Theorem imsdval2 22140

Description: Value of the distance function of the induced metric of a normed complex vector space. Equation 1 of [Kreyszig] p. 59. (Contributed by NM, 28-Nov-2006.) (New usage is discouraged.)

Hypotheses

Ref	Expression
imsdval2.1	|- X = ( BaseSet ` U )
imsdval2.2	|- G = ( +v ` U )
imsdval2.4	|- S = ( .s OLD ` U )
imsdval2.6	|- N = ( normCV ` U )
imsdval2.8	|- D = ( IndMet ` U )

Assertion

Ref	Expression
imsdval2	|- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( A D B ) = ( N ` ( A G ( -u 1 S B ) ) ) )

Proof of Theorem imsdval2

Step	Hyp	Ref	Expression
1		imsdval2.1	. . 3 |- X = ( BaseSet ` U )
2		eqid 2412	. . 3 |- ( -v ` U ) = ( -v ` U )
3		imsdval2.6	. . 3 |- N = ( normCV ` U )
4		imsdval2.8	. . 3 |- D = ( IndMet ` U )
5	1, 2, 3, 4	imsdval 22139	. 2 |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( A D B ) = ( N ` ( A ( -v ` U ) B ) ) )
6		imsdval2.2	. . . 4 |- G = ( +v ` U )
7		imsdval2.4	. . . 4 |- S = ( .s OLD ` U )
8	1, 6, 7, 2	nvmval 22084	. . 3 |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( A ( -v ` U ) B ) = ( A G ( -u 1 S B ) ) )
9	8	fveq2d 5699	. 2 |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( N ` ( A ( -v ` U ) B ) ) = ( N ` ( A G ( -u 1 S B ) ) ) )
10	5, 9	eqtrd 2444	1 |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( A D B ) = ( N ` ( A G ( -u 1 S B ) ) ) )

Syntax hints:   -> wi 4   /\ w3a 936   = wceq 1649   e. wcel 1721   ` cfv 5421  (class class class)co 6048   1c1 8955   -ucneg 9256   NrmCVeccnv 22024   +vcpv 22025   BaseSetcba 22026   .s OLDcns 22027   -vcnsb 22029   norm_CVcnmcv 22030   IndMetcims 22031

This theorem is referenced by:  imsmetlem  22143  nmcvcn  22152  smcnlem  22154

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-po 4471  df-so 4472  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-pnf 9086  df-mnf 9087  df-ltxr 9089  df-sub 9257  df-neg 9258  df-grpo 21740  df-gid 21741  df-ginv 21742  df-gdiv 21743  df-ablo 21831  df-vc 21986  df-nv 22032  df-va 22035  df-ba 22036  df-sm 22037  df-0v 22038  df-vs 22039  df-nmcv 22040  df-ims 22041