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Theorem ipval2lem5 22211
Description: Lemma for ipval3 22210. (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
dipfval.1  |-  X  =  ( BaseSet `  U )
dipfval.2  |-  G  =  ( +v `  U
)
dipfval.4  |-  S  =  ( .s OLD `  U
)
dipfval.6  |-  N  =  ( normCV `  U )
dipfval.7  |-  P  =  ( .i OLD `  U
)
ipval3.3  |-  M  =  ( -v `  U
)
Assertion
Ref Expression
ipval2lem5  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  /\  C  e.  CC )  ->  ( ( N `
 ( A M ( C S B ) ) ) ^
2 )  e.  RR )

Proof of Theorem ipval2lem5
StepHypRef Expression
1 simpl1 961 . . 3  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  /\  C  e.  CC )  ->  U  e.  NrmCVec )
2 simpl2 962 . . . 4  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  /\  C  e.  CC )  ->  A  e.  X
)
3 dipfval.1 . . . . . . . 8  |-  X  =  ( BaseSet `  U )
4 dipfval.4 . . . . . . . 8  |-  S  =  ( .s OLD `  U
)
53, 4nvscl 22112 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  C  e.  CC  /\  B  e.  X )  ->  ( C S B )  e.  X )
653com23 1160 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  B  e.  X  /\  C  e.  CC )  ->  ( C S B )  e.  X )
763expa 1154 . . . . 5  |-  ( ( ( U  e.  NrmCVec  /\  B  e.  X )  /\  C  e.  CC )  ->  ( C S B )  e.  X
)
873adantl2 1115 . . . 4  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  /\  C  e.  CC )  ->  ( C S B )  e.  X
)
9 ipval3.3 . . . . 5  |-  M  =  ( -v `  U
)
103, 9nvmcl 22133 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  ( C S B )  e.  X )  ->  ( A M ( C S B ) )  e.  X )
111, 2, 8, 10syl3anc 1185 . . 3  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  /\  C  e.  CC )  ->  ( A M ( C S B ) )  e.  X
)
12 dipfval.6 . . . 4  |-  N  =  ( normCV `  U )
133, 12nvcl 22153 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( A M ( C S B ) )  e.  X )  ->  ( N `  ( A M ( C S B ) ) )  e.  RR )
141, 11, 13syl2anc 644 . 2  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  /\  C  e.  CC )  ->  ( N `  ( A M ( C S B ) ) )  e.  RR )
1514resqcld 11554 1  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  /\  C  e.  CC )  ->  ( ( N `
 ( A M ( C S B ) ) ) ^
2 )  e.  RR )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   ` cfv 5457  (class class class)co 6084   CCcc 8993   RRcr 8994   2c2 10054   ^cexp 11387   NrmCVeccnv 22068   +vcpv 22069   BaseSetcba 22070   .s
OLDcns 22071   -vcnsb 22073   normCVcnmcv 22074   .i OLDcdip 22201
This theorem is referenced by:  ipval2lem6  22212  4ipval3  22213
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-2 10063  df-n0 10227  df-z 10288  df-uz 10494  df-seq 11329  df-exp 11388  df-grpo 21784  df-gid 21785  df-ginv 21786  df-gdiv 21787  df-ablo 21875  df-vc 22030  df-nv 22076  df-va 22079  df-ba 22080  df-sm 22081  df-0v 22082  df-vs 22083  df-nmcv 22084
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