Theorem seq3distr 10762

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 276	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝐶 ∈ 𝑆)
7		seq3distr.t	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑇𝑦) ∈ 𝑆)
8	7	ralrimivva 2612	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝑇𝑦) ∈ 𝑆)
9		oveq1 6014	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → (𝑥𝑇𝑦) = (𝑎𝑇𝑦))
10	9	eleq1d 2298	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → ((𝑥𝑇𝑦) ∈ 𝑆 ↔ (𝑎𝑇𝑦) ∈ 𝑆))
11		oveq2 6015	. . . . . . . . . 10 ⊢ (𝑦 = 𝑏 → (𝑎𝑇𝑦) = (𝑎𝑇𝑏))
12	11	eleq1d 2298	. . . . . . . . 9 ⊢ (𝑦 = 𝑏 → ((𝑎𝑇𝑦) ∈ 𝑆 ↔ (𝑎𝑇𝑏) ∈ 𝑆))
13	10, 12	cbvral2v 2778	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝑇𝑦) ∈ 𝑆 ↔ ∀𝑎 ∈ 𝑆 ∀𝑏 ∈ 𝑆 (𝑎𝑇𝑏) ∈ 𝑆)
14	8, 13	sylib 122	. . . . . . 7 ⊢ (𝜑 → ∀𝑎 ∈ 𝑆 ∀𝑏 ∈ 𝑆 (𝑎𝑇𝑏) ∈ 𝑆)
15	14	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ∀𝑎 ∈ 𝑆 ∀𝑏 ∈ 𝑆 (𝑎𝑇𝑏) ∈ 𝑆)
16		oveq1 6014	. . . . . . . 8 ⊢ (𝑎 = 𝐶 → (𝑎𝑇𝑏) = (𝐶𝑇𝑏))
17	16	eleq1d 2298	. . . . . . 7 ⊢ (𝑎 = 𝐶 → ((𝑎𝑇𝑏) ∈ 𝑆 ↔ (𝐶𝑇𝑏) ∈ 𝑆))
18		oveq2 6015	. . . . . . . 8 ⊢ (𝑏 = (𝑥 + 𝑦) → (𝐶𝑇𝑏) = (𝐶𝑇(𝑥 + 𝑦)))
19	18	eleq1d 2298	. . . . . . 7 ⊢ (𝑏 = (𝑥 + 𝑦) → ((𝐶𝑇𝑏) ∈ 𝑆 ↔ (𝐶𝑇(𝑥 + 𝑦)) ∈ 𝑆))
20	17, 19	rspc2va 2921	. . . . . 6 ⊢ (((𝐶 ∈ 𝑆 ∧ (𝑥 + 𝑦) ∈ 𝑆) ∧ ∀𝑎 ∈ 𝑆 ∀𝑏 ∈ 𝑆 (𝑎𝑇𝑏) ∈ 𝑆) → (𝐶𝑇(𝑥 + 𝑦)) ∈ 𝑆)
21	6, 1, 15, 20	syl21anc 1270	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝐶𝑇(𝑥 + 𝑦)) ∈ 𝑆)
22		oveq2 6015	. . . . . 6 ⊢ (𝑧 = (𝑥 + 𝑦) → (𝐶𝑇𝑧) = (𝐶𝑇(𝑥 + 𝑦)))
23		eqid 2229	. . . . . 6 ⊢ (𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧)) = (𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))
24	22, 23	fvmptg 5712	. . . . 5 ⊢ (((𝑥 + 𝑦) ∈ 𝑆 ∧ (𝐶𝑇(𝑥 + 𝑦)) ∈ 𝑆) → ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘(𝑥 + 𝑦)) = (𝐶𝑇(𝑥 + 𝑦)))
25	1, 21, 24	syl2anc 411	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘(𝑥 + 𝑦)) = (𝐶𝑇(𝑥 + 𝑦)))
26		simprl 529	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑥 ∈ 𝑆)
27		oveq2 6015	. . . . . . . . 9 ⊢ (𝑏 = 𝑥 → (𝐶𝑇𝑏) = (𝐶𝑇𝑥))
28	27	eleq1d 2298	. . . . . . . 8 ⊢ (𝑏 = 𝑥 → ((𝐶𝑇𝑏) ∈ 𝑆 ↔ (𝐶𝑇𝑥) ∈ 𝑆))
29	17, 28	rspc2va 2921	. . . . . . 7 ⊢ (((𝐶 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆) ∧ ∀𝑎 ∈ 𝑆 ∀𝑏 ∈ 𝑆 (𝑎𝑇𝑏) ∈ 𝑆) → (𝐶𝑇𝑥) ∈ 𝑆)
30	6, 26, 15, 29	syl21anc 1270	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝐶𝑇𝑥) ∈ 𝑆)
31		oveq2 6015	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝐶𝑇𝑧) = (𝐶𝑇𝑥))
32	31, 23	fvmptg 5712	. . . . . 6 ⊢ ((𝑥 ∈ 𝑆 ∧ (𝐶𝑇𝑥) ∈ 𝑆) → ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘𝑥) = (𝐶𝑇𝑥))
33	26, 30, 32	syl2anc 411	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘𝑥) = (𝐶𝑇𝑥))
34		simprr 531	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ 𝑆)
35		oveq2 6015	. . . . . . . . 9 ⊢ (𝑏 = 𝑦 → (𝐶𝑇𝑏) = (𝐶𝑇𝑦))
36	35	eleq1d 2298	. . . . . . . 8 ⊢ (𝑏 = 𝑦 → ((𝐶𝑇𝑏) ∈ 𝑆 ↔ (𝐶𝑇𝑦) ∈ 𝑆))
37	17, 36	rspc2va 2921	. . . . . . 7 ⊢ (((𝐶 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ ∀𝑎 ∈ 𝑆 ∀𝑏 ∈ 𝑆 (𝑎𝑇𝑏) ∈ 𝑆) → (𝐶𝑇𝑦) ∈ 𝑆)
38	6, 34, 15, 37	syl21anc 1270	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝐶𝑇𝑦) ∈ 𝑆)
39		oveq2 6015	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝐶𝑇𝑧) = (𝐶𝑇𝑦))
40	39, 23	fvmptg 5712	. . . . . 6 ⊢ ((𝑦 ∈ 𝑆 ∧ (𝐶𝑇𝑦) ∈ 𝑆) → ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘𝑦) = (𝐶𝑇𝑦))
41	34, 38, 40	syl2anc 411	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘𝑦) = (𝐶𝑇𝑦))
42	33, 41	oveq12d 6025	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘𝑥) + ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘𝑦)) = ((𝐶𝑇𝑥) + (𝐶𝑇𝑦)))
43	4, 25, 42	3eqtr4d 2272	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘(𝑥 + 𝑦)) = (((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘𝑥) + ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘𝑦)))
44	5	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → 𝐶 ∈ 𝑆)
45	14	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → ∀𝑎 ∈ 𝑆 ∀𝑏 ∈ 𝑆 (𝑎𝑇𝑏) ∈ 𝑆)
46		oveq2 6015	. . . . . . . 8 ⊢ (𝑏 = (𝐺‘𝑥) → (𝐶𝑇𝑏) = (𝐶𝑇(𝐺‘𝑥)))
47	46	eleq1d 2298	. . . . . . 7 ⊢ (𝑏 = (𝐺‘𝑥) → ((𝐶𝑇𝑏) ∈ 𝑆 ↔ (𝐶𝑇(𝐺‘𝑥)) ∈ 𝑆))
48	17, 47	rspc2va 2921	. . . . . 6 ⊢ (((𝐶 ∈ 𝑆 ∧ (𝐺‘𝑥) ∈ 𝑆) ∧ ∀𝑎 ∈ 𝑆 ∀𝑏 ∈ 𝑆 (𝑎𝑇𝑏) ∈ 𝑆) → (𝐶𝑇(𝐺‘𝑥)) ∈ 𝑆)
49	44, 2, 45, 48	syl21anc 1270	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐶𝑇(𝐺‘𝑥)) ∈ 𝑆)
50		oveq2 6015	. . . . . 6 ⊢ (𝑧 = (𝐺‘𝑥) → (𝐶𝑇𝑧) = (𝐶𝑇(𝐺‘𝑥)))
51	50, 23	fvmptg 5712	. . . . 5 ⊢ (((𝐺‘𝑥) ∈ 𝑆 ∧ (𝐶𝑇(𝐺‘𝑥)) ∈ 𝑆) → ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘(𝐺‘𝑥)) = (𝐶𝑇(𝐺‘𝑥)))
52	2, 49, 51	syl2anc 411	. . . 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 2265	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘(𝐺‘𝑥)) = (𝐹‘𝑥))
55	53, 49	eqeltrd 2306	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
56	1, 2, 3, 43, 54, 55, 1	seq3homo 10757	. 2 ⊢ (𝜑 → ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘(seq𝑀( + , 𝐺)‘𝑁)) = (seq𝑀( + , 𝐹)‘𝑁))
57		eqid 2229	. . . . 5 ⊢ (ℤ≥‘𝑀) = (ℤ≥‘𝑀)
58		eluzel2 9735	. . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
59	3, 58	syl 14	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
60	57, 59, 2, 1	seqf 10694	. . . 4 ⊢ (𝜑 → seq𝑀( + , 𝐺):(ℤ≥‘𝑀)⟶𝑆)
61	60, 3	ffvelcdmd 5773	. . 3 ⊢ (𝜑 → (seq𝑀( + , 𝐺)‘𝑁) ∈ 𝑆)
62	7, 5, 61	caovcld 6165	. . 3 ⊢ (𝜑 → (𝐶𝑇(seq𝑀( + , 𝐺)‘𝑁)) ∈ 𝑆)
63		oveq2 6015	. . . 4 ⊢ (𝑧 = (seq𝑀( + , 𝐺)‘𝑁) → (𝐶𝑇𝑧) = (𝐶𝑇(seq𝑀( + , 𝐺)‘𝑁)))
64	63, 23	fvmptg 5712	. . 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 411	. 2 ⊢ (𝜑 → ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘(seq𝑀( + , 𝐺)‘𝑁)) = (𝐶𝑇(seq𝑀( + , 𝐺)‘𝑁)))
66	56, 65	eqtr3d 2264	1 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (𝐶𝑇(seq𝑀( + , 𝐺)‘𝑁)))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395   ∈ wcel 2200  ∀wral 2508   ↦ cmpt 4145  ‘cfv 5318  (class class class)co 6007  ℤcz 9454  ℤ_≥cuz 9730  seqcseq 10677

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8098  ax-resscn 8099  ax-1cn 8100  ax-1re 8101  ax-icn 8102  ax-addcl 8103  ax-addrcl 8104  ax-mulcl 8105  ax-addcom 8107  ax-addass 8109  ax-distr 8111  ax-i2m1 8112  ax-0lt1 8113  ax-0id 8115  ax-rnegex 8116  ax-cnre 8118  ax-pre-ltirr 8119  ax-pre-ltwlin 8120  ax-pre-lttrn 8121  ax-pre-ltadd 8123

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-frec 6543  df-pnf 8191  df-mnf 8192  df-xr 8193  df-ltxr 8194  df-le 8195  df-sub 8327  df-neg 8328  df-inn 9119  df-n0 9378  df-z 9455  df-uz 9731  df-seqfrec 10678

This theorem is referenced by:  isermulc2  11859  fsummulc2  11967