Theorem dipcj 30914

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 2762	. . . 4 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈)
3		eqid 2762	. . . 4 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈)
4		eqid 2762	. . . 4 ⊢ (normCV‘𝑈) = (normCV‘𝑈)
5		ipcl.7	. . . 4 ⊢ 𝑃 = (·𝑖OLD‘𝑈)
6	1, 2, 3, 4, 5	ipval2 30907	. . 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 6871	. 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 30907	. . . 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 1139	. . 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 30905	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)𝐵))↑2) ∈ ℝ)
11	10	recnd 11210	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)𝐵))↑2) ∈ ℂ)
12		neg1cn 12180	. . . . . . . 8 ⊢ -1 ∈ ℂ
13	1, 2, 3, 4, 5	ipval2lem4 30906	. . . . . . . 8 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ -1 ∈ ℂ) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵)))↑2) ∈ ℂ)
14	12, 13	mpan2 701	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵)))↑2) ∈ ℂ)
15	11, 14	subcld 11542	. . . . . 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 11132	. . . . . . 7 ⊢ i ∈ ℂ
17	1, 2, 3, 4, 5	ipval2lem4 30906	. . . . . . . . 9 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ i ∈ ℂ) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2) ∈ ℂ)
18	16, 17	mpan2 701	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2) ∈ ℂ)
19		negicn 11431	. . . . . . . . 9 ⊢ -i ∈ ℂ
20	1, 2, 3, 4, 5	ipval2lem4 30906	. . . . . . . . 9 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ -i ∈ ℂ) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵)))↑2) ∈ ℂ)
21	19, 20	mpan2 701	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵)))↑2) ∈ ℂ)
22	18, 21	subcld 11542	. . . . . . 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 11157	. . . . . . 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 596	. . . . . 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 11201	. . . . 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 12303	. . . . . 6 ⊢ 4 ∈ ℂ
27		4ne0 12329	. . . . . 6 ⊢ 4 ≠ 0
28		cjdiv 15191	. . . . . 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 1474	. . . . 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 12302	. . . . . . 7 ⊢ 4 ∈ ℝ
32		cjre 15166	. . . . . . 7 ⊢ (4 ∈ ℝ → (∗‘4) = 4)
33	31, 32	ax-mp 5	. . . . . 6 ⊢ (∗‘4) = 4
34	33	oveq2i 7407	. . . . 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 30904	. . . . . . . . . 10 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ -1 ∈ ℂ) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵)))↑2) ∈ ℝ)
36	12, 35	mpan2 701	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵)))↑2) ∈ ℝ)
37	10, 36	resubcld 11615	. . . . . . . 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 30904	. . . . . . . . . 10 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ i ∈ ℂ) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(i( ·𝑠OLD ‘𝑈)𝐵)))↑2) ∈ ℝ)
39	16, 38	mpan2 701	. . . . . . . . 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 30904	. . . . . . . . . 10 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ -i ∈ ℂ) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵)))↑2) ∈ ℝ)
41	19, 40	mpan2 701	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)(-i( ·𝑠OLD ‘𝑈)𝐵)))↑2) ∈ ℝ)
42	39, 41	resubcld 11615	. . . . . . . 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 15187	. . . . . . . 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 593	. . . . . . 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 11627	. . . . . . . . 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 1470	. . . . . . . 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 593	. . . . . . 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 30821	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴( +𝑣 ‘𝑈)𝐵) = (𝐵( +𝑣 ‘𝑈)𝐴))
49	48	fveq2d 6871	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((normCV‘𝑈)‘(𝐴( +𝑣 ‘𝑈)𝐵)) = ((normCV‘𝑈)‘(𝐵( +𝑣 ‘𝑈)𝐴)))
50	49	oveq1d 7411	. . . . . . . . 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 30866	. . . . . . . . . 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 7411	. . . . . . . . 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 7414	. . . . . . . 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 11558	. . . . . . . . . 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 30867	. . . . . . . . . . . . . 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 1139	. . . . . . . . . . . . 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 2768	. . . . . . . . . . . 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 7411	. . . . . . . . . . 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 30867	. . . . . . . . . . . 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 7411	. . . . . . . . . . 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 7414	. . . . . . . . . 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 2797	. . . . . . . . 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 7412	. . . . . . . 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 7414	. . . . . . 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 2801	. . . . . 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 7411	. . . . 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 2809	. . . 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 2797	. . 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 2800	. 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 2800	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∗‘(𝐴𝑃𝐵)) = (𝐵𝑃𝐴))

Syntax hints:   → wi 4   ∧ w3a 1098   = wceq 1560   ∈ wcel 2142   ≠ wne 2957  ‘cfv 6521  (class class class)co 7396  ℂcc 11071  ℝcr 11072  0cc0 11073  1c1 11074  ici 11075   + caddc 11076   · cmul 11078   − cmin 11414  -cneg 11415   / cdiv 11844  2c2 12272  4c4 12274  ↑cexp 14074  ∗ccj 15123  NrmCVeccnv 30784   +_𝑣 cpv 30785  BaseSetcba 30786   ·_𝑠OLD cns 30787  norm_CVcnmcv 30790  ·_𝑖OLDcdip 30900

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718  ax-inf2 9596  ax-cnex 11129  ax-resscn 11130  ax-1cn 11131  ax-icn 11132  ax-addcl 11133  ax-addrcl 11134  ax-mulcl 11135  ax-mulrcl 11136  ax-mulcom 11137  ax-addass 11138  ax-mulass 11139  ax-distr 11140  ax-i2m1 11141  ax-1ne0 11142  ax-1rid 11143  ax-rnegex 11144  ax-rrecex 11145  ax-cnre 11146  ax-pre-lttri 11147  ax-pre-lttrn 11148  ax-pre-ltadd 11149  ax-pre-mulgt0 11150  ax-pre-sup 11151

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4906  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-se 5601  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-isom 6530  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7847  df-1st 7970  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-1o 8437  df-er 8678  df-en 8928  df-dom 8929  df-sdom 8930  df-fin 8931  df-sup 9388  df-oi 9458  df-card 9897  df-pnf 11218  df-mnf 11219  df-xr 11220  df-ltxr 11221  df-le 11222  df-sub 11416  df-neg 11417  df-div 11845  df-nn 12211  df-2 12280  df-3 12281  df-4 12282  df-n0 12482  df-z 12569  df-uz 12840  df-rp 12994  df-fz 13513  df-fzo 13660  df-seq 14015  df-exp 14075  df-hash 14344  df-cj 15126  df-re 15127  df-im 15128  df-sqrt 15262  df-abs 15263  df-clim 15515  df-sum 15714  df-grpo 30693  df-gid 30694  df-ginv 30695  df-ablo 30745  df-vc 30759  df-nv 30792  df-va 30795  df-ba 30796  df-sm 30797  df-0v 30798  df-nmcv 30800  df-dip 30901

This theorem is referenced by:  ipipcj  30915  diporthcom  30916  dip0l  30918  ipasslem10  31039  dipdi  31043  dipassr  31046  dipsubdi  31049  siii  31053  hlipcj  31111