Theorem seq3distr 10317

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 2517	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝑇𝑦) ∈ 𝑆)
9		oveq1 5789	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → (𝑥𝑇𝑦) = (𝑎𝑇𝑦))
10	9	eleq1d 2209	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → ((𝑥𝑇𝑦) ∈ 𝑆 ↔ (𝑎𝑇𝑦) ∈ 𝑆))
11		oveq2 5790	. . . . . . . . . 10 ⊢ (𝑦 = 𝑏 → (𝑎𝑇𝑦) = (𝑎𝑇𝑏))
12	11	eleq1d 2209	. . . . . . . . 9 ⊢ (𝑦 = 𝑏 → ((𝑎𝑇𝑦) ∈ 𝑆 ↔ (𝑎𝑇𝑏) ∈ 𝑆))
13	10, 12	cbvral2v 2668	. . . . . . . 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 5789	. . . . . . . 8 ⊢ (𝑎 = 𝐶 → (𝑎𝑇𝑏) = (𝐶𝑇𝑏))
17	16	eleq1d 2209	. . . . . . 7 ⊢ (𝑎 = 𝐶 → ((𝑎𝑇𝑏) ∈ 𝑆 ↔ (𝐶𝑇𝑏) ∈ 𝑆))
18		oveq2 5790	. . . . . . . 8 ⊢ (𝑏 = (𝑥 + 𝑦) → (𝐶𝑇𝑏) = (𝐶𝑇(𝑥 + 𝑦)))
19	18	eleq1d 2209	. . . . . . 7 ⊢ (𝑏 = (𝑥 + 𝑦) → ((𝐶𝑇𝑏) ∈ 𝑆 ↔ (𝐶𝑇(𝑥 + 𝑦)) ∈ 𝑆))
20	17, 19	rspc2va 2807	. . . . . 6 ⊢ (((𝐶 ∈ 𝑆 ∧ (𝑥 + 𝑦) ∈ 𝑆) ∧ ∀𝑎 ∈ 𝑆 ∀𝑏 ∈ 𝑆 (𝑎𝑇𝑏) ∈ 𝑆) → (𝐶𝑇(𝑥 + 𝑦)) ∈ 𝑆)
21	6, 1, 15, 20	syl21anc 1216	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝐶𝑇(𝑥 + 𝑦)) ∈ 𝑆)
22		oveq2 5790	. . . . . 6 ⊢ (𝑧 = (𝑥 + 𝑦) → (𝐶𝑇𝑧) = (𝐶𝑇(𝑥 + 𝑦)))
23		eqid 2140	. . . . . 6 ⊢ (𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧)) = (𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))
24	22, 23	fvmptg 5505	. . . . 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 521	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑥 ∈ 𝑆)
27		oveq2 5790	. . . . . . . . 9 ⊢ (𝑏 = 𝑥 → (𝐶𝑇𝑏) = (𝐶𝑇𝑥))
28	27	eleq1d 2209	. . . . . . . 8 ⊢ (𝑏 = 𝑥 → ((𝐶𝑇𝑏) ∈ 𝑆 ↔ (𝐶𝑇𝑥) ∈ 𝑆))
29	17, 28	rspc2va 2807	. . . . . . 7 ⊢ (((𝐶 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆) ∧ ∀𝑎 ∈ 𝑆 ∀𝑏 ∈ 𝑆 (𝑎𝑇𝑏) ∈ 𝑆) → (𝐶𝑇𝑥) ∈ 𝑆)
30	6, 26, 15, 29	syl21anc 1216	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝐶𝑇𝑥) ∈ 𝑆)
31		oveq2 5790	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝐶𝑇𝑧) = (𝐶𝑇𝑥))
32	31, 23	fvmptg 5505	. . . . . 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 522	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ 𝑆)
35		oveq2 5790	. . . . . . . . 9 ⊢ (𝑏 = 𝑦 → (𝐶𝑇𝑏) = (𝐶𝑇𝑦))
36	35	eleq1d 2209	. . . . . . . 8 ⊢ (𝑏 = 𝑦 → ((𝐶𝑇𝑏) ∈ 𝑆 ↔ (𝐶𝑇𝑦) ∈ 𝑆))
37	17, 36	rspc2va 2807	. . . . . . 7 ⊢ (((𝐶 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ ∀𝑎 ∈ 𝑆 ∀𝑏 ∈ 𝑆 (𝑎𝑇𝑏) ∈ 𝑆) → (𝐶𝑇𝑦) ∈ 𝑆)
38	6, 34, 15, 37	syl21anc 1216	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝐶𝑇𝑦) ∈ 𝑆)
39		oveq2 5790	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝐶𝑇𝑧) = (𝐶𝑇𝑦))
40	39, 23	fvmptg 5505	. . . . . 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 5800	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘𝑥) + ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘𝑦)) = ((𝐶𝑇𝑥) + (𝐶𝑇𝑦)))
43	4, 25, 42	3eqtr4d 2183	. . 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 5790	. . . . . . . 8 ⊢ (𝑏 = (𝐺‘𝑥) → (𝐶𝑇𝑏) = (𝐶𝑇(𝐺‘𝑥)))
47	46	eleq1d 2209	. . . . . . 7 ⊢ (𝑏 = (𝐺‘𝑥) → ((𝐶𝑇𝑏) ∈ 𝑆 ↔ (𝐶𝑇(𝐺‘𝑥)) ∈ 𝑆))
48	17, 47	rspc2va 2807	. . . . . 6 ⊢ (((𝐶 ∈ 𝑆 ∧ (𝐺‘𝑥) ∈ 𝑆) ∧ ∀𝑎 ∈ 𝑆 ∀𝑏 ∈ 𝑆 (𝑎𝑇𝑏) ∈ 𝑆) → (𝐶𝑇(𝐺‘𝑥)) ∈ 𝑆)
49	44, 2, 45, 48	syl21anc 1216	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐶𝑇(𝐺‘𝑥)) ∈ 𝑆)
50		oveq2 5790	. . . . . 6 ⊢ (𝑧 = (𝐺‘𝑥) → (𝐶𝑇𝑧) = (𝐶𝑇(𝐺‘𝑥)))
51	50, 23	fvmptg 5505	. . . . 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 2176	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘(𝐺‘𝑥)) = (𝐹‘𝑥))
55	53, 49	eqeltrd 2217	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
56	1, 2, 3, 43, 54, 55, 1	seq3homo 10314	. 2 ⊢ (𝜑 → ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘(seq𝑀( + , 𝐺)‘𝑁)) = (seq𝑀( + , 𝐹)‘𝑁))
57		eqid 2140	. . . . 5 ⊢ (ℤ≥‘𝑀) = (ℤ≥‘𝑀)
58		eluzel2 9355	. . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
59	3, 58	syl 14	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
60	57, 59, 2, 1	seqf 10265	. . . 4 ⊢ (𝜑 → seq𝑀( + , 𝐺):(ℤ≥‘𝑀)⟶𝑆)
61	60, 3	ffvelrnd 5564	. . 3 ⊢ (𝜑 → (seq𝑀( + , 𝐺)‘𝑁) ∈ 𝑆)
62	7, 5, 61	caovcld 5932	. . 3 ⊢ (𝜑 → (𝐶𝑇(seq𝑀( + , 𝐺)‘𝑁)) ∈ 𝑆)
63		oveq2 5790	. . . 4 ⊢ (𝑧 = (seq𝑀( + , 𝐺)‘𝑁) → (𝐶𝑇𝑧) = (𝐶𝑇(seq𝑀( + , 𝐺)‘𝑁)))
64	63, 23	fvmptg 5505	. . 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 2175	1 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (𝐶𝑇(seq𝑀( + , 𝐺)‘𝑁)))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1332   ∈ wcel 1481  ∀wral 2417   ↦ cmpt 3997  ‘cfv 5131  (class class class)co 5782  ℤcz 9078  ℤ_≥cuz 9350  seqcseq 10249

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-addcom 7744  ax-addass 7746  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-0id 7752  ax-rnegex 7753  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-ltadd 7760

This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-iord 4296  df-on 4298  df-ilim 4299  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-frec 6296  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-inn 8745  df-n0 9002  df-z 9079  df-uz 9351  df-seqfrec 10250

This theorem is referenced by:  isermulc2  11141  fsummulc2  11249