Theorem seq3distr 10469

Description: The distributive property for series. (Contributed by Jim Kingdon, 10-Oct-2022.)

Hypotheses

Ref	Expression
seq3distr.1	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
seq3distr.2	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝐶𝑇(𝑥 + 𝑦)) = ((𝐶𝑇𝑥) + (𝐶𝑇𝑦)))
seq3distr.3	⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
seq3distr.4	⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆)
seq3distr.5	⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) = (𝐶𝑇(𝐺‘𝑥)))
seq3distr.t	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑇𝑦) ∈ 𝑆)
seq3distr.c	⊢ (𝜑 → 𝐶 ∈ 𝑆)

Assertion

Ref	Expression
seq3distr	⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (𝐶𝑇(seq𝑀( + , 𝐺)‘𝑁)))

Distinct variable groups:   𝑥, + ,𝑦   𝑥,𝐶,𝑦   𝑥,𝐹,𝑦   𝑥,𝐺,𝑦   𝑥,𝑀,𝑦   𝑥,𝑁,𝑦   𝑥,𝑆,𝑦   𝑥,𝑇,𝑦   𝜑,𝑥,𝑦

Proof of Theorem seq3distr

Dummy variables 𝑏 𝑧 𝑎 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		seq3distr.1	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
2		seq3distr.4	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆)
3		seq3distr.3	. . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
4		seq3distr.2	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝐶𝑇(𝑥 + 𝑦)) = ((𝐶𝑇𝑥) + (𝐶𝑇𝑦)))
5		seq3distr.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ 𝑆)
6	5	adantr 274	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝐶 ∈ 𝑆)
7		seq3distr.t	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑇𝑦) ∈ 𝑆)
8	7	ralrimivva 2552	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝑇𝑦) ∈ 𝑆)
9		oveq1 5860	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → (𝑥𝑇𝑦) = (𝑎𝑇𝑦))
10	9	eleq1d 2239	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → ((𝑥𝑇𝑦) ∈ 𝑆 ↔ (𝑎𝑇𝑦) ∈ 𝑆))
11		oveq2 5861	. . . . . . . . . 10 ⊢ (𝑦 = 𝑏 → (𝑎𝑇𝑦) = (𝑎𝑇𝑏))
12	11	eleq1d 2239	. . . . . . . . 9 ⊢ (𝑦 = 𝑏 → ((𝑎𝑇𝑦) ∈ 𝑆 ↔ (𝑎𝑇𝑏) ∈ 𝑆))
13	10, 12	cbvral2v 2709	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝑇𝑦) ∈ 𝑆 ↔ ∀𝑎 ∈ 𝑆 ∀𝑏 ∈ 𝑆 (𝑎𝑇𝑏) ∈ 𝑆)
14	8, 13	sylib 121	. . . . . . 7 ⊢ (𝜑 → ∀𝑎 ∈ 𝑆 ∀𝑏 ∈ 𝑆 (𝑎𝑇𝑏) ∈ 𝑆)
15	14	adantr 274	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ∀𝑎 ∈ 𝑆 ∀𝑏 ∈ 𝑆 (𝑎𝑇𝑏) ∈ 𝑆)
16		oveq1 5860	. . . . . . . 8 ⊢ (𝑎 = 𝐶 → (𝑎𝑇𝑏) = (𝐶𝑇𝑏))
17	16	eleq1d 2239	. . . . . . 7 ⊢ (𝑎 = 𝐶 → ((𝑎𝑇𝑏) ∈ 𝑆 ↔ (𝐶𝑇𝑏) ∈ 𝑆))
18		oveq2 5861	. . . . . . . 8 ⊢ (𝑏 = (𝑥 + 𝑦) → (𝐶𝑇𝑏) = (𝐶𝑇(𝑥 + 𝑦)))
19	18	eleq1d 2239	. . . . . . 7 ⊢ (𝑏 = (𝑥 + 𝑦) → ((𝐶𝑇𝑏) ∈ 𝑆 ↔ (𝐶𝑇(𝑥 + 𝑦)) ∈ 𝑆))
20	17, 19	rspc2va 2848	. . . . . 6 ⊢ (((𝐶 ∈ 𝑆 ∧ (𝑥 + 𝑦) ∈ 𝑆) ∧ ∀𝑎 ∈ 𝑆 ∀𝑏 ∈ 𝑆 (𝑎𝑇𝑏) ∈ 𝑆) → (𝐶𝑇(𝑥 + 𝑦)) ∈ 𝑆)
21	6, 1, 15, 20	syl21anc 1232	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝐶𝑇(𝑥 + 𝑦)) ∈ 𝑆)
22		oveq2 5861	. . . . . 6 ⊢ (𝑧 = (𝑥 + 𝑦) → (𝐶𝑇𝑧) = (𝐶𝑇(𝑥 + 𝑦)))
23		eqid 2170	. . . . . 6 ⊢ (𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧)) = (𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))
24	22, 23	fvmptg 5572	. . . . 5 ⊢ (((𝑥 + 𝑦) ∈ 𝑆 ∧ (𝐶𝑇(𝑥 + 𝑦)) ∈ 𝑆) → ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘(𝑥 + 𝑦)) = (𝐶𝑇(𝑥 + 𝑦)))
25	1, 21, 24	syl2anc 409	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘(𝑥 + 𝑦)) = (𝐶𝑇(𝑥 + 𝑦)))
26		simprl 526	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑥 ∈ 𝑆)
27		oveq2 5861	. . . . . . . . 9 ⊢ (𝑏 = 𝑥 → (𝐶𝑇𝑏) = (𝐶𝑇𝑥))
28	27	eleq1d 2239	. . . . . . . 8 ⊢ (𝑏 = 𝑥 → ((𝐶𝑇𝑏) ∈ 𝑆 ↔ (𝐶𝑇𝑥) ∈ 𝑆))
29	17, 28	rspc2va 2848	. . . . . . 7 ⊢ (((𝐶 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆) ∧ ∀𝑎 ∈ 𝑆 ∀𝑏 ∈ 𝑆 (𝑎𝑇𝑏) ∈ 𝑆) → (𝐶𝑇𝑥) ∈ 𝑆)
30	6, 26, 15, 29	syl21anc 1232	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝐶𝑇𝑥) ∈ 𝑆)
31		oveq2 5861	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝐶𝑇𝑧) = (𝐶𝑇𝑥))
32	31, 23	fvmptg 5572	. . . . . 6 ⊢ ((𝑥 ∈ 𝑆 ∧ (𝐶𝑇𝑥) ∈ 𝑆) → ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘𝑥) = (𝐶𝑇𝑥))
33	26, 30, 32	syl2anc 409	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘𝑥) = (𝐶𝑇𝑥))
34		simprr 527	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ 𝑆)
35		oveq2 5861	. . . . . . . . 9 ⊢ (𝑏 = 𝑦 → (𝐶𝑇𝑏) = (𝐶𝑇𝑦))
36	35	eleq1d 2239	. . . . . . . 8 ⊢ (𝑏 = 𝑦 → ((𝐶𝑇𝑏) ∈ 𝑆 ↔ (𝐶𝑇𝑦) ∈ 𝑆))
37	17, 36	rspc2va 2848	. . . . . . 7 ⊢ (((𝐶 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ ∀𝑎 ∈ 𝑆 ∀𝑏 ∈ 𝑆 (𝑎𝑇𝑏) ∈ 𝑆) → (𝐶𝑇𝑦) ∈ 𝑆)
38	6, 34, 15, 37	syl21anc 1232	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝐶𝑇𝑦) ∈ 𝑆)
39		oveq2 5861	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝐶𝑇𝑧) = (𝐶𝑇𝑦))
40	39, 23	fvmptg 5572	. . . . . 6 ⊢ ((𝑦 ∈ 𝑆 ∧ (𝐶𝑇𝑦) ∈ 𝑆) → ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘𝑦) = (𝐶𝑇𝑦))
41	34, 38, 40	syl2anc 409	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘𝑦) = (𝐶𝑇𝑦))
42	33, 41	oveq12d 5871	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘𝑥) + ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘𝑦)) = ((𝐶𝑇𝑥) + (𝐶𝑇𝑦)))
43	4, 25, 42	3eqtr4d 2213	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘(𝑥 + 𝑦)) = (((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘𝑥) + ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘𝑦)))
44	5	adantr 274	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → 𝐶 ∈ 𝑆)
45	14	adantr 274	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → ∀𝑎 ∈ 𝑆 ∀𝑏 ∈ 𝑆 (𝑎𝑇𝑏) ∈ 𝑆)
46		oveq2 5861	. . . . . . . 8 ⊢ (𝑏 = (𝐺‘𝑥) → (𝐶𝑇𝑏) = (𝐶𝑇(𝐺‘𝑥)))
47	46	eleq1d 2239	. . . . . . 7 ⊢ (𝑏 = (𝐺‘𝑥) → ((𝐶𝑇𝑏) ∈ 𝑆 ↔ (𝐶𝑇(𝐺‘𝑥)) ∈ 𝑆))
48	17, 47	rspc2va 2848	. . . . . 6 ⊢ (((𝐶 ∈ 𝑆 ∧ (𝐺‘𝑥) ∈ 𝑆) ∧ ∀𝑎 ∈ 𝑆 ∀𝑏 ∈ 𝑆 (𝑎𝑇𝑏) ∈ 𝑆) → (𝐶𝑇(𝐺‘𝑥)) ∈ 𝑆)
49	44, 2, 45, 48	syl21anc 1232	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐶𝑇(𝐺‘𝑥)) ∈ 𝑆)
50		oveq2 5861	. . . . . 6 ⊢ (𝑧 = (𝐺‘𝑥) → (𝐶𝑇𝑧) = (𝐶𝑇(𝐺‘𝑥)))
51	50, 23	fvmptg 5572	. . . . 5 ⊢ (((𝐺‘𝑥) ∈ 𝑆 ∧ (𝐶𝑇(𝐺‘𝑥)) ∈ 𝑆) → ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘(𝐺‘𝑥)) = (𝐶𝑇(𝐺‘𝑥)))
52	2, 49, 51	syl2anc 409	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘(𝐺‘𝑥)) = (𝐶𝑇(𝐺‘𝑥)))
53		seq3distr.5	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) = (𝐶𝑇(𝐺‘𝑥)))
54	52, 53	eqtr4d 2206	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘(𝐺‘𝑥)) = (𝐹‘𝑥))
55	53, 49	eqeltrd 2247	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
56	1, 2, 3, 43, 54, 55, 1	seq3homo 10466	. 2 ⊢ (𝜑 → ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘(seq𝑀( + , 𝐺)‘𝑁)) = (seq𝑀( + , 𝐹)‘𝑁))
57		eqid 2170	. . . . 5 ⊢ (ℤ≥‘𝑀) = (ℤ≥‘𝑀)
58		eluzel2 9492	. . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
59	3, 58	syl 14	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
60	57, 59, 2, 1	seqf 10417	. . . 4 ⊢ (𝜑 → seq𝑀( + , 𝐺):(ℤ≥‘𝑀)⟶𝑆)
61	60, 3	ffvelrnd 5632	. . 3 ⊢ (𝜑 → (seq𝑀( + , 𝐺)‘𝑁) ∈ 𝑆)
62	7, 5, 61	caovcld 6006	. . 3 ⊢ (𝜑 → (𝐶𝑇(seq𝑀( + , 𝐺)‘𝑁)) ∈ 𝑆)
63		oveq2 5861	. . . 4 ⊢ (𝑧 = (seq𝑀( + , 𝐺)‘𝑁) → (𝐶𝑇𝑧) = (𝐶𝑇(seq𝑀( + , 𝐺)‘𝑁)))
64	63, 23	fvmptg 5572	. . 3 ⊢ (((seq𝑀( + , 𝐺)‘𝑁) ∈ 𝑆 ∧ (𝐶𝑇(seq𝑀( + , 𝐺)‘𝑁)) ∈ 𝑆) → ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘(seq𝑀( + , 𝐺)‘𝑁)) = (𝐶𝑇(seq𝑀( + , 𝐺)‘𝑁)))
65	61, 62, 64	syl2anc 409	. 2 ⊢ (𝜑 → ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘(seq𝑀( + , 𝐺)‘𝑁)) = (𝐶𝑇(seq𝑀( + , 𝐺)‘𝑁)))
66	56, 65	eqtr3d 2205	1 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (𝐶𝑇(seq𝑀( + , 𝐺)‘𝑁)))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1348   ∈ wcel 2141  ∀wral 2448   ↦ cmpt 4050  ‘cfv 5198  (class class class)co 5853  ℤcz 9212  ℤ_≥cuz 9487  seqcseq 10401

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-ltadd 7890

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-inn 8879  df-n0 9136  df-z 9213  df-uz 9488  df-seqfrec 10402

This theorem is referenced by:  isermulc2  11303  fsummulc2  11411