Theorem dipcj 30643

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 2729	. . . 4 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈)
3		eqid 2729	. . . 4 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈)
4		eqid 2729	. . . 4 ⊢ (normCV‘𝑈) = (normCV‘𝑈)
5		ipcl.7	. . . 4 ⊢ 𝑃 = (·𝑖OLD‘𝑈)
6	1, 2, 3, 4, 5	ipval2 30636	. . 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 6862	. 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 30636	. . . 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 1126	. . 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 30634	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)𝐵))↑2) ∈ ℝ)
11	10	recnd 11202	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)𝐵))↑2) ∈ ℂ)
12		neg1cn 12171	. . . . . . . 8 ⊢ -1 ∈ ℂ
13	1, 2, 3, 4, 5	ipval2lem4 30635	. . . . . . . 8 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ -1 ∈ ℂ) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵)))↑2) ∈ ℂ)
14	12, 13	mpan2 691	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵)))↑2) ∈ ℂ)
15	11, 14	subcld 11533	. . . . . 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 11127	. . . . . . 7 ⊢ i ∈ ℂ
17	1, 2, 3, 4, 5	ipval2lem4 30635	. . . . . . . . 9 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ i ∈ ℂ) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2) ∈ ℂ)
18	16, 17	mpan2 691	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2) ∈ ℂ)
19		negicn 11422	. . . . . . . . 9 ⊢ -i ∈ ℂ
20	1, 2, 3, 4, 5	ipval2lem4 30635	. . . . . . . . 9 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ -i ∈ ℂ) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵)))↑2) ∈ ℂ)
21	19, 20	mpan2 691	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵)))↑2) ∈ ℂ)
22	18, 21	subcld 11533	. . . . . . 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 11152	. . . . . . 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 587	. . . . . 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 11193	. . . . 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 12271	. . . . . 6 ⊢ 4 ∈ ℂ
27		4ne0 12294	. . . . . 6 ⊢ 4 ≠ 0
28		cjdiv 15130	. . . . . 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 1455	. . . . 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 12270	. . . . . . 7 ⊢ 4 ∈ ℝ
32		cjre 15105	. . . . . . 7 ⊢ (4 ∈ ℝ → (∗‘4) = 4)
33	31, 32	ax-mp 5	. . . . . 6 ⊢ (∗‘4) = 4
34	33	oveq2i 7398	. . . . 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 30633	. . . . . . . . . 10 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ -1 ∈ ℂ) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵)))↑2) ∈ ℝ)
36	12, 35	mpan2 691	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵)))↑2) ∈ ℝ)
37	10, 36	resubcld 11606	. . . . . . . 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 30633	. . . . . . . . . 10 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ i ∈ ℂ) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2) ∈ ℝ)
39	16, 38	mpan2 691	. . . . . . . . 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 30633	. . . . . . . . . 10 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ -i ∈ ℂ) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵)))↑2) ∈ ℝ)
41	19, 40	mpan2 691	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵)))↑2) ∈ ℝ)
42	39, 41	resubcld 11606	. . . . . . . 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 15126	. . . . . . . 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 584	. . . . . . 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 11618	. . . . . . . . 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 1451	. . . . . . . 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 584	. . . . . . 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 30550	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴( +𝑣 ‘𝑈)𝐵) = (𝐵( +𝑣 ‘𝑈)𝐴))
49	48	fveq2d 6862	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)𝐵)) = ((normCV‘𝑈)‘(𝐵( +𝑣 ‘𝑈)𝐴)))
50	49	oveq1d 7402	. . . . . . . . 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 30595	. . . . . . . . . 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 7402	. . . . . . . . 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 7405	. . . . . . . 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 11549	. . . . . . . . . 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 30596	. . . . . . . . . . . . . 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 1126	. . . . . . . . . . . . 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 2735	. . . . . . . . . . . 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 7402	. . . . . . . . . . 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 30596	. . . . . . . . . . . 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 7402	. . . . . . . . . . 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 7405	. . . . . . . . . 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 2764	. . . . . . . . 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 7403	. . . . . . . 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 7405	. . . . . . 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 2768	. . . . . 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 7402	. . . . 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	eqtrid 2776	. . . 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 2764	. . 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 2767	. 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 2767	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∗‘(𝐴𝑃𝐵)) = (𝐵𝑃𝐴))

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ‘cfv 6511  (class class class)co 7387  ℂcc 11066  ℝcr 11067  0cc0 11068  1c1 11069  ici 11070   + caddc 11071   · cmul 11073   − cmin 11405  -cneg 11406   / cdiv 11835  2c2 12241  4c4 12243  ↑cexp 14026  ∗ccj 15062  NrmCVeccnv 30513   +_𝑣 cpv 30514  BaseSetcba 30515   ·_𝑠OLD cns 30516  norm_CVcnmcv 30519  ·_𝑖OLDcdip 30629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-inf2 9594  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145  ax-pre-sup 11146

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-isom 6520  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-er 8671  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-sup 9393  df-oi 9463  df-card 9892  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-div 11836  df-nn 12187  df-2 12249  df-3 12250  df-4 12251  df-n0 12443  df-z 12530  df-uz 12794  df-rp 12952  df-fz 13469  df-fzo 13616  df-seq 13967  df-exp 14027  df-hash 14296  df-cj 15065  df-re 15066  df-im 15067  df-sqrt 15201  df-abs 15202  df-clim 15454  df-sum 15653  df-grpo 30422  df-gid 30423  df-ginv 30424  df-ablo 30474  df-vc 30488  df-nv 30521  df-va 30524  df-ba 30525  df-sm 30526  df-0v 30527  df-nmcv 30529  df-dip 30630

This theorem is referenced by:  ipipcj  30644  diporthcom  30645  dip0l  30647  ipasslem10  30768  dipdi  30772  dipassr  30775  dipsubdi  30778  siii  30782  hlipcj  30840