑘 + 1)) → ((seq𝑀(𝑄, 𝐹)‘𝑘) + 𝑦) = ((seq𝑀(𝑄, 𝐹)‘𝑘) + (𝐹‘(𝑘 + 1))))
41		oveq2 5930	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐹‘(𝑘 + 1)) → ((seq𝑀(𝑄, 𝐹)‘𝑘)𝑄𝑦) = ((seq𝑀(𝑄, 𝐹)‘𝑘)𝑄(𝐹‘(𝑘 + 1))))
42	40, 41	eqeq12d 2211	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘(𝑘 + 1)) → (((seq𝑀(𝑄, 𝐹)‘𝑘) + 𝑦) = ((seq𝑀(𝑄, 𝐹)‘𝑘)𝑄𝑦) ↔ ((seq𝑀(𝑄, 𝐹)‘𝑘) + (𝐹‘(𝑘 + 1))) = ((seq𝑀(𝑄, 𝐹)‘𝑘)𝑄(𝐹‘(𝑘 + 1)))))
43		oveq1 5929	. . . . . . . . . . . . 13 ⊢ (𝑥 = (seq𝑀(𝑄, 𝐹)‘𝑘) → (𝑥 + 𝑦) = ((seq𝑀(𝑄, 𝐹)‘𝑘) + 𝑦))
44		oveq1 5929	. . . . . . . . . . . . 13 ⊢ (𝑥 = (seq𝑀(𝑄, 𝐹)‘𝑘) → (𝑥𝑄𝑦) = ((seq𝑀(𝑄, 𝐹)‘𝑘)𝑄𝑦))
45	43, 44	eqeq12d 2211	. . . . . . . . . . . 12 ⊢ (𝑥 = (seq𝑀(𝑄, 𝐹)‘𝑘) → ((𝑥 + 𝑦) = (𝑥𝑄𝑦) ↔ ((seq𝑀(𝑄, 𝐹)‘𝑘) + 𝑦) = ((seq𝑀(𝑄, 𝐹)‘𝑘)𝑄𝑦)))
46	45	ralbidv 2497	. . . . . . . . . . 11 ⊢ (𝑥 = (seq𝑀(𝑄, 𝐹)‘𝑘) → (∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) = (𝑥𝑄𝑦) ↔ ∀𝑦 ∈ 𝑆 ((seq𝑀(𝑄, 𝐹)‘𝑘) + 𝑦) = ((seq𝑀(𝑄, 𝐹)‘𝑘)𝑄𝑦)))
47		seqfeq4.id	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑥𝑄𝑦))
48	47	ralrimivva 2579	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) = (𝑥𝑄𝑦))
49	48	adantr 276	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) = (𝑥𝑄𝑦))
50		elfzouz 10226	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (𝑀..^𝑁) → 𝑘 ∈ (ℤ≥‘𝑀))
51	50	adantl 277	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑘 ∈ (ℤ≥‘𝑀))
52	26	adantlr 477	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ V)
53		simpll 527	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑘)) → 𝜑)
54	21	ad2antrr 488	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑘)) → 𝑀 ∈ ℤ)
55	3	elfzelzd 10101	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ ℤ)
56	55	ad2antrr 488	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑘)) → 𝑁 ∈ ℤ)
57		elfzelz 10100	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝑀...𝑘) → 𝑥 ∈ ℤ)
58	57	adantl 277	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑘)) → 𝑥 ∈ ℤ)
59		elfzle1 10102	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝑀...𝑘) → 𝑀 ≤ 𝑥)
60	59	adantl 277	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑘)) → 𝑀 ≤ 𝑥)
61	58	zred 9448	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑘)) → 𝑥 ∈ ℝ)
62		elfzoelz 10222	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (𝑀..^𝑁) → 𝑘 ∈ ℤ)
63	62	ad2antlr 489	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑘)) → 𝑘 ∈ ℤ)
64	63	zred 9448	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑘)) → 𝑘 ∈ ℝ)
65	56	zred 9448	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑘)) → 𝑁 ∈ ℝ)
66		elfzle2 10103	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (𝑀...𝑘) → 𝑥 ≤ 𝑘)
67	66	adantl 277	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑘)) → 𝑥 ≤ 𝑘)
68		elfzofz 10238	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (𝑀..^𝑁) → 𝑘 ∈ (𝑀...𝑁))
69		elfzle2 10103	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ≤ 𝑁)
70	68, 69	syl 14	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (𝑀..^𝑁) → 𝑘 ≤ 𝑁)
71	70	ad2antlr 489	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑘)) → 𝑘 ≤ 𝑁)
72	61, 64, 65, 67, 71	letrd 8150	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑘)) → 𝑥 ≤ 𝑁)
73	54, 56, 58, 60, 72	elfzd 10091	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑘)) → 𝑥 ∈ (𝑀...𝑁))
74		seqfeq4.f	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ 𝑆)
75	53, 73, 74	syl2anc 411	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑘)) → (𝐹‘𝑥) ∈ 𝑆)
76		seqfeq4.cl	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
77	47, 76	eqeltrrd 2274	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)
78	77	adantlr 477	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)
79		ssv 3205	. . . . . . . . . . . . 13 ⊢ 𝑆 ⊆ V
80	79	a1i 9	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑆 ⊆ V)
81	34	adantlr 477	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → (𝑥𝑄𝑦) ∈ V)
82	51, 52, 75, 78, 80, 81	seq3clss 10563	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (seq𝑀(𝑄, 𝐹)‘𝑘) ∈ 𝑆)
83	46, 49, 82	rspcdva 2873	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ∀𝑦 ∈ 𝑆 ((seq𝑀(𝑄, 𝐹)‘𝑘) + 𝑦) = ((seq𝑀(𝑄, 𝐹)‘𝑘)𝑄𝑦))
84		fveq2 5558	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑘 + 1) → (𝐹‘𝑥) = (𝐹‘(𝑘 + 1)))
85	84	eleq1d 2265	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑘 + 1) → ((𝐹‘𝑥) ∈ 𝑆 ↔ (𝐹‘(𝑘 + 1)) ∈ 𝑆))
86	74	ralrimiva 2570	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑥 ∈ (𝑀...𝑁)(𝐹‘𝑥) ∈ 𝑆)
87	86	adantr 276	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ∀𝑥 ∈ (𝑀...𝑁)(𝐹‘𝑥) ∈ 𝑆)
88		fzofzp1 10303	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (𝑀..^𝑁) → (𝑘 + 1) ∈ (𝑀...𝑁))
89	88	adantl 277	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ∈ (𝑀...𝑁))
90	85, 87, 89	rspcdva 2873	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝐹‘(𝑘 + 1)) ∈ 𝑆)
91	42, 83, 90	rspcdva 2873	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((seq𝑀(𝑄, 𝐹)‘𝑘) + (𝐹‘(𝑘 + 1))) = ((seq𝑀(𝑄, 𝐹)‘𝑘)𝑄(𝐹‘(𝑘 + 1))))
92	91	adantr 276	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ (seq𝑀( + , 𝐹)‘𝑘) = (seq𝑀(𝑄, 𝐹)‘𝑘)) → ((seq𝑀(𝑄, 𝐹)‘𝑘) + (𝐹‘(𝑘 + 1))) = ((seq𝑀(𝑄, 𝐹)‘𝑘)𝑄(𝐹‘(𝑘 + 1))))
93	39, 92	eqtrd 2229	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ (seq𝑀( + , 𝐹)‘𝑘) = (seq𝑀(𝑄, 𝐹)‘𝑘)) → ((seq𝑀( + , 𝐹)‘𝑘) + (𝐹‘(𝑘 + 1))) = ((seq𝑀(𝑄, 𝐹)‘𝑘)𝑄(𝐹‘(𝑘 + 1))))
94	50	ad2antlr 489	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ (seq𝑀( + , 𝐹)‘𝑘) = (seq𝑀(𝑄, 𝐹)‘𝑘)) → 𝑘 ∈ (ℤ≥‘𝑀))
95	26	ad4ant14 514	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ (seq𝑀( + , 𝐹)‘𝑘) = (seq𝑀(𝑄, 𝐹)‘𝑘)) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ V)
96	30	ad4ant14 514	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ (seq𝑀( + , 𝐹)‘𝑘) = (seq𝑀(𝑄, 𝐹)‘𝑘)) ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → (𝑥 + 𝑦) ∈ V)
97	94, 95, 96	seq3p1 10557	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ (seq𝑀( + , 𝐹)‘𝑘) = (seq𝑀(𝑄, 𝐹)‘𝑘)) → (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = ((seq𝑀( + , 𝐹)‘𝑘) + (𝐹‘(𝑘 + 1))))
98	34	ad4ant14 514	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ (seq𝑀( + , 𝐹)‘𝑘) = (seq𝑀(𝑄, 𝐹)‘𝑘)) ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → (𝑥𝑄𝑦) ∈ V)
99	94, 95, 98	seq3p1 10557	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ (seq𝑀( + , 𝐹)‘𝑘) = (seq𝑀(𝑄, 𝐹)‘𝑘)) → (seq𝑀(𝑄, 𝐹)‘(𝑘 + 1)) = ((seq𝑀(𝑄, 𝐹)‘𝑘)𝑄(𝐹‘(𝑘 + 1))))
100	93, 97, 99	3eqtr4d 2239	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ (seq𝑀( + , 𝐹)‘𝑘) = (seq𝑀(𝑄, 𝐹)‘𝑘)) → (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = (seq𝑀(𝑄, 𝐹)‘(𝑘 + 1)))
101	100	ex 115	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((seq𝑀( + , 𝐹)‘𝑘) = (seq𝑀(𝑄, 𝐹)‘𝑘) → (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = (seq𝑀(𝑄, 𝐹)‘(𝑘 + 1))))
102	101	expcom 116	. . . 4 ⊢ (𝑘 ∈ (𝑀..^𝑁) → (𝜑 → ((seq𝑀( + , 𝐹)‘𝑘) = (seq𝑀(𝑄, 𝐹)‘𝑘) → (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = (seq𝑀(𝑄, 𝐹)‘(𝑘 + 1)))))
103	102	a2d 26	. . 3 ⊢ (𝑘 ∈ (𝑀..^𝑁) → ((𝜑 → (seq𝑀( + , 𝐹)‘𝑘) = (seq𝑀(𝑄, 𝐹)‘𝑘)) → (𝜑 → (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = (seq𝑀(𝑄, 𝐹)‘(𝑘 + 1)))))
104	7, 11, 15, 19, 37, 103	fzind2 10315	. 2 ⊢ (𝑁 ∈ (𝑀...𝑁) → (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq𝑀(𝑄, 𝐹)‘𝑁)))
105	3, 104	mpcom 36	1 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq𝑀(𝑄, 𝐹)‘𝑁))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2167  ∀wral 2475  Vcvv 2763   ⊆ wss 3157   class class class wbr 4033  ‘cfv 5258  (class class class)co 5922  1c1 7880   + caddc 7882   ≤ cle 8062  ℤcz 9326  ℤ_≥cuz 9601  ...cfz 10083  ..^cfzo 10217  seqcseq 10539

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 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-ltadd 7995

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 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-inn 8991  df-n0 9250  df-z 9327  df-uz 9602  df-fz 10084  df-fzo 10218  df-seqfrec 10540

This theorem is referenced by:  gsumpropd2  13036