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Theorem cphsqrcl3 19023
Description: If the scalar field contains  _i, it is completely closed under square roots (i.e. it is quadratically closed). (Contributed by Mario Carneiro, 11-Oct-2015.)
Hypotheses
Ref Expression
cphsca.f  |-  F  =  (Scalar `  W )
cphsca.k  |-  K  =  ( Base `  F
)
Assertion
Ref Expression
cphsqrcl3  |-  ( ( W  e.  CPreHil  /\  _i  e.  K  /\  A  e.  K )  ->  ( sqr `  A )  e.  K )

Proof of Theorem cphsqrcl3
StepHypRef Expression
1 simpl1 960 . . . . . . . . . 10  |-  ( ( ( W  e.  CPreHil  /\  _i  e.  K  /\  A  e.  K )  /\  -u A  e.  RR+ )  ->  W  e.  CPreHil )
2 cphsca.f . . . . . . . . . . 11  |-  F  =  (Scalar `  W )
3 cphsca.k . . . . . . . . . . 11  |-  K  =  ( Base `  F
)
42, 3cphsubrg 19016 . . . . . . . . . 10  |-  ( W  e.  CPreHil  ->  K  e.  (SubRing ` fld ) )
51, 4syl 16 . . . . . . . . 9  |-  ( ( ( W  e.  CPreHil  /\  _i  e.  K  /\  A  e.  K )  /\  -u A  e.  RR+ )  ->  K  e.  (SubRing ` fld ) )
6 cnfldbas 16632 . . . . . . . . . 10  |-  CC  =  ( Base ` fld )
76subrgss 15798 . . . . . . . . 9  |-  ( K  e.  (SubRing ` fld )  ->  K  C_  CC )
85, 7syl 16 . . . . . . . 8  |-  ( ( ( W  e.  CPreHil  /\  _i  e.  K  /\  A  e.  K )  /\  -u A  e.  RR+ )  ->  K  C_  CC )
9 simpl3 962 . . . . . . . 8  |-  ( ( ( W  e.  CPreHil  /\  _i  e.  K  /\  A  e.  K )  /\  -u A  e.  RR+ )  ->  A  e.  K )
108, 9sseldd 3294 . . . . . . 7  |-  ( ( ( W  e.  CPreHil  /\  _i  e.  K  /\  A  e.  K )  /\  -u A  e.  RR+ )  ->  A  e.  CC )
1110negnegd 9336 . . . . . 6  |-  ( ( ( W  e.  CPreHil  /\  _i  e.  K  /\  A  e.  K )  /\  -u A  e.  RR+ )  ->  -u -u A  =  A )
1211fveq2d 5674 . . . . 5  |-  ( ( ( W  e.  CPreHil  /\  _i  e.  K  /\  A  e.  K )  /\  -u A  e.  RR+ )  ->  ( sqr `  -u -u A )  =  ( sqr `  A
) )
13 rpre 10552 . . . . . . 7  |-  ( -u A  e.  RR+  ->  -u A  e.  RR )
1413adantl 453 . . . . . 6  |-  ( ( ( W  e.  CPreHil  /\  _i  e.  K  /\  A  e.  K )  /\  -u A  e.  RR+ )  ->  -u A  e.  RR )
15 rpge0 10558 . . . . . . 7  |-  ( -u A  e.  RR+  ->  0  <_ 
-u A )
1615adantl 453 . . . . . 6  |-  ( ( ( W  e.  CPreHil  /\  _i  e.  K  /\  A  e.  K )  /\  -u A  e.  RR+ )  ->  0  <_ 
-u A )
1714, 16sqrnegd 12153 . . . . 5  |-  ( ( ( W  e.  CPreHil  /\  _i  e.  K  /\  A  e.  K )  /\  -u A  e.  RR+ )  ->  ( sqr `  -u -u A )  =  ( _i  x.  ( sqr `  -u A ) ) )
1812, 17eqtr3d 2423 . . . 4  |-  ( ( ( W  e.  CPreHil  /\  _i  e.  K  /\  A  e.  K )  /\  -u A  e.  RR+ )  ->  ( sqr `  A )  =  ( _i  x.  ( sqr `  -u A ) ) )
19 simpl2 961 . . . . 5  |-  ( ( ( W  e.  CPreHil  /\  _i  e.  K  /\  A  e.  K )  /\  -u A  e.  RR+ )  ->  _i  e.  K )
20 cnfldneg 16652 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( inv g ` fld ) `  A )  =  -u A )
2110, 20syl 16 . . . . . . 7  |-  ( ( ( W  e.  CPreHil  /\  _i  e.  K  /\  A  e.  K )  /\  -u A  e.  RR+ )  ->  (
( inv g ` fld ) `  A )  =  -u A )
22 subrgsubg 15803 . . . . . . . . 9  |-  ( K  e.  (SubRing ` fld )  ->  K  e.  (SubGrp ` fld ) )
235, 22syl 16 . . . . . . . 8  |-  ( ( ( W  e.  CPreHil  /\  _i  e.  K  /\  A  e.  K )  /\  -u A  e.  RR+ )  ->  K  e.  (SubGrp ` fld ) )
24 eqid 2389 . . . . . . . . 9  |-  ( inv g ` fld )  =  ( inv g ` fld )
2524subginvcl 14882 . . . . . . . 8  |-  ( ( K  e.  (SubGrp ` fld )  /\  A  e.  K
)  ->  ( ( inv g ` fld ) `  A )  e.  K )
2623, 9, 25syl2anc 643 . . . . . . 7  |-  ( ( ( W  e.  CPreHil  /\  _i  e.  K  /\  A  e.  K )  /\  -u A  e.  RR+ )  ->  (
( inv g ` fld ) `  A )  e.  K
)
2721, 26eqeltrrd 2464 . . . . . 6  |-  ( ( ( W  e.  CPreHil  /\  _i  e.  K  /\  A  e.  K )  /\  -u A  e.  RR+ )  ->  -u A  e.  K )
282, 3cphsqrcl 19020 . . . . . 6  |-  ( ( W  e.  CPreHil  /\  ( -u A  e.  K  /\  -u A  e.  RR  /\  0  <_  -u A ) )  ->  ( sqr `  -u A
)  e.  K )
291, 27, 14, 16, 28syl13anc 1186 . . . . 5  |-  ( ( ( W  e.  CPreHil  /\  _i  e.  K  /\  A  e.  K )  /\  -u A  e.  RR+ )  ->  ( sqr `  -u A )  e.  K )
30 cnfldmul 16634 . . . . . 6  |-  x.  =  ( .r ` fld )
3130subrgmcl 15809 . . . . 5  |-  ( ( K  e.  (SubRing ` fld )  /\  _i  e.  K  /\  ( sqr `  -u A
)  e.  K )  ->  ( _i  x.  ( sqr `  -u A
) )  e.  K
)
325, 19, 29, 31syl3anc 1184 . . . 4  |-  ( ( ( W  e.  CPreHil  /\  _i  e.  K  /\  A  e.  K )  /\  -u A  e.  RR+ )  ->  (
_i  x.  ( sqr `  -u A ) )  e.  K )
3318, 32eqeltrd 2463 . . 3  |-  ( ( ( W  e.  CPreHil  /\  _i  e.  K  /\  A  e.  K )  /\  -u A  e.  RR+ )  ->  ( sqr `  A )  e.  K )
3433ex 424 . 2  |-  ( ( W  e.  CPreHil  /\  _i  e.  K  /\  A  e.  K )  ->  ( -u A  e.  RR+  ->  ( sqr `  A )  e.  K ) )
352, 3cphsqrcl2 19022 . . . 4  |-  ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  ->  ( sqr `  A )  e.  K )
36353expia 1155 . . 3  |-  ( ( W  e.  CPreHil  /\  A  e.  K )  ->  ( -.  -u A  e.  RR+  ->  ( sqr `  A
)  e.  K ) )
37363adant2 976 . 2  |-  ( ( W  e.  CPreHil  /\  _i  e.  K  /\  A  e.  K )  ->  ( -.  -u A  e.  RR+  ->  ( sqr `  A
)  e.  K ) )
3834, 37pm2.61d 152 1  |-  ( ( W  e.  CPreHil  /\  _i  e.  K  /\  A  e.  K )  ->  ( sqr `  A )  e.  K )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    C_ wss 3265   class class class wbr 4155   ` cfv 5396  (class class class)co 6022   CCcc 8923   RRcr 8924   0cc0 8925   _ici 8927    x. cmul 8930    <_ cle 9056   -ucneg 9226   RR+crp 10546   sqrcsqr 11967   Basecbs 13398  Scalarcsca 13461   inv gcminusg 14615  SubGrpcsubg 14867  SubRingcsubrg 15793  ℂfldccnfld 16628   CPreHilccph 19002
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002  ax-pre-sup 9003  ax-addf 9004  ax-mulf 9005
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-tpos 6417  df-riota 6487  df-recs 6571  df-rdg 6606  df-1o 6662  df-oadd 6666  df-er 6843  df-map 6958  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-sup 7383  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-div 9612  df-nn 9935  df-2 9992  df-3 9993  df-4 9994  df-5 9995  df-6 9996  df-7 9997  df-8 9998  df-9 9999  df-10 10000  df-n0 10156  df-z 10217  df-dec 10317  df-uz 10423  df-rp 10547  df-ico 10856  df-fz 10978  df-seq 11253  df-exp 11312  df-cj 11833  df-re 11834  df-im 11835  df-sqr 11969  df-abs 11970  df-struct 13400  df-ndx 13401  df-slot 13402  df-base 13403  df-sets 13404  df-ress 13405  df-plusg 13471  df-mulr 13472  df-starv 13473  df-tset 13477  df-ple 13478  df-ds 13480  df-unif 13481  df-0g 13656  df-mnd 14619  df-mhm 14667  df-grp 14741  df-minusg 14742  df-subg 14870  df-ghm 14933  df-cmn 15343  df-mgp 15578  df-rng 15592  df-cring 15593  df-ur 15594  df-oppr 15657  df-dvdsr 15675  df-unit 15676  df-invr 15706  df-dvr 15717  df-rnghom 15748  df-drng 15766  df-subrg 15795  df-staf 15862  df-srng 15863  df-lvec 16104  df-cnfld 16629  df-phl 16782  df-cph 19004
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