Theorem imsval 8312

Description: Value of the induced metric of a normed complex vector space.

Hypotheses

Ref	Expression
imsval.3	|- M = (-v` U)
imsval.6	|- N = (norm` U)
imsval.8	|- D = (IndMet` U)

Assertion

Ref	Expression
imsval	|- (U e. NrmCVec -> D = (N o. M))

Proof of Theorem imsval

Step	Hyp	Ref	Expression
1		fveq2 3730	. . . . . 6 |- (u = U -> (norm` u) = (norm` U))
2		imsval.6	. . . . . 6 |- N = (norm` U)
3	1, 2	syl6eqr 1528	. . . . 5 |- (u = U -> (norm` u) = N)
4	3	coeq1d 3291	. . . 4 |- (u = U -> ((norm` u) o. (-v` u)) = (N o. (-v` u)))
5		fveq2 3730	. . . . . 6 |- (u = U -> (-v` u) = (-v` U))
6		imsval.3	. . . . . 6 |- M = (-v` U)
7	5, 6	syl6eqr 1528	. . . . 5 |- (u = U -> (-v` u) = M)
8	7	coeq2d 3292	. . . 4 |- (u = U -> (N o. (-v` u)) = (N o. M))
9	4, 8	eqtrd 1510	. . 3 |- (u = U -> ((norm` u) o. (-v` u)) = (N o. M))
10		df-ims 8216	. . 3 |- IndMet = {<.u, d>. | (u e. NrmCVec /\ d = ((norm` u) o. (-v` u)))}
11		fvex 3738	. . . . 5 |- (norm` U) e. V
12	2, 11	eqeltr 1547	. . . 4 |- N e. V
13		fvex 3738	. . . . 5 |- (-v` U) e. V
14	6, 13	eqeltr 1547	. . . 4 |- M e. V
15	12, 14	coex 3531	. . 3 |- (N o. M) e. V
16	9, 10, 15	fvopab4 3786	. 2 |- (U e. NrmCVec -> (IndMet` U) = (N o. M))
17		imsval.8	. 2 |- D = (IndMet` U)
18	16, 17	syl5eq 1522	1 |- (U e. NrmCVec -> D = (N o. M))

Syntax hints:   -> wi 3   = wceq 958   e. wcel 960  Vcvv 1814   o. ccom 3180  ` cfv 3188  NrmCVeccnv 8199  -vcnsb 8204  normcnm 8205  IndMetcims 8206

This theorem is referenced by:  imsdval 8313  imsdf 8316  cnims 8330  hhims 9034  hhssims 9140

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fv 3204  df-ims 8216

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