Theorem dipcj 31373

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 2763	. . . 4 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈)
3		eqid 2763	. . . 4 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈)
4		eqid 2763	. . . 4 ⊢ (normCV‘𝑈) = (normCV‘𝑈)
5		ipcl.7	. . . 4 ⊢ 𝑃 = (·𝑖OLD‘𝑈)
6	1, 2, 3, 4, 5	ipval2 31366	. . 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 6920	. 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 31366	. . . 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 31364	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)𝐵))↑2) ∈ ℝ)
11	10	recnd 11376	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)𝐵))↑2) ∈ ℂ)
12		neg1cn 12345	. . . . . . . 8 ⊢ -1 ∈ ℂ
13	1, 2, 3, 4, 5	ipval2lem4 31365	. . . . . . . 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 11708	. . . . . 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 11298	. . . . . . 7 ⊢ i ∈ ℂ
17	1, 2, 3, 4, 5	ipval2lem4 31365	. . . . . . . . 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 11597	. . . . . . . . 9 ⊢ -i ∈ ℂ
20	1, 2, 3, 4, 5	ipval2lem4 31365	. . . . . . . . 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 11708	. . . . . . 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 11323	. . . . . . 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 11367	. . . . 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 12468	. . . . . 6 ⊢ 4 ∈ ℂ
27		4ne0 12494	. . . . . 6 ⊢ 4 ≠ 0
28		cjdiv 15374	. . . . . 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 1483	. . . . 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 12467	. . . . . . 7 ⊢ 4 ∈ ℝ
32		cjre 15349	. . . . . . 7 ⊢ (4 ∈ ℝ → (∗‘4) = 4)
33	31, 32	ax-mp 5	. . . . . 6 ⊢ (∗‘4) = 4
34	33	oveq2i 7462	. . . . 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 31363	. . . . . . . . . 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 11783	. . . . . . . 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 31363	. . . . . . . . . 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 31363	. . . . . . . . . 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 11783	. . . . . . . 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 15370	. . . . . . . 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 11795	. . . . . . . . 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 1479	. . . . . . . 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 31280	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴( +𝑣 ‘𝑈)𝐵) = (𝐵( +𝑣 ‘𝑈)𝐴))
49	48	fveq2d 6920	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)𝐵)) = ((normCV‘𝑈)‘(𝐵( +𝑣 ‘𝑈)𝐴)))
50	49	oveq1d 7466	. . . . . . . . 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 31325	. . . . . . . . . 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 7466	. . . . . . . . 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 7469	. . . . . . . 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 11724	. . . . . . . . . 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 31326	. . . . . . . . . . . . . 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 2769	. . . . . . . . . . . 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 7466	. . . . . . . . . . 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 31326	. . . . . . . . . . . 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 7466	. . . . . . . . . . 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 7469	. . . . . . . . . 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 2798	. . . . . . . . 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 7467	. . . . . . . 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 7469	. . . . . . 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 2802	. . . . . 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 7466	. . . . 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 2810	. . . 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 2798	. . 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 2801	. 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 2801	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∗‘(𝐴𝑃𝐵)) = (𝐵𝑃𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ‘cfv 6566  (class class class)co 7451  ℂcc 11237  ℝcr 11238  0cc0 11239  1c1 11240  ici 11241   + caddc 11242   · cmul 11244   − cmin 11580   -cneg 11581   / cdiv 12013  2c2 12437  4c4 12439  ↑cexp 14244  ∗ccj 15306  NrmCVeccnv 31243   +_𝑣 cpv 31244  BaseSetcba 31245   ·_𝑠OLD cns 31246  norm_CVcnmcv 31249  ·_𝑖OLDcdip 31359

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-inf2 9679  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316  ax-pre-sup 11317

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-isom 6575  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-er 8755  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-sup 9471  df-oi 9541  df-card 10057  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-div 12014  df-nn 12376  df-2 12445  df-3 12446  df-4 12447  df-n0 12647  df-z 12734  df-uz 13006  df-rp 13161  df-fz 13680  df-fzo 13829  df-seq 14185  df-exp 14245  df-hash 14516  df-cj 15309  df-re 15310  df-im 15311  df-sqrt 15445  df-abs 15446  df-clim 15699  df-sum 15898  df-grpo 31152  df-gid 31153  df-ginv 31154  df-ablo 31204  df-vc 31218  df-nv 31251  df-va 31254  df-ba 31255  df-sm 31256  df-0v 31257  df-nmcv 31259  df-dip 31360

This theorem is used by:  ipipcj  31374  diporthcom  31375  dip0l  31377  ipasslem10  31498  dipdi  31502  dipassr  31505  dipsubdi  31508  siii  31512  hlipcj  31570