Theorem ipasslem1 31120

Description: Lemma for ipassi 31130. Show the inner product associative law for nonnegative integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)

Hypotheses

Ref	Expression
ip1i.1	⊢ 𝑋 = (BaseSet‘𝑈)
ip1i.2	⊢ 𝐺 = ( +𝑣 ‘𝑈)
ip1i.4	⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)
ip1i.7	⊢ 𝑃 = (·𝑖OLD‘𝑈)
ip1i.9	⊢ 𝑈 ∈ CPreHilOLD
ipasslem1.b	⊢ 𝐵 ∈ 𝑋

Assertion

Ref	Expression
ipasslem1	⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝑋) → ((𝑁𝑆𝐴)𝑃𝐵) = (𝑁 · (𝐴𝑃𝐵)))

Proof of Theorem ipasslem1

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

Step	Hyp	Ref	Expression
1		nn0cn 12510	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℂ)
2		ax-1cn 11154	. . . . . . . . . . . 12 ⊢ 1 ∈ ℂ
3		ip1i.9	. . . . . . . . . . . . . 14 ⊢ 𝑈 ∈ CPreHilOLD
4	3	phnvi 31105	. . . . . . . . . . . . 13 ⊢ 𝑈 ∈ NrmCVec
5		ip1i.1	. . . . . . . . . . . . . 14 ⊢ 𝑋 = (BaseSet‘𝑈)
6		ip1i.2	. . . . . . . . . . . . . 14 ⊢ 𝐺 = ( +𝑣 ‘𝑈)
7		ip1i.4	. . . . . . . . . . . . . 14 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)
8	5, 6, 7	nvdir 30920	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝑘 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝐴 ∈ 𝑋)) → ((𝑘 + 1)𝑆𝐴) = ((𝑘𝑆𝐴)𝐺(1𝑆𝐴)))
9	4, 8	mpan 702	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝐴 ∈ 𝑋) → ((𝑘 + 1)𝑆𝐴) = ((𝑘𝑆𝐴)𝐺(1𝑆𝐴)))
10	2, 9	mp3an2 1475	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℂ ∧ 𝐴 ∈ 𝑋) → ((𝑘 + 1)𝑆𝐴) = ((𝑘𝑆𝐴)𝐺(1𝑆𝐴)))
11	1, 10	sylan 591	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝐴 ∈ 𝑋) → ((𝑘 + 1)𝑆𝐴) = ((𝑘𝑆𝐴)𝐺(1𝑆𝐴)))
12	5, 7	nvsid 30916	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (1𝑆𝐴) = 𝐴)
13	4, 12	mpan 702	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑋 → (1𝑆𝐴) = 𝐴)
14	13	adantl 486	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝐴 ∈ 𝑋) → (1𝑆𝐴) = 𝐴)
15	14	oveq2d 7424	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝐴 ∈ 𝑋) → ((𝑘𝑆𝐴)𝐺(1𝑆𝐴)) = ((𝑘𝑆𝐴)𝐺𝐴))
16	11, 15	eqtrd 2804	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝐴 ∈ 𝑋) → ((𝑘 + 1)𝑆𝐴) = ((𝑘𝑆𝐴)𝐺𝐴))
17	16	oveq1d 7423	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝐴 ∈ 𝑋) → (((𝑘 + 1)𝑆𝐴)𝑃𝐵) = (((𝑘𝑆𝐴)𝐺𝐴)𝑃𝐵))
18		ipasslem1.b	. . . . . . . . . . . . 13 ⊢ 𝐵 ∈ 𝑋
19		ip1i.7	. . . . . . . . . . . . . 14 ⊢ 𝑃 = (·𝑖OLD‘𝑈)
20	5, 19	dipcl 31001	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑃𝐵) ∈ ℂ)
21	4, 18, 20	mp3an13 1478	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑋 → (𝐴𝑃𝐵) ∈ ℂ)
22	21	mullidd 11223	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑋 → (1 · (𝐴𝑃𝐵)) = (𝐴𝑃𝐵))
23	22	adantl 486	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝐴 ∈ 𝑋) → (1 · (𝐴𝑃𝐵)) = (𝐴𝑃𝐵))
24	23	oveq2d 7424	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝐴 ∈ 𝑋) → (((𝑘𝑆𝐴)𝑃𝐵) + (1 · (𝐴𝑃𝐵))) = (((𝑘𝑆𝐴)𝑃𝐵) + (𝐴𝑃𝐵)))
25	5, 7	nvscl 30915	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑘 ∈ ℂ ∧ 𝐴 ∈ 𝑋) → (𝑘𝑆𝐴) ∈ 𝑋)
26	4, 25	mp3an1 1474	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℂ ∧ 𝐴 ∈ 𝑋) → (𝑘𝑆𝐴) ∈ 𝑋)
27	1, 26	sylan 591	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝐴 ∈ 𝑋) → (𝑘𝑆𝐴) ∈ 𝑋)
28	5, 6, 7, 19, 3	ipdiri 31119	. . . . . . . . . . 11 ⊢ (((𝑘𝑆𝐴) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑘𝑆𝐴)𝐺𝐴)𝑃𝐵) = (((𝑘𝑆𝐴)𝑃𝐵) + (𝐴𝑃𝐵)))
29	18, 28	mp3an3 1476	. . . . . . . . . 10 ⊢ (((𝑘𝑆𝐴) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (((𝑘𝑆𝐴)𝐺𝐴)𝑃𝐵) = (((𝑘𝑆𝐴)𝑃𝐵) + (𝐴𝑃𝐵)))
30	27, 29	sylancom 599	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝐴 ∈ 𝑋) → (((𝑘𝑆𝐴)𝐺𝐴)𝑃𝐵) = (((𝑘𝑆𝐴)𝑃𝐵) + (𝐴𝑃𝐵)))
31	24, 30	eqtr4d 2807	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝐴 ∈ 𝑋) → (((𝑘𝑆𝐴)𝑃𝐵) + (1 · (𝐴𝑃𝐵))) = (((𝑘𝑆𝐴)𝐺𝐴)𝑃𝐵))
32	17, 31	eqtr4d 2807	. . . . . . 7 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝐴 ∈ 𝑋) → (((𝑘 + 1)𝑆𝐴)𝑃𝐵) = (((𝑘𝑆𝐴)𝑃𝐵) + (1 · (𝐴𝑃𝐵))))
33		oveq1 7415	. . . . . . 7 ⊢ (((𝑘𝑆𝐴)𝑃𝐵) = (𝑘 · (𝐴𝑃𝐵)) → (((𝑘𝑆𝐴)𝑃𝐵) + (1 · (𝐴𝑃𝐵))) = ((𝑘 · (𝐴𝑃𝐵)) + (1 · (𝐴𝑃𝐵))))
34	32, 33	sylan9eq 2824	. . . . . 6 ⊢ (((𝑘 ∈ ℕ0 ∧ 𝐴 ∈ 𝑋) ∧ ((𝑘𝑆𝐴)𝑃𝐵) = (𝑘 · (𝐴𝑃𝐵))) → (((𝑘 + 1)𝑆𝐴)𝑃𝐵) = ((𝑘 · (𝐴𝑃𝐵)) + (1 · (𝐴𝑃𝐵))))
35		adddir 11193	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ ∧ (𝐴𝑃𝐵) ∈ ℂ) → ((𝑘 + 1) · (𝐴𝑃𝐵)) = ((𝑘 · (𝐴𝑃𝐵)) + (1 · (𝐴𝑃𝐵))))
36	2, 35	mp3an2 1475	. . . . . . . 8 ⊢ ((𝑘 ∈ ℂ ∧ (𝐴𝑃𝐵) ∈ ℂ) → ((𝑘 + 1) · (𝐴𝑃𝐵)) = ((𝑘 · (𝐴𝑃𝐵)) + (1 · (𝐴𝑃𝐵))))
37	1, 21, 36	syl2an 607	. . . . . . 7 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝐴 ∈ 𝑋) → ((𝑘 + 1) · (𝐴𝑃𝐵)) = ((𝑘 · (𝐴𝑃𝐵)) + (1 · (𝐴𝑃𝐵))))
38	37	adantr 485	. . . . . 6 ⊢ (((𝑘 ∈ ℕ0 ∧ 𝐴 ∈ 𝑋) ∧ ((𝑘𝑆𝐴)𝑃𝐵) = (𝑘 · (𝐴𝑃𝐵))) → ((𝑘 + 1) · (𝐴𝑃𝐵)) = ((𝑘 · (𝐴𝑃𝐵)) + (1 · (𝐴𝑃𝐵))))
39	34, 38	eqtr4d 2807	. . . . 5 ⊢ (((𝑘 ∈ ℕ0 ∧ 𝐴 ∈ 𝑋) ∧ ((𝑘𝑆𝐴)𝑃𝐵) = (𝑘 · (𝐴𝑃𝐵))) → (((𝑘 + 1)𝑆𝐴)𝑃𝐵) = ((𝑘 + 1) · (𝐴𝑃𝐵)))
40	39	exp31 424	. . . 4 ⊢ (𝑘 ∈ ℕ0 → (𝐴 ∈ 𝑋 → (((𝑘𝑆𝐴)𝑃𝐵) = (𝑘 · (𝐴𝑃𝐵)) → (((𝑘 + 1)𝑆𝐴)𝑃𝐵) = ((𝑘 + 1) · (𝐴𝑃𝐵)))))
41	40	a2d 30	. . 3 ⊢ (𝑘 ∈ ℕ0 → ((𝐴 ∈ 𝑋 → ((𝑘𝑆𝐴)𝑃𝐵) = (𝑘 · (𝐴𝑃𝐵))) → (𝐴 ∈ 𝑋 → (((𝑘 + 1)𝑆𝐴)𝑃𝐵) = ((𝑘 + 1) · (𝐴𝑃𝐵)))))
42		eqid 2769	. . . . . 6 ⊢ (0vec‘𝑈) = (0vec‘𝑈)
43	5, 42, 19	dip0l 31007	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋) → ((0vec‘𝑈)𝑃𝐵) = 0)
44	4, 18, 43	mp2an 704	. . . 4 ⊢ ((0vec‘𝑈)𝑃𝐵) = 0
45	5, 7, 42	nv0 30926	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (0𝑆𝐴) = (0vec‘𝑈))
46	4, 45	mpan 702	. . . . 5 ⊢ (𝐴 ∈ 𝑋 → (0𝑆𝐴) = (0vec‘𝑈))
47	46	oveq1d 7423	. . . 4 ⊢ (𝐴 ∈ 𝑋 → ((0𝑆𝐴)𝑃𝐵) = ((0vec‘𝑈)𝑃𝐵))
48	21	mul02d 11404	. . . 4 ⊢ (𝐴 ∈ 𝑋 → (0 · (𝐴𝑃𝐵)) = 0)
49	44, 47, 48	3eqtr4a 2830	. . 3 ⊢ (𝐴 ∈ 𝑋 → ((0𝑆𝐴)𝑃𝐵) = (0 · (𝐴𝑃𝐵)))
50		oveq1 7415	. . . . . 6 ⊢ (𝑗 = 0 → (𝑗𝑆𝐴) = (0𝑆𝐴))
51	50	oveq1d 7423	. . . . 5 ⊢ (𝑗 = 0 → ((𝑗𝑆𝐴)𝑃𝐵) = ((0𝑆𝐴)𝑃𝐵))
52		oveq1 7415	. . . . 5 ⊢ (𝑗 = 0 → (𝑗 · (𝐴𝑃𝐵)) = (0 · (𝐴𝑃𝐵)))
53	51, 52	eqeq12d 2785	. . . 4 ⊢ (𝑗 = 0 → (((𝑗𝑆𝐴)𝑃𝐵) = (𝑗 · (𝐴𝑃𝐵)) ↔ ((0𝑆𝐴)𝑃𝐵) = (0 · (𝐴𝑃𝐵))))
54	53	imbi2d 343	. . 3 ⊢ (𝑗 = 0 → ((𝐴 ∈ 𝑋 → ((𝑗𝑆𝐴)𝑃𝐵) = (𝑗 · (𝐴𝑃𝐵))) ↔ (𝐴 ∈ 𝑋 → ((0𝑆𝐴)𝑃𝐵) = (0 · (𝐴𝑃𝐵)))))
55		oveq1 7415	. . . . . 6 ⊢ (𝑗 = 𝑘 → (𝑗𝑆𝐴) = (𝑘𝑆𝐴))
56	55	oveq1d 7423	. . . . 5 ⊢ (𝑗 = 𝑘 → ((𝑗𝑆𝐴)𝑃𝐵) = ((𝑘𝑆𝐴)𝑃𝐵))
57		oveq1 7415	. . . . 5 ⊢ (𝑗 = 𝑘 → (𝑗 · (𝐴𝑃𝐵)) = (𝑘 · (𝐴𝑃𝐵)))
58	56, 57	eqeq12d 2785	. . . 4 ⊢ (𝑗 = 𝑘 → (((𝑗𝑆𝐴)𝑃𝐵) = (𝑗 · (𝐴𝑃𝐵)) ↔ ((𝑘𝑆𝐴)𝑃𝐵) = (𝑘 · (𝐴𝑃𝐵))))
59	58	imbi2d 343	. . 3 ⊢ (𝑗 = 𝑘 → ((𝐴 ∈ 𝑋 → ((𝑗𝑆𝐴)𝑃𝐵) = (𝑗 · (𝐴𝑃𝐵))) ↔ (𝐴 ∈ 𝑋 → ((𝑘𝑆𝐴)𝑃𝐵) = (𝑘 · (𝐴𝑃𝐵)))))
60		oveq1 7415	. . . . . 6 ⊢ (𝑗 = (𝑘 + 1) → (𝑗𝑆𝐴) = ((𝑘 + 1)𝑆𝐴))
61	60	oveq1d 7423	. . . . 5 ⊢ (𝑗 = (𝑘 + 1) → ((𝑗𝑆𝐴)𝑃𝐵) = (((𝑘 + 1)𝑆𝐴)𝑃𝐵))
62		oveq1 7415	. . . . 5 ⊢ (𝑗 = (𝑘 + 1) → (𝑗 · (𝐴𝑃𝐵)) = ((𝑘 + 1) · (𝐴𝑃𝐵)))
63	61, 62	eqeq12d 2785	. . . 4 ⊢ (𝑗 = (𝑘 + 1) → (((𝑗𝑆𝐴)𝑃𝐵) = (𝑗 · (𝐴𝑃𝐵)) ↔ (((𝑘 + 1)𝑆𝐴)𝑃𝐵) = ((𝑘 + 1) · (𝐴𝑃𝐵))))
64	63	imbi2d 343	. . 3 ⊢ (𝑗 = (𝑘 + 1) → ((𝐴 ∈ 𝑋 → ((𝑗𝑆𝐴)𝑃𝐵) = (𝑗 · (𝐴𝑃𝐵))) ↔ (𝐴 ∈ 𝑋 → (((𝑘 + 1)𝑆𝐴)𝑃𝐵) = ((𝑘 + 1) · (𝐴𝑃𝐵)))))
65		oveq1 7415	. . . . . 6 ⊢ (𝑗 = 𝑁 → (𝑗𝑆𝐴) = (𝑁𝑆𝐴))
66	65	oveq1d 7423	. . . . 5 ⊢ (𝑗 = 𝑁 → ((𝑗𝑆𝐴)𝑃𝐵) = ((𝑁𝑆𝐴)𝑃𝐵))
67		oveq1 7415	. . . . 5 ⊢ (𝑗 = 𝑁 → (𝑗 · (𝐴𝑃𝐵)) = (𝑁 · (𝐴𝑃𝐵)))
68	66, 67	eqeq12d 2785	. . . 4 ⊢ (𝑗 = 𝑁 → (((𝑗𝑆𝐴)𝑃𝐵) = (𝑗 · (𝐴𝑃𝐵)) ↔ ((𝑁𝑆𝐴)𝑃𝐵) = (𝑁 · (𝐴𝑃𝐵))))
69	68	imbi2d 343	. . 3 ⊢ (𝑗 = 𝑁 → ((𝐴 ∈ 𝑋 → ((𝑗𝑆𝐴)𝑃𝐵) = (𝑗 · (𝐴𝑃𝐵))) ↔ (𝐴 ∈ 𝑋 → ((𝑁𝑆𝐴)𝑃𝐵) = (𝑁 · (𝐴𝑃𝐵)))))
70	41, 49, 54, 59, 64, 69	nn0indALT 12688	. 2 ⊢ (𝑁 ∈ ℕ0 → (𝐴 ∈ 𝑋 → ((𝑁𝑆𝐴)𝑃𝐵) = (𝑁 · (𝐴𝑃𝐵))))
71	70	imp 411	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝑋) → ((𝑁𝑆𝐴)𝑃𝐵) = (𝑁 · (𝐴𝑃𝐵)))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ‘cfv 6533  (class class class)co 7408  ℂcc 11094  0cc0 11096  1c1 11097   + caddc 11099   · cmul 11101  ℕ₀cn0 12500  NrmCVeccnv 30873   +_𝑣 cpv 30874  BaseSetcba 30875   ·_𝑠OLD cns 30876  0_veccn0v 30877  ·_𝑖OLDcdip 30989  CPreHil_OLDccphlo 31101

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730  ax-inf2 9606  ax-cnex 11152  ax-resscn 11153  ax-1cn 11154  ax-icn 11155  ax-addcl 11156  ax-addrcl 11157  ax-mulcl 11158  ax-mulrcl 11159  ax-mulcom 11160  ax-addass 11161  ax-mulass 11162  ax-distr 11163  ax-i2m1 11164  ax-1ne0 11165  ax-1rid 11166  ax-rnegex 11167  ax-rrecex 11168  ax-cnre 11169  ax-pre-lttri 11170  ax-pre-lttrn 11171  ax-pre-ltadd 11172  ax-pre-mulgt0 11173  ax-pre-sup 11174

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-er 8690  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9398  df-oi 9468  df-card 9921  df-pnf 11241  df-mnf 11242  df-xr 11243  df-ltxr 11244  df-le 11245  df-sub 11439  df-neg 11440  df-div 11868  df-nn 12230  df-2 12299  df-3 12300  df-4 12301  df-n0 12501  df-z 12588  df-uz 12859  df-rp 13013  df-fz 13532  df-fzo 13679  df-seq 14034  df-exp 14094  df-hash 14363  df-cj 15146  df-re 15147  df-im 15148  df-sqrt 15282  df-abs 15283  df-clim 15535  df-sum 15734  df-grpo 30782  df-gid 30783  df-ginv 30784  df-ablo 30834  df-vc 30848  df-nv 30881  df-va 30884  df-ba 30885  df-sm 30886  df-0v 30887  df-nmcv 30889  df-dip 30990  df-ph 31102

This theorem is referenced by:  ipasslem2  31121  ipasslem3  31122  ipasslem4  31123