Theorem dipcj 27569

Description: The complex conjugate of an inner product reverses its arguments. Equation I1 of [Ponnusamy] p. 362. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.)

Hypotheses

Ref	Expression
ipcl.1	⊢ 𝑋 = (BaseSet‘𝑈)
ipcl.7	⊢ 𝑃 = (·𝑖OLD‘𝑈)

Assertion

Ref	Expression
dipcj	⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∗‘(𝐴𝑃𝐵)) = (𝐵𝑃𝐴))

Proof of Theorem dipcj

Step	Hyp	Ref	Expression
1		ipcl.1	. . . 4 ⊢ 𝑋 = (BaseSet‘𝑈)
2		eqid 2622	. . . 4 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈)
3		eqid 2622	. . . 4 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈)
4		eqid 2622	. . . 4 ⊢ (normCV‘𝑈) = (normCV‘𝑈)
5		ipcl.7	. . . 4 ⊢ 𝑃 = (·𝑖OLD‘𝑈)
6	1, 2, 3, 4, 5	ipval2 27562	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑃𝐵) = ((((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)𝐵))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵)))↑2)) + (i · ((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵)))↑2)))) / 4))
7	6	fveq2d 6195	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∗‘(𝐴𝑃𝐵)) = (∗‘((((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)𝐵))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵)))↑2)) + (i · ((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵)))↑2)))) / 4)))
8	1, 2, 3, 4, 5	ipval2 27562	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐵𝑃𝐴) = ((((((normCV‘𝑈)‘(𝐵( +𝑣 ‘𝑈)𝐴))↑2) − (((normCV‘𝑈)‘(𝐵( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐴)))↑2)) + (i · ((((normCV‘𝑈)‘(𝐵( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐴)))↑2) − (((normCV‘𝑈)‘(𝐵( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐴)))↑2)))) / 4))
9	8	3com23 1271	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐵𝑃𝐴) = ((((((normCV‘𝑈)‘(𝐵( +𝑣 ‘𝑈)𝐴))↑2) − (((normCV‘𝑈)‘(𝐵( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐴)))↑2)) + (i · ((((normCV‘𝑈)‘(𝐵( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐴)))↑2) − (((normCV‘𝑈)‘(𝐵( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐴)))↑2)))) / 4))
10	1, 2, 3, 4, 5	ipval2lem3 27560	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)𝐵))↑2) ∈ ℝ)
11	10	recnd 10068	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)𝐵))↑2) ∈ ℂ)
12		neg1cn 11124	. . . . . . . 8 ⊢ -1 ∈ ℂ
13	1, 2, 3, 4, 5	ipval2lem4 27561	. . . . . . . 8 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ -1 ∈ ℂ) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵)))↑2) ∈ ℂ)
14	12, 13	mpan2 707	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵)))↑2) ∈ ℂ)
15	11, 14	subcld 10392	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)𝐵))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵)))↑2)) ∈ ℂ)
16		ax-icn 9995	. . . . . . 7 ⊢ i ∈ ℂ
17	1, 2, 3, 4, 5	ipval2lem4 27561	. . . . . . . . 9 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ i ∈ ℂ) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2) ∈ ℂ)
18	16, 17	mpan2 707	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2) ∈ ℂ)
19		negicn 10282	. . . . . . . . 9 ⊢ -i ∈ ℂ
20	1, 2, 3, 4, 5	ipval2lem4 27561	. . . . . . . . 9 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ -i ∈ ℂ) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵)))↑2) ∈ ℂ)
21	19, 20	mpan2 707	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵)))↑2) ∈ ℂ)
22	18, 21	subcld 10392	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵)))↑2)) ∈ ℂ)
23		mulcl 10020	. . . . . . 7 ⊢ ((i ∈ ℂ ∧ ((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵)))↑2)) ∈ ℂ) → (i · ((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵)))↑2))) ∈ ℂ)
24	16, 22, 23	sylancr 695	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (i · ((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵)))↑2))) ∈ ℂ)
25	15, 24	addcld 10059	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)𝐵))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵)))↑2)) + (i · ((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵)))↑2)))) ∈ ℂ)
26		4cn 11098	. . . . . 6 ⊢ 4 ∈ ℂ
27		4ne0 11117	. . . . . 6 ⊢ 4 ≠ 0
28		cjdiv 13904	. . . . . 6 ⊢ (((((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)𝐵))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵)))↑2)) + (i · ((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵)))↑2)))) ∈ ℂ ∧ 4 ∈ ℂ ∧ 4 ≠ 0) → (∗‘((((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)𝐵))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵)))↑2)) + (i · ((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵)))↑2)))) / 4)) = ((∗‘(((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)𝐵))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵)))↑2)) + (i · ((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵)))↑2))))) / (∗‘4)))
29	26, 27, 28	mp3an23 1416	. . . . 5 ⊢ ((((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)𝐵))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵)))↑2)) + (i · ((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵)))↑2)))) ∈ ℂ → (∗‘((((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)𝐵))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵)))↑2)) + (i · ((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵)))↑2)))) / 4)) = ((∗‘(((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)𝐵))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵)))↑2)) + (i · ((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵)))↑2))))) / (∗‘4)))
30	25, 29	syl 17	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∗‘((((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)𝐵))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵)))↑2)) + (i · ((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵)))↑2)))) / 4)) = ((∗‘(((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)𝐵))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵)))↑2)) + (i · ((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵)))↑2))))) / (∗‘4)))
31		4re 11097	. . . . . . 7 ⊢ 4 ∈ ℝ
32		cjre 13879	. . . . . . 7 ⊢ (4 ∈ ℝ → (∗‘4) = 4)
33	31, 32	ax-mp 5	. . . . . 6 ⊢ (∗‘4) = 4
34	33	oveq2i 6661	. . . . 5 ⊢ ((∗‘(((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)𝐵))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵)))↑2)) + (i · ((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵)))↑2))))) / (∗‘4)) = ((∗‘(((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)𝐵))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵)))↑2)) + (i · ((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵)))↑2))))) / 4)
35	1, 2, 3, 4, 5	ipval2lem2 27559	. . . . . . . . . 10 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ -1 ∈ ℂ) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵)))↑2) ∈ ℝ)
36	12, 35	mpan2 707	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵)))↑2) ∈ ℝ)
37	10, 36	resubcld 10458	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)𝐵))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵)))↑2)) ∈ ℝ)
38	1, 2, 3, 4, 5	ipval2lem2 27559	. . . . . . . . . 10 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ i ∈ ℂ) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2) ∈ ℝ)
39	16, 38	mpan2 707	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2) ∈ ℝ)
40	1, 2, 3, 4, 5	ipval2lem2 27559	. . . . . . . . . 10 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ -i ∈ ℂ) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵)))↑2) ∈ ℝ)
41	19, 40	mpan2 707	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵)))↑2) ∈ ℝ)
42	39, 41	resubcld 10458	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵)))↑2)) ∈ ℝ)
43		cjreim 13900	. . . . . . . 8 ⊢ ((((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)𝐵))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵)))↑2)) ∈ ℝ ∧ ((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵)))↑2)) ∈ ℝ) → (∗‘(((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)𝐵))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵)))↑2)) + (i · ((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵)))↑2))))) = (((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)𝐵))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵)))↑2)) − (i · ((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵)))↑2)))))
44	37, 42, 43	syl2anc 693	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∗‘(((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)𝐵))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵)))↑2)) + (i · ((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵)))↑2))))) = (((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)𝐵))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵)))↑2)) − (i · ((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵)))↑2)))))
45		submul2 10470	. . . . . . . . 9 ⊢ ((((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)𝐵))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵)))↑2)) ∈ ℂ ∧ i ∈ ℂ ∧ ((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵)))↑2)) ∈ ℂ) → (((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)𝐵))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵)))↑2)) − (i · ((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵)))↑2)))) = (((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)𝐵))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵)))↑2)) + (i · -((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵)))↑2)))))
46	16, 45	mp3an2 1412	. . . . . . . 8 ⊢ ((((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)𝐵))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵)))↑2)) ∈ ℂ ∧ ((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵)))↑2)) ∈ ℂ) → (((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)𝐵))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵)))↑2)) − (i · ((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵)))↑2)))) = (((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)𝐵))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵)))↑2)) + (i · -((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵)))↑2)))))
47	15, 22, 46	syl2anc 693	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)𝐵))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵)))↑2)) − (i · ((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵)))↑2)))) = (((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)𝐵))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵)))↑2)) + (i · -((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵)))↑2)))))
48	1, 2	nvcom 27476	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴( +𝑣 ‘𝑈)𝐵) = (𝐵( +𝑣 ‘𝑈)𝐴))
49	48	fveq2d 6195	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)𝐵)) = ((normCV‘𝑈)‘(𝐵( +𝑣 ‘𝑈)𝐴)))
50	49	oveq1d 6665	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)𝐵))↑2) = (((normCV‘𝑈)‘(𝐵( +𝑣 ‘𝑈)𝐴))↑2))
51	1, 2, 3, 4	nvdif 27521	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵))) = ((normCV‘𝑈)‘(𝐵( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐴))))
52	51	oveq1d 6665	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵)))↑2) = (((normCV‘𝑈)‘(𝐵( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐴)))↑2))
53	50, 52	oveq12d 6668	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)𝐵))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵)))↑2)) = ((((normCV‘𝑈)‘(𝐵( +𝑣 ‘𝑈)𝐴))↑2) − (((normCV‘𝑈)‘(𝐵( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐴)))↑2)))
54	18, 21	negsubdi2d 10408	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → -((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵)))↑2)) = ((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵)))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2)))
55	1, 2, 3, 4	nvpi 27522	. . . . . . . . . . . . . 14 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((normCV‘𝑈)‘(𝐵( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐴))) = ((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵))))
56	55	3com23 1271	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((normCV‘𝑈)‘(𝐵( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐴))) = ((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵))))
57	56	eqcomd 2628	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵))) = ((normCV‘𝑈)‘(𝐵( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐴))))
58	57	oveq1d 6665	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵)))↑2) = (((normCV‘𝑈)‘(𝐵( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐴)))↑2))
59	1, 2, 3, 4	nvpi 27522	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵))) = ((normCV‘𝑈)‘(𝐵( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐴))))
60	59	oveq1d 6665	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2) = (((normCV‘𝑈)‘(𝐵( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐴)))↑2))
61	58, 60	oveq12d 6668	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵)))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2)) = ((((normCV‘𝑈)‘(𝐵( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐴)))↑2) − (((normCV‘𝑈)‘(𝐵( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐴)))↑2)))
62	54, 61	eqtrd 2656	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → -((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵)))↑2)) = ((((normCV‘𝑈)‘(𝐵( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐴)))↑2) − (((normCV‘𝑈)‘(𝐵( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐴)))↑2)))
63	62	oveq2d 6666	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (i · -((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵)))↑2))) = (i · ((((normCV‘𝑈)‘(𝐵( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐴)))↑2) − (((normCV‘𝑈)‘(𝐵( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐴)))↑2))))
64	53, 63	oveq12d 6668	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)𝐵))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵)))↑2)) + (i · -((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵)))↑2)))) = (((((normCV‘𝑈)‘(𝐵( +𝑣 ‘𝑈)𝐴))↑2) − (((normCV‘𝑈)‘(𝐵( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐴)))↑2)) + (i · ((((normCV‘𝑈)‘(𝐵( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐴)))↑2) − (((normCV‘𝑈)‘(𝐵( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐴)))↑2)))))
65	44, 47, 64	3eqtrd 2660	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∗‘(((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)𝐵))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵)))↑2)) + (i · ((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵)))↑2))))) = (((((normCV‘𝑈)‘(𝐵( +𝑣 ‘𝑈)𝐴))↑2) − (((normCV‘𝑈)‘(𝐵( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐴)))↑2)) + (i · ((((normCV‘𝑈)‘(𝐵( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐴)))↑2) − (((normCV‘𝑈)‘(𝐵( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐴)))↑2)))))
66	65	oveq1d 6665	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((∗‘(((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)𝐵))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵)))↑2)) + (i · ((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵)))↑2))))) / 4) = ((((((normCV‘𝑈)‘(𝐵( +𝑣 ‘𝑈)𝐴))↑2) − (((normCV‘𝑈)‘(𝐵( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐴)))↑2)) + (i · ((((normCV‘𝑈)‘(𝐵( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐴)))↑2) − (((normCV‘𝑈)‘(𝐵( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐴)))↑2)))) / 4))
67	34, 66	syl5eq 2668	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((∗‘(((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)𝐵))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵)))↑2)) + (i · ((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵)))↑2))))) / (∗‘4)) = ((((((normCV‘𝑈)‘(𝐵( +𝑣 ‘𝑈)𝐴))↑2) − (((normCV‘𝑈)‘(𝐵( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐴)))↑2)) + (i · ((((normCV‘𝑈)‘(𝐵( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐴)))↑2) − (((normCV‘𝑈)‘(𝐵( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐴)))↑2)))) / 4))
68	30, 67	eqtrd 2656	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∗‘((((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)𝐵))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵)))↑2)) + (i · ((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵)))↑2)))) / 4)) = ((((((normCV‘𝑈)‘(𝐵( +𝑣 ‘𝑈)𝐴))↑2) − (((normCV‘𝑈)‘(𝐵( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐴)))↑2)) + (i · ((((normCV‘𝑈)‘(𝐵( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐴)))↑2) − (((normCV‘𝑈)‘(𝐵( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐴)))↑2)))) / 4))
69	9, 68	eqtr4d 2659	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐵𝑃𝐴) = (∗‘((((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)𝐵))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵)))↑2)) + (i · ((((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2) − (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵)))↑2)))) / 4)))
70	7, 69	eqtr4d 2659	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∗‘(𝐴𝑃𝐵)) = (𝐵𝑃𝐴))

Syntax hints:   → wi 4   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ‘cfv 5888  (class class class)co 6650  ℂcc 9934  ℝcr 9935  0cc0 9936  1c1 9937  ici 9938   + caddc 9939   · cmul 9941   − cmin 10266  -cneg 10267   / cdiv 10684  2c2 11070  4c4 11072  ↑cexp 12860  ∗ccj 13836  NrmCVeccnv 27439   +_𝑣 cpv 27440  BaseSetcba 27441   ·_𝑠OLD cns 27442  norm_CVcnmcv 27445  ·_𝑖OLDcdip 27555

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

This theorem is referenced by:  ipipcj  27570  diporthcom  27571  dip0l  27573  ipasslem10  27694  dipdi  27698  dipassr  27701  dipsubdi  27704  siii  27708  hlipcj  27767