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Theorem ipdirilem 27684

Description: Lemma for ipdiri 27685. (Contributed by NM, 26-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 ∈ CPreHil_OLD
ipdiri.8	⊢ 𝐴 ∈ 𝑋
ipdiri.9	⊢ 𝐵 ∈ 𝑋
ipdiri.10	⊢ 𝐶 ∈ 𝑋

**Assertion**
Ref	Expression
ipdirilem	⊢ ((𝐴𝐺𝐵)𝑃𝐶) = ((𝐴𝑃𝐶) + (𝐵𝑃𝐶))

**Proof of Theorem ipdirilem**
Step	Hyp	Ref	Expression
1		2cn 11091	. . . . . . 7 ⊢ 2 ∈ ℂ
2		2ne0 11113	. . . . . . 7 ⊢ 2 ≠ 0
3	1, 2	recidi 10756	. . . . . 6 ⊢ (2 · (1 / 2)) = 1
4	3	oveq1i 6660	. . . . 5 ⊢ ((2 · (1 / 2))𝑆(𝐴𝐺𝐵)) = (1𝑆(𝐴𝐺𝐵))
5		ip1i.9	. . . . . . 7 ⊢ 𝑈 ∈ CPreHil_OLD
6	5	phnvi 27671	. . . . . 6 ⊢ 𝑈 ∈ NrmCVec
7		halfcn 11247	. . . . . . 7 ⊢ (1 / 2) ∈ ℂ
8		ipdiri.8	. . . . . . . 8 ⊢ 𝐴 ∈ 𝑋
9		ipdiri.9	. . . . . . . 8 ⊢ 𝐵 ∈ 𝑋
10		ip1i.1	. . . . . . . . 9 ⊢ 𝑋 = (BaseSet‘𝑈)
11		ip1i.2	. . . . . . . . 9 ⊢ 𝐺 = ( +_𝑣 ‘𝑈)
12	10, 11	nvgcl 27475	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
13	6, 8, 9, 12	mp3an 1424	. . . . . . 7 ⊢ (𝐴𝐺𝐵) ∈ 𝑋
14	1, 7, 13	3pm3.2i 1239	. . . . . 6 ⊢ (2 ∈ ℂ ∧ (1 / 2) ∈ ℂ ∧ (𝐴𝐺𝐵) ∈ 𝑋)
15		ip1i.4	. . . . . . 7 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
16	10, 15	nvsass 27483	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ (2 ∈ ℂ ∧ (1 / 2) ∈ ℂ ∧ (𝐴𝐺𝐵) ∈ 𝑋)) → ((2 · (1 / 2))𝑆(𝐴𝐺𝐵)) = (2𝑆((1 / 2)𝑆(𝐴𝐺𝐵))))
17	6, 14, 16	mp2an 708	. . . . 5 ⊢ ((2 · (1 / 2))𝑆(𝐴𝐺𝐵)) = (2𝑆((1 / 2)𝑆(𝐴𝐺𝐵)))
18	10, 15	nvsid 27482	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴𝐺𝐵) ∈ 𝑋) → (1𝑆(𝐴𝐺𝐵)) = (𝐴𝐺𝐵))
19	6, 13, 18	mp2an 708	. . . . 5 ⊢ (1𝑆(𝐴𝐺𝐵)) = (𝐴𝐺𝐵)
20	4, 17, 19	3eqtr3i 2652	. . . 4 ⊢ (2𝑆((1 / 2)𝑆(𝐴𝐺𝐵))) = (𝐴𝐺𝐵)
21	20	oveq1i 6660	. . 3 ⊢ ((2𝑆((1 / 2)𝑆(𝐴𝐺𝐵)))𝑃𝐶) = ((𝐴𝐺𝐵)𝑃𝐶)
22		ip1i.7	. . . 4 ⊢ 𝑃 = (·_𝑖OLD‘𝑈)
23	10, 15	nvscl 27481	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ (1 / 2) ∈ ℂ ∧ (𝐴𝐺𝐵) ∈ 𝑋) → ((1 / 2)𝑆(𝐴𝐺𝐵)) ∈ 𝑋)
24	6, 7, 13, 23	mp3an 1424	. . . 4 ⊢ ((1 / 2)𝑆(𝐴𝐺𝐵)) ∈ 𝑋
25		ipdiri.10	. . . 4 ⊢ 𝐶 ∈ 𝑋
26	10, 11, 15, 22, 5, 24, 25	ip2i 27683	. . 3 ⊢ ((2𝑆((1 / 2)𝑆(𝐴𝐺𝐵)))𝑃𝐶) = (2 · (((1 / 2)𝑆(𝐴𝐺𝐵))𝑃𝐶))
27	21, 26	eqtr3i 2646	. 2 ⊢ ((𝐴𝐺𝐵)𝑃𝐶) = (2 · (((1 / 2)𝑆(𝐴𝐺𝐵))𝑃𝐶))
28		neg1cn 11124	. . . . . 6 ⊢ -1 ∈ ℂ
29	10, 15	nvscl 27481	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ -1 ∈ ℂ ∧ 𝐵 ∈ 𝑋) → (-1𝑆𝐵) ∈ 𝑋)
30	6, 28, 9, 29	mp3an 1424	. . . . 5 ⊢ (-1𝑆𝐵) ∈ 𝑋
31	10, 11	nvgcl 27475	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ (-1𝑆𝐵) ∈ 𝑋) → (𝐴𝐺(-1𝑆𝐵)) ∈ 𝑋)
32	6, 8, 30, 31	mp3an 1424	. . . 4 ⊢ (𝐴𝐺(-1𝑆𝐵)) ∈ 𝑋
33	10, 15	nvscl 27481	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ (1 / 2) ∈ ℂ ∧ (𝐴𝐺(-1𝑆𝐵)) ∈ 𝑋) → ((1 / 2)𝑆(𝐴𝐺(-1𝑆𝐵))) ∈ 𝑋)
34	6, 7, 32, 33	mp3an 1424	. . 3 ⊢ ((1 / 2)𝑆(𝐴𝐺(-1𝑆𝐵))) ∈ 𝑋
35	10, 11, 15, 22, 5, 24, 34, 25	ip1i 27682	. 2 ⊢ (((((1 / 2)𝑆(𝐴𝐺𝐵))𝐺((1 / 2)𝑆(𝐴𝐺(-1𝑆𝐵))))𝑃𝐶) + ((((1 / 2)𝑆(𝐴𝐺𝐵))𝐺(-1𝑆((1 / 2)𝑆(𝐴𝐺(-1𝑆𝐵)))))𝑃𝐶)) = (2 · (((1 / 2)𝑆(𝐴𝐺𝐵))𝑃𝐶))
36		eqid 2622	. . . . . . . . . . . 12 ⊢ (1^st ‘𝑈) = (1^st ‘𝑈)
37	36	nvvc 27470	. . . . . . . . . . 11 ⊢ (𝑈 ∈ NrmCVec → (1^st ‘𝑈) ∈ CVec_OLD)
38	6, 37	ax-mp 5	. . . . . . . . . 10 ⊢ (1^st ‘𝑈) ∈ CVec_OLD
39	11	vafval 27458	. . . . . . . . . . 11 ⊢ 𝐺 = (1^st ‘(1^st ‘𝑈))
40	39	vcablo 27424	. . . . . . . . . 10 ⊢ ((1^st ‘𝑈) ∈ CVec_OLD → 𝐺 ∈ AbelOp)
41	38, 40	ax-mp 5	. . . . . . . . 9 ⊢ 𝐺 ∈ AbelOp
42	8, 9	pm3.2i 471	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)
43	8, 30	pm3.2i 471	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑋 ∧ (-1𝑆𝐵) ∈ 𝑋)
44	10, 11	bafval 27459	. . . . . . . . . 10 ⊢ 𝑋 = ran 𝐺
45	44	ablo4 27404	. . . . . . . . 9 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ (-1𝑆𝐵) ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺(𝐴𝐺(-1𝑆𝐵))) = ((𝐴𝐺𝐴)𝐺(𝐵𝐺(-1𝑆𝐵))))
46	41, 42, 43, 45	mp3an 1424	. . . . . . . 8 ⊢ ((𝐴𝐺𝐵)𝐺(𝐴𝐺(-1𝑆𝐵))) = ((𝐴𝐺𝐴)𝐺(𝐵𝐺(-1𝑆𝐵)))
47	15	smfval 27460	. . . . . . . . . . 11 ⊢ 𝑆 = (2^nd ‘(1^st ‘𝑈))
48	39, 47, 44	vc2OLD 27423	. . . . . . . . . 10 ⊢ (((1^st ‘𝑈) ∈ CVec_OLD ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺𝐴) = (2𝑆𝐴))
49	38, 8, 48	mp2an 708	. . . . . . . . 9 ⊢ (𝐴𝐺𝐴) = (2𝑆𝐴)
50		eqid 2622	. . . . . . . . . . 11 ⊢ (0_vec‘𝑈) = (0_vec‘𝑈)
51	10, 11, 15, 50	nvrinv 27506	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋) → (𝐵𝐺(-1𝑆𝐵)) = (0_vec‘𝑈))
52	6, 9, 51	mp2an 708	. . . . . . . . 9 ⊢ (𝐵𝐺(-1𝑆𝐵)) = (0_vec‘𝑈)
53	49, 52	oveq12i 6662	. . . . . . . 8 ⊢ ((𝐴𝐺𝐴)𝐺(𝐵𝐺(-1𝑆𝐵))) = ((2𝑆𝐴)𝐺(0_vec‘𝑈))
54	10, 15	nvscl 27481	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 2 ∈ ℂ ∧ 𝐴 ∈ 𝑋) → (2𝑆𝐴) ∈ 𝑋)
55	6, 1, 8, 54	mp3an 1424	. . . . . . . . 9 ⊢ (2𝑆𝐴) ∈ 𝑋
56	10, 11, 50	nv0rid 27490	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ (2𝑆𝐴) ∈ 𝑋) → ((2𝑆𝐴)𝐺(0_vec‘𝑈)) = (2𝑆𝐴))
57	6, 55, 56	mp2an 708	. . . . . . . 8 ⊢ ((2𝑆𝐴)𝐺(0_vec‘𝑈)) = (2𝑆𝐴)
58	46, 53, 57	3eqtri 2648	. . . . . . 7 ⊢ ((𝐴𝐺𝐵)𝐺(𝐴𝐺(-1𝑆𝐵))) = (2𝑆𝐴)
59	58	oveq2i 6661	. . . . . 6 ⊢ ((1 / 2)𝑆((𝐴𝐺𝐵)𝐺(𝐴𝐺(-1𝑆𝐵)))) = ((1 / 2)𝑆(2𝑆𝐴))
60	7, 1, 8	3pm3.2i 1239	. . . . . . 7 ⊢ ((1 / 2) ∈ ℂ ∧ 2 ∈ ℂ ∧ 𝐴 ∈ 𝑋)
61	10, 15	nvsass 27483	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ ((1 / 2) ∈ ℂ ∧ 2 ∈ ℂ ∧ 𝐴 ∈ 𝑋)) → (((1 / 2) · 2)𝑆𝐴) = ((1 / 2)𝑆(2𝑆𝐴)))
62	6, 60, 61	mp2an 708	. . . . . 6 ⊢ (((1 / 2) · 2)𝑆𝐴) = ((1 / 2)𝑆(2𝑆𝐴))
63	59, 62	eqtr4i 2647	. . . . 5 ⊢ ((1 / 2)𝑆((𝐴𝐺𝐵)𝐺(𝐴𝐺(-1𝑆𝐵)))) = (((1 / 2) · 2)𝑆𝐴)
64	7, 13, 32	3pm3.2i 1239	. . . . . 6 ⊢ ((1 / 2) ∈ ℂ ∧ (𝐴𝐺𝐵) ∈ 𝑋 ∧ (𝐴𝐺(-1𝑆𝐵)) ∈ 𝑋)
65	10, 11, 15	nvdi 27485	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ ((1 / 2) ∈ ℂ ∧ (𝐴𝐺𝐵) ∈ 𝑋 ∧ (𝐴𝐺(-1𝑆𝐵)) ∈ 𝑋)) → ((1 / 2)𝑆((𝐴𝐺𝐵)𝐺(𝐴𝐺(-1𝑆𝐵)))) = (((1 / 2)𝑆(𝐴𝐺𝐵))𝐺((1 / 2)𝑆(𝐴𝐺(-1𝑆𝐵)))))
66	6, 64, 65	mp2an 708	. . . . 5 ⊢ ((1 / 2)𝑆((𝐴𝐺𝐵)𝐺(𝐴𝐺(-1𝑆𝐵)))) = (((1 / 2)𝑆(𝐴𝐺𝐵))𝐺((1 / 2)𝑆(𝐴𝐺(-1𝑆𝐵))))
67		ax-1cn 9994	. . . . . . . 8 ⊢ 1 ∈ ℂ
68	67, 1, 2	divcan1i 10769	. . . . . . 7 ⊢ ((1 / 2) · 2) = 1
69	68	oveq1i 6660	. . . . . 6 ⊢ (((1 / 2) · 2)𝑆𝐴) = (1𝑆𝐴)
70	10, 15	nvsid 27482	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (1𝑆𝐴) = 𝐴)
71	6, 8, 70	mp2an 708	. . . . . 6 ⊢ (1𝑆𝐴) = 𝐴
72	69, 71	eqtri 2644	. . . . 5 ⊢ (((1 / 2) · 2)𝑆𝐴) = 𝐴
73	63, 66, 72	3eqtr3i 2652	. . . 4 ⊢ (((1 / 2)𝑆(𝐴𝐺𝐵))𝐺((1 / 2)𝑆(𝐴𝐺(-1𝑆𝐵)))) = 𝐴
74	73	oveq1i 6660	. . 3 ⊢ ((((1 / 2)𝑆(𝐴𝐺𝐵))𝐺((1 / 2)𝑆(𝐴𝐺(-1𝑆𝐵))))𝑃𝐶) = (𝐴𝑃𝐶)
75	28, 7	mulcomi 10046	. . . . . . . . 9 ⊢ (-1 · (1 / 2)) = ((1 / 2) · -1)
76	75	oveq1i 6660	. . . . . . . 8 ⊢ ((-1 · (1 / 2))𝑆(𝐴𝐺(-1𝑆𝐵))) = (((1 / 2) · -1)𝑆(𝐴𝐺(-1𝑆𝐵)))
77	28, 7, 32	3pm3.2i 1239	. . . . . . . . 9 ⊢ (-1 ∈ ℂ ∧ (1 / 2) ∈ ℂ ∧ (𝐴𝐺(-1𝑆𝐵)) ∈ 𝑋)
78	10, 15	nvsass 27483	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ (-1 ∈ ℂ ∧ (1 / 2) ∈ ℂ ∧ (𝐴𝐺(-1𝑆𝐵)) ∈ 𝑋)) → ((-1 · (1 / 2))𝑆(𝐴𝐺(-1𝑆𝐵))) = (-1𝑆((1 / 2)𝑆(𝐴𝐺(-1𝑆𝐵)))))
79	6, 77, 78	mp2an 708	. . . . . . . 8 ⊢ ((-1 · (1 / 2))𝑆(𝐴𝐺(-1𝑆𝐵))) = (-1𝑆((1 / 2)𝑆(𝐴𝐺(-1𝑆𝐵))))
80	7, 28, 32	3pm3.2i 1239	. . . . . . . . . 10 ⊢ ((1 / 2) ∈ ℂ ∧ -1 ∈ ℂ ∧ (𝐴𝐺(-1𝑆𝐵)) ∈ 𝑋)
81	10, 15	nvsass 27483	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ ((1 / 2) ∈ ℂ ∧ -1 ∈ ℂ ∧ (𝐴𝐺(-1𝑆𝐵)) ∈ 𝑋)) → (((1 / 2) · -1)𝑆(𝐴𝐺(-1𝑆𝐵))) = ((1 / 2)𝑆(-1𝑆(𝐴𝐺(-1𝑆𝐵)))))
82	6, 80, 81	mp2an 708	. . . . . . . . 9 ⊢ (((1 / 2) · -1)𝑆(𝐴𝐺(-1𝑆𝐵))) = ((1 / 2)𝑆(-1𝑆(𝐴𝐺(-1𝑆𝐵))))
83	28, 8, 30	3pm3.2i 1239	. . . . . . . . . . . 12 ⊢ (-1 ∈ ℂ ∧ 𝐴 ∈ 𝑋 ∧ (-1𝑆𝐵) ∈ 𝑋)
84	10, 11, 15	nvdi 27485	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ NrmCVec ∧ (-1 ∈ ℂ ∧ 𝐴 ∈ 𝑋 ∧ (-1𝑆𝐵) ∈ 𝑋)) → (-1𝑆(𝐴𝐺(-1𝑆𝐵))) = ((-1𝑆𝐴)𝐺(-1𝑆(-1𝑆𝐵))))
85	6, 83, 84	mp2an 708	. . . . . . . . . . 11 ⊢ (-1𝑆(𝐴𝐺(-1𝑆𝐵))) = ((-1𝑆𝐴)𝐺(-1𝑆(-1𝑆𝐵)))
86		neg1mulneg1e1 11245	. . . . . . . . . . . . . 14 ⊢ (-1 · -1) = 1
87	86	oveq1i 6660	. . . . . . . . . . . . 13 ⊢ ((-1 · -1)𝑆𝐵) = (1𝑆𝐵)
88	28, 28, 9	3pm3.2i 1239	. . . . . . . . . . . . . 14 ⊢ (-1 ∈ ℂ ∧ -1 ∈ ℂ ∧ 𝐵 ∈ 𝑋)
89	10, 15	nvsass 27483	. . . . . . . . . . . . . 14 ⊢ ((𝑈 ∈ NrmCVec ∧ (-1 ∈ ℂ ∧ -1 ∈ ℂ ∧ 𝐵 ∈ 𝑋)) → ((-1 · -1)𝑆𝐵) = (-1𝑆(-1𝑆𝐵)))
90	6, 88, 89	mp2an 708	. . . . . . . . . . . . 13 ⊢ ((-1 · -1)𝑆𝐵) = (-1𝑆(-1𝑆𝐵))
91	10, 15	nvsid 27482	. . . . . . . . . . . . . 14 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋) → (1𝑆𝐵) = 𝐵)
92	6, 9, 91	mp2an 708	. . . . . . . . . . . . 13 ⊢ (1𝑆𝐵) = 𝐵
93	87, 90, 92	3eqtr3i 2652	. . . . . . . . . . . 12 ⊢ (-1𝑆(-1𝑆𝐵)) = 𝐵
94	93	oveq2i 6661	. . . . . . . . . . 11 ⊢ ((-1𝑆𝐴)𝐺(-1𝑆(-1𝑆𝐵))) = ((-1𝑆𝐴)𝐺𝐵)
95	85, 94	eqtri 2644	. . . . . . . . . 10 ⊢ (-1𝑆(𝐴𝐺(-1𝑆𝐵))) = ((-1𝑆𝐴)𝐺𝐵)
96	95	oveq2i 6661	. . . . . . . . 9 ⊢ ((1 / 2)𝑆(-1𝑆(𝐴𝐺(-1𝑆𝐵)))) = ((1 / 2)𝑆((-1𝑆𝐴)𝐺𝐵))
97	82, 96	eqtri 2644	. . . . . . . 8 ⊢ (((1 / 2) · -1)𝑆(𝐴𝐺(-1𝑆𝐵))) = ((1 / 2)𝑆((-1𝑆𝐴)𝐺𝐵))
98	76, 79, 97	3eqtr3i 2652	. . . . . . 7 ⊢ (-1𝑆((1 / 2)𝑆(𝐴𝐺(-1𝑆𝐵)))) = ((1 / 2)𝑆((-1𝑆𝐴)𝐺𝐵))
99	98	oveq2i 6661	. . . . . 6 ⊢ (((1 / 2)𝑆(𝐴𝐺𝐵))𝐺(-1𝑆((1 / 2)𝑆(𝐴𝐺(-1𝑆𝐵))))) = (((1 / 2)𝑆(𝐴𝐺𝐵))𝐺((1 / 2)𝑆((-1𝑆𝐴)𝐺𝐵)))
100	10, 15	nvscl 27481	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ -1 ∈ ℂ ∧ 𝐴 ∈ 𝑋) → (-1𝑆𝐴) ∈ 𝑋)
101	6, 28, 8, 100	mp3an 1424	. . . . . . . . 9 ⊢ (-1𝑆𝐴) ∈ 𝑋
102	10, 11	nvgcl 27475	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ (-1𝑆𝐴) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((-1𝑆𝐴)𝐺𝐵) ∈ 𝑋)
103	6, 101, 9, 102	mp3an 1424	. . . . . . . 8 ⊢ ((-1𝑆𝐴)𝐺𝐵) ∈ 𝑋
104	7, 13, 103	3pm3.2i 1239	. . . . . . 7 ⊢ ((1 / 2) ∈ ℂ ∧ (𝐴𝐺𝐵) ∈ 𝑋 ∧ ((-1𝑆𝐴)𝐺𝐵) ∈ 𝑋)
105	10, 11, 15	nvdi 27485	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ ((1 / 2) ∈ ℂ ∧ (𝐴𝐺𝐵) ∈ 𝑋 ∧ ((-1𝑆𝐴)𝐺𝐵) ∈ 𝑋)) → ((1 / 2)𝑆((𝐴𝐺𝐵)𝐺((-1𝑆𝐴)𝐺𝐵))) = (((1 / 2)𝑆(𝐴𝐺𝐵))𝐺((1 / 2)𝑆((-1𝑆𝐴)𝐺𝐵))))
106	6, 104, 105	mp2an 708	. . . . . 6 ⊢ ((1 / 2)𝑆((𝐴𝐺𝐵)𝐺((-1𝑆𝐴)𝐺𝐵))) = (((1 / 2)𝑆(𝐴𝐺𝐵))𝐺((1 / 2)𝑆((-1𝑆𝐴)𝐺𝐵)))
107	99, 106	eqtr4i 2647	. . . . 5 ⊢ (((1 / 2)𝑆(𝐴𝐺𝐵))𝐺(-1𝑆((1 / 2)𝑆(𝐴𝐺(-1𝑆𝐵))))) = ((1 / 2)𝑆((𝐴𝐺𝐵)𝐺((-1𝑆𝐴)𝐺𝐵)))
108	101, 9	pm3.2i 471	. . . . . . . . 9 ⊢ ((-1𝑆𝐴) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)
109	44	ablo4 27404	. . . . . . . . 9 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((-1𝑆𝐴) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺((-1𝑆𝐴)𝐺𝐵)) = ((𝐴𝐺(-1𝑆𝐴))𝐺(𝐵𝐺𝐵)))
110	41, 42, 108, 109	mp3an 1424	. . . . . . . 8 ⊢ ((𝐴𝐺𝐵)𝐺((-1𝑆𝐴)𝐺𝐵)) = ((𝐴𝐺(-1𝑆𝐴))𝐺(𝐵𝐺𝐵))
111	10, 11, 15, 50	nvrinv 27506	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺(-1𝑆𝐴)) = (0_vec‘𝑈))
112	6, 8, 111	mp2an 708	. . . . . . . . . 10 ⊢ (𝐴𝐺(-1𝑆𝐴)) = (0_vec‘𝑈)
113	112	oveq1i 6660	. . . . . . . . 9 ⊢ ((𝐴𝐺(-1𝑆𝐴))𝐺(𝐵𝐺𝐵)) = ((0_vec‘𝑈)𝐺(𝐵𝐺𝐵))
114	10, 11	nvgcl 27475	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐵𝐺𝐵) ∈ 𝑋)
115	6, 9, 9, 114	mp3an 1424	. . . . . . . . . 10 ⊢ (𝐵𝐺𝐵) ∈ 𝑋
116	10, 11, 50	nv0lid 27491	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐵𝐺𝐵) ∈ 𝑋) → ((0_vec‘𝑈)𝐺(𝐵𝐺𝐵)) = (𝐵𝐺𝐵))
117	6, 115, 116	mp2an 708	. . . . . . . . 9 ⊢ ((0_vec‘𝑈)𝐺(𝐵𝐺𝐵)) = (𝐵𝐺𝐵)
118	113, 117	eqtri 2644	. . . . . . . 8 ⊢ ((𝐴𝐺(-1𝑆𝐴))𝐺(𝐵𝐺𝐵)) = (𝐵𝐺𝐵)
119	39, 47, 44	vc2OLD 27423	. . . . . . . . 9 ⊢ (((1^st ‘𝑈) ∈ CVec_OLD ∧ 𝐵 ∈ 𝑋) → (𝐵𝐺𝐵) = (2𝑆𝐵))
120	38, 9, 119	mp2an 708	. . . . . . . 8 ⊢ (𝐵𝐺𝐵) = (2𝑆𝐵)
121	110, 118, 120	3eqtri 2648	. . . . . . 7 ⊢ ((𝐴𝐺𝐵)𝐺((-1𝑆𝐴)𝐺𝐵)) = (2𝑆𝐵)
122	121	oveq2i 6661	. . . . . 6 ⊢ ((1 / 2)𝑆((𝐴𝐺𝐵)𝐺((-1𝑆𝐴)𝐺𝐵))) = ((1 / 2)𝑆(2𝑆𝐵))
123	7, 1, 9	3pm3.2i 1239	. . . . . . 7 ⊢ ((1 / 2) ∈ ℂ ∧ 2 ∈ ℂ ∧ 𝐵 ∈ 𝑋)
124	10, 15	nvsass 27483	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ ((1 / 2) ∈ ℂ ∧ 2 ∈ ℂ ∧ 𝐵 ∈ 𝑋)) → (((1 / 2) · 2)𝑆𝐵) = ((1 / 2)𝑆(2𝑆𝐵)))
125	6, 123, 124	mp2an 708	. . . . . 6 ⊢ (((1 / 2) · 2)𝑆𝐵) = ((1 / 2)𝑆(2𝑆𝐵))
126	68	oveq1i 6660	. . . . . 6 ⊢ (((1 / 2) · 2)𝑆𝐵) = (1𝑆𝐵)
127	122, 125, 126	3eqtr2i 2650	. . . . 5 ⊢ ((1 / 2)𝑆((𝐴𝐺𝐵)𝐺((-1𝑆𝐴)𝐺𝐵))) = (1𝑆𝐵)
128	107, 127, 92	3eqtri 2648	. . . 4 ⊢ (((1 / 2)𝑆(𝐴𝐺𝐵))𝐺(-1𝑆((1 / 2)𝑆(𝐴𝐺(-1𝑆𝐵))))) = 𝐵
129	128	oveq1i 6660	. . 3 ⊢ ((((1 / 2)𝑆(𝐴𝐺𝐵))𝐺(-1𝑆((1 / 2)𝑆(𝐴𝐺(-1𝑆𝐵)))))𝑃𝐶) = (𝐵𝑃𝐶)
130	74, 129	oveq12i 6662	. 2 ⊢ (((((1 / 2)𝑆(𝐴𝐺𝐵))𝐺((1 / 2)𝑆(𝐴𝐺(-1𝑆𝐵))))𝑃𝐶) + ((((1 / 2)𝑆(𝐴𝐺𝐵))𝐺(-1𝑆((1 / 2)𝑆(𝐴𝐺(-1𝑆𝐵)))))𝑃𝐶)) = ((𝐴𝑃𝐶) + (𝐵𝑃𝐶))
131	27, 35, 130	3eqtr2i 2650	1 ⊢ ((𝐴𝐺𝐵)𝑃𝐶) = ((𝐴𝑃𝐶) + (𝐵𝑃𝐶))

Colors of variables: wff setvar class

Syntax hints: ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ‘cfv 5888 (class class class)co 6650 1^st c1st 7166 ℂcc 9934 1c1 9937 + caddc 9939 · cmul 9941 -cneg 10267 / cdiv 10684 2c2 11070 AbelOpcablo 27398 CVec_OLDcvc 27413 NrmCVeccnv 27439 +_𝑣 cpv 27440 BaseSetcba 27441 ·_𝑠OLD cns 27442 0_veccn0v 27443 ·_𝑖OLDcdip 27555 CPreHil_OLDccphlo 27667

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 df-grpo 27347 df-gid 27348 df-ginv 27349 df-ablo 27399 df-vc 27414 df-nv 27447 df-va 27450 df-ba 27451 df-sm 27452 df-0v 27453 df-nmcv 27455 df-dip 27556 df-ph 27668

This theorem is referenced by: ipdiri 27685

W3C validator