Theorem seqfeq4g 10602

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 10098	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
3	1, 2	syl 14	. 2 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
4		fveq2 5554	. . . . 5 ⊢ (𝑤 = 𝑀 → (seq𝑀( + , 𝐹)‘𝑤) = (seq𝑀( + , 𝐹)‘𝑀))
5		fveq2 5554	. . . . 5 ⊢ (𝑤 = 𝑀 → (seq𝑀(𝑄, 𝐹)‘𝑤) = (seq𝑀(𝑄, 𝐹)‘𝑀))
6	4, 5	eqeq12d 2208	. . . 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 5554	. . . . 5 ⊢ (𝑤 = 𝑘 → (seq𝑀( + , 𝐹)‘𝑤) = (seq𝑀( + , 𝐹)‘𝑘))
9		fveq2 5554	. . . . 5 ⊢ (𝑤 = 𝑘 → (seq𝑀(𝑄, 𝐹)‘𝑤) = (seq𝑀(𝑄, 𝐹)‘𝑘))
10	8, 9	eqeq12d 2208	. . . 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 5554	. . . . 5 ⊢ (𝑤 = (𝑘 + 1) → (seq𝑀( + , 𝐹)‘𝑤) = (seq𝑀( + , 𝐹)‘(𝑘 + 1)))
13		fveq2 5554	. . . . 5 ⊢ (𝑤 = (𝑘 + 1) → (seq𝑀(𝑄, 𝐹)‘𝑤) = (seq𝑀(𝑄, 𝐹)‘(𝑘 + 1)))
14	12, 13	eqeq12d 2208	. . . 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 5554	. . . . 5 ⊢ (𝑤 = 𝑁 → (seq𝑀( + , 𝐹)‘𝑤) = (seq𝑀( + , 𝐹)‘𝑁))
17		fveq2 5554	. . . . 5 ⊢ (𝑤 = 𝑁 → (seq𝑀(𝑄, 𝐹)‘𝑤) = (seq𝑀(𝑄, 𝐹)‘𝑁))
18	16, 17	eqeq12d 2208	. . . 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 9597	. . . . . . 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 2763	. . . . . . 7 ⊢ 𝑥 ∈ V
25		fvexg 5573	. . . . . . 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 5952	. . . . . . 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 10533	. . . . 5 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀))
32		seqfeq4g.q	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ 𝑋)
33		ovexg 5952	. . . . . . 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 10533	. . . . 5 ⊢ (𝜑 → (seq𝑀(𝑄, 𝐹)‘𝑀) = (𝐹‘𝑀))
36	31, 35	eqtr4d 2229	. . . 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 5933	. . . . . . . 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 5926	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐹‘(𝑘 + 1)) → ((seq𝑀(𝑄, 𝐹)‘𝑘) + 𝑦) = ((seq𝑀(𝑄, 𝐹)‘𝑘) + (𝐹‘(𝑘 + 1))))
41		oveq2 5926	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐹‘(𝑘 + 1)) → ((seq𝑀(𝑄, 𝐹)‘𝑘)𝑄𝑦) = ((seq𝑀(𝑄, 𝐹)‘𝑘)𝑄(𝐹‘(𝑘 + 1))))
42	40, 41	eqeq12d 2208	. . . . . . . . . 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 5925	. . . . . . . . . . . . 13 ⊢ (𝑥 = (seq𝑀(𝑄, 𝐹)‘𝑘) → (𝑥 + 𝑦) = ((seq𝑀(𝑄, 𝐹)‘𝑘) + 𝑦))
44		oveq1 5925	. . . . . . . . . . . . 13 ⊢ (𝑥 = (seq𝑀(𝑄, 𝐹)‘𝑘) → (𝑥𝑄𝑦) = ((seq𝑀(𝑄, 𝐹)‘𝑘)𝑄𝑦))
45	43, 44	eqeq12d 2208	. . . . . . . . . . . 12 ⊢ (𝑥 = (seq𝑀(𝑄, 𝐹)‘𝑘) → ((𝑥 + 𝑦) = (𝑥𝑄𝑦) ↔ ((seq𝑀(𝑄, 𝐹)‘𝑘) + 𝑦) = ((seq𝑀(𝑄, 𝐹)‘𝑘)𝑄𝑦)))
46	45	ralbidv 2494	. . . . . . . . . . 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 2576	. . . . . . . . . . . 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 10217	. . . . . . . . . . . . 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 10092	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ ℤ)
56	55	ad2antrr 488	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑘)) → 𝑁 ∈ ℤ)
57		elfzelz 10091	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝑀...𝑘) → 𝑥 ∈ ℤ)
58	57	adantl 277	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑘)) → 𝑥 ∈ ℤ)
59		elfzle1 10093	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝑀...𝑘) → 𝑀 ≤ 𝑥)
60	59	adantl 277	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑘)) → 𝑀 ≤ 𝑥)
61	58	zred 9439	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑘)) → 𝑥 ∈ ℝ)
62		elfzoelz 10213	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (𝑀..^𝑁) → 𝑘 ∈ ℤ)
63	62	ad2antlr 489	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑘)) → 𝑘 ∈ ℤ)
64	63	zred 9439	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑘)) → 𝑘 ∈ ℝ)
65	56	zred 9439	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑘)) → 𝑁 ∈ ℝ)
66		elfzle2 10094	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (𝑀...𝑘) → 𝑥 ≤ 𝑘)
67	66	adantl 277	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑘)) → 𝑥 ≤ 𝑘)
68		elfzofz 10229	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (𝑀..^𝑁) → 𝑘 ∈ (𝑀...𝑁))
69		elfzle2 10094	. . . . . . . . . . . . . . . . 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 8143	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑘)) → 𝑥 ≤ 𝑁)
73	54, 56, 58, 60, 72	elfzd 10082	. . . . . . . . . . . . 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 2271	. . . . . . . . . . . . 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 3201	. . . . . . . . . . . . 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 10542	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (seq𝑀(𝑄, 𝐹)‘𝑘) ∈ 𝑆)
83	46, 49, 82	rspcdva 2869	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ∀𝑦 ∈ 𝑆 ((seq𝑀(𝑄, 𝐹)‘𝑘) + 𝑦) = ((seq𝑀(𝑄, 𝐹)‘𝑘)𝑄𝑦))
84		fveq2 5554	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑘 + 1) → (𝐹‘𝑥) = (𝐹‘(𝑘 + 1)))
85	84	eleq1d 2262	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑘 + 1) → ((𝐹‘𝑥) ∈ 𝑆 ↔ (𝐹‘(𝑘 + 1)) ∈ 𝑆))
86	74	ralrimiva 2567	. . . . . . . . . . . 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 10294	. . . . . . . . . . . 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 2869	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝐹‘(𝑘 + 1)) ∈ 𝑆)
91	42, 83, 90	rspcdva 2869	. . . . . . . . 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 2226	. . . . . . 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 10536	. . . . . . 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 10536	. . . . . . 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 2236	. . . . . 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 10306	. 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 2164  ∀wral 2472  Vcvv 2760   ⊆ wss 3153   class class class wbr 4029  ‘cfv 5254  (class class class)co 5918  1c1 7873   + caddc 7875   ≤ cle 8055  ℤcz 9317  ℤ_≥cuz 9592  ...cfz 10074  ..^cfzo 10208  seqcseq 10518

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-ltadd 7988

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-inn 8983  df-n0 9241  df-z 9318  df-uz 9593  df-fz 10075  df-fzo 10209  df-seqfrec 10519

This theorem is referenced by:  gsumpropd2  12976