Theorem dipcj 31325

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 2760	. . . 4 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈)
3		eqid 2760	. . . 4 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈)
4		eqid 2760	. . . 4 ⊢ (normCV‘𝑈) = (normCV‘𝑈)
5		ipcl.7	. . . 4 ⊢ 𝑃 = (·𝑖OLD‘𝑈)
6	1, 2, 3, 4, 5	ipval2 31318	. . 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 6894	. 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 31318	. . . 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 1144	. . 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 31316	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)𝐵))↑2) ∈ ℝ)
11	10	recnd 11344	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)𝐵))↑2) ∈ ℂ)
12		neg1cn 12312	. . . . . . . 8 ⊢ -1 ∈ ℂ
13	1, 2, 3, 4, 5	ipval2lem4 31317	. . . . . . . 8 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ -1 ∈ ℂ) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)( -1( ·𝑠OLD ‘𝑈)𝐵)))↑2) ∈ ℂ)
14	12, 13	mpan2 704	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)( -1( ·𝑠OLD ‘𝑈)𝐵)))↑2) ∈ ℂ)
15	11, 14	subcld 11676	. . . . . 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 11266	. . . . . . 7 ⊢ i ∈ ℂ
17	1, 2, 3, 4, 5	ipval2lem4 31317	. . . . . . . . 9 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ i ∈ ℂ) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2) ∈ ℂ)
18	16, 17	mpan2 704	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2) ∈ ℂ)
19		negicn 11565	. . . . . . . . 9 ⊢ -i ∈ ℂ
20	1, 2, 3, 4, 5	ipval2lem4 31317	. . . . . . . . 9 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ -i ∈ ℂ) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)( -i( ·𝑠OLD ‘𝑈)𝐵)))↑2) ∈ ℂ)
21	19, 20	mpan2 704	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)( -i( ·𝑠OLD ‘𝑈)𝐵)))↑2) ∈ ℂ)
22	18, 21	subcld 11676	. . . . . . 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 11291	. . . . . . 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 599	. . . . . 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 11335	. . . . 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 12435	. . . . . 6 ⊢ 4 ∈ ℂ
27		4ne0 12461	. . . . . 6 ⊢ 4 ≠ 0
28		cjdiv 15340	. . . . . 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 1482	. . . . 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 18	. . . 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 12434	. . . . . . 7 ⊢ 4 ∈ ℝ
32		cjre 15315	. . . . . . 7 ⊢ (4 ∈ ℝ → (∗‘4) = 4)
33	31, 32	ax-mp 5	. . . . . 6 ⊢ (∗‘4) = 4
34	33	oveq2i 7435	. . . . 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 31315	. . . . . . . . . 10 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ -1 ∈ ℂ) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)( -1( ·𝑠OLD ‘𝑈)𝐵)))↑2) ∈ ℝ)
36	12, 35	mpan2 704	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)( -1( ·𝑠OLD ‘𝑈)𝐵)))↑2) ∈ ℝ)
37	10, 36	resubcld 11751	. . . . . . . 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 31315	. . . . . . . . . 10 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ i ∈ ℂ) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2) ∈ ℝ)
39	16, 38	mpan2 704	. . . . . . . . 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 31315	. . . . . . . . . 10 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ -i ∈ ℂ) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)( -i( ·𝑠OLD ‘𝑈)𝐵)))↑2) ∈ ℝ)
41	19, 40	mpan2 704	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)( -i( ·𝑠OLD ‘𝑈)𝐵)))↑2) ∈ ℝ)
42	39, 41	resubcld 11751	. . . . . . . 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 15336	. . . . . . . 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 596	. . . . . . 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 11763	. . . . . . . . 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 1478	. . . . . . . 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 596	. . . . . . 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 31232	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴( +𝑣 ‘𝑈)𝐵) = (𝐵( +𝑣 ‘𝑈)𝐴))
49	48	fveq2d 6894	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)𝐵)) = ((normCV‘𝑈)‘(𝐵( +𝑣 ‘𝑈)𝐴)))
50	49	oveq1d 7439	. . . . . . . . 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 31277	. . . . . . . . . 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 7439	. . . . . . . . 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 7442	. . . . . . . 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 11692	. . . . . . . . . 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 31278	. . . . . . . . . . . . . 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 1144	. . . . . . . . . . . . 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 2766	. . . . . . . . . . . 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 7439	. . . . . . . . . . 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 31278	. . . . . . . . . . . 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 7439	. . . . . . . . . . 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 7442	. . . . . . . . . 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 2795	. . . . . . . . 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 7440	. . . . . . . 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 7442	. . . . . . 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 2799	. . . . . 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 7439	. . . . 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 2807	. . . 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 2795	. . 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 2798	. 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 2798	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∗‘(𝐴𝑃𝐵)) = (𝐵𝑃𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ‘cfv 6543  (class class class)co 7424  ℂcc 11205  ℝcr 11206  0cc0 11207  1c1 11208  ici 11209   + caddc 11210   · cmul 11212   − cmin 11548   -cneg 11549   / cdiv 11980  2c2 12404  4c4 12406  ↑cexp 14211  ∗ccj 15272  NrmCVeccnv 31195   +_𝑣 cpv 31196  BaseSetcba 31197   ·_𝑠OLD cns 31198  norm_CVcnmcv 31201  ·_𝑖OLDcdip 31311

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755  ax-inf2 9649  ax-cnex 11263  ax-resscn 11264  ax-1cn 11265  ax-icn 11266  ax-addcl 11267  ax-addrcl 11268  ax-mulcl 11269  ax-mulrcl 11270  ax-mulcom 11271  ax-addass 11272  ax-mulass 11273  ax-distr 11274  ax-i2m1 11275  ax-1ne0 11276  ax-1rid 11277  ax-rnegex 11278  ax-rrecex 11279  ax-cnre 11280  ax-pre-lttri 11281  ax-pre-lttrn 11282  ax-pre-ltadd 11283  ax-pre-mulgt0 11284  ax-pre-sup 11285

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7882  df-1st 8005  df-2nd 8006  df-frecs 8307  df-wrecs 8338  df-recs 8387  df-rdg 8426  df-1o 8484  df-er 8725  df-en 8982  df-dom 8983  df-sdom 8984  df-fin 8985  df-sup 9441  df-oi 9511  df-card 10027  df-pnf 11352  df-mnf 11353  df-xr 11354  df-ltxr 11355  df-le 11356  df-sub 11550  df-neg 11551  df-div 11981  df-nn 12343  df-2 12412  df-3 12413  df-4 12414  df-n0 12614  df-z 12701  df-uz 12973  df-rp 13128  df-fz 13647  df-fzo 13796  df-seq 14152  df-exp 14212  df-hash 14482  df-cj 15275  df-re 15276  df-im 15277  df-sqrt 15411  df-abs 15412  df-clim 15664  df-sum 15863  df-grpo 31104  df-gid 31105  df-ginv 31106  df-ablo 31156  df-vc 31170  df-nv 31203  df-va 31206  df-ba 31207  df-sm 31208  df-0v 31209  df-nmcv 31211  df-dip 31312

This theorem is used by:  ipipcj  31326  diporthcom  31327  dip0l  31329  ipasslem10  31450  dipdi  31454  dipassr  31457  dipsubdi  31460  siii  31464  hlipcj  31522