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Theorem cphsqrcl3 20824
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 991 . . . . . . . . . 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 20817 . . . . . . . . . 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 17933 . . . . . . . . . 10  |-  CC  =  ( Base ` fld )
76subrgss 16974 . . . . . . . . 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 993 . . . . . . . 8  |-  ( ( ( W  e.  CPreHil  /\  _i  e.  K  /\  A  e.  K )  /\  -u A  e.  RR+ )  ->  A  e.  K )
108, 9sseldd 3457 . . . . . . 7  |-  ( ( ( W  e.  CPreHil  /\  _i  e.  K  /\  A  e.  K )  /\  -u A  e.  RR+ )  ->  A  e.  CC )
1110negnegd 9813 . . . . . 6  |-  ( ( ( W  e.  CPreHil  /\  _i  e.  K  /\  A  e.  K )  /\  -u A  e.  RR+ )  ->  -u -u A  =  A )
1211fveq2d 5795 . . . . 5  |-  ( ( ( W  e.  CPreHil  /\  _i  e.  K  /\  A  e.  K )  /\  -u A  e.  RR+ )  ->  ( sqr `  -u -u A )  =  ( sqr `  A
) )
13 rpre 11100 . . . . . . 7  |-  ( -u A  e.  RR+  ->  -u A  e.  RR )
1413adantl 466 . . . . . 6  |-  ( ( ( W  e.  CPreHil  /\  _i  e.  K  /\  A  e.  K )  /\  -u A  e.  RR+ )  ->  -u A  e.  RR )
15 rpge0 11106 . . . . . . 7  |-  ( -u A  e.  RR+  ->  0  <_ 
-u A )
1615adantl 466 . . . . . 6  |-  ( ( ( W  e.  CPreHil  /\  _i  e.  K  /\  A  e.  K )  /\  -u A  e.  RR+ )  ->  0  <_ 
-u A )
1714, 16sqrnegd 13012 . . . . 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 2494 . . . 4  |-  ( ( ( W  e.  CPreHil  /\  _i  e.  K  /\  A  e.  K )  /\  -u A  e.  RR+ )  ->  ( sqr `  A )  =  ( _i  x.  ( sqr `  -u A ) ) )
19 simpl2 992 . . . . 5  |-  ( ( ( W  e.  CPreHil  /\  _i  e.  K  /\  A  e.  K )  /\  -u A  e.  RR+ )  ->  _i  e.  K )
20 cnfldneg 17953 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( invg ` fld ) `  A )  =  -u A )
2110, 20syl 16 . . . . . . 7  |-  ( ( ( W  e.  CPreHil  /\  _i  e.  K  /\  A  e.  K )  /\  -u A  e.  RR+ )  ->  (
( invg ` fld ) `  A )  =  -u A )
22 subrgsubg 16979 . . . . . . . . 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 2451 . . . . . . . . 9  |-  ( invg ` fld )  =  ( invg ` fld )
2524subginvcl 15794 . . . . . . . 8  |-  ( ( K  e.  (SubGrp ` fld )  /\  A  e.  K
)  ->  ( ( invg ` fld ) `  A )  e.  K )
2623, 9, 25syl2anc 661 . . . . . . 7  |-  ( ( ( W  e.  CPreHil  /\  _i  e.  K  /\  A  e.  K )  /\  -u A  e.  RR+ )  ->  (
( invg ` fld ) `  A )  e.  K
)
2721, 26eqeltrrd 2540 . . . . . 6  |-  ( ( ( W  e.  CPreHil  /\  _i  e.  K  /\  A  e.  K )  /\  -u A  e.  RR+ )  ->  -u A  e.  K )
282, 3cphsqrcl 20821 . . . . . 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 1221 . . . . 5  |-  ( ( ( W  e.  CPreHil  /\  _i  e.  K  /\  A  e.  K )  /\  -u A  e.  RR+ )  ->  ( sqr `  -u A )  e.  K )
30 cnfldmul 17935 . . . . . 6  |-  x.  =  ( .r ` fld )
3130subrgmcl 16985 . . . . 5  |-  ( ( K  e.  (SubRing ` fld )  /\  _i  e.  K  /\  ( sqr `  -u A
)  e.  K )  ->  ( _i  x.  ( sqr `  -u A
) )  e.  K
)
325, 19, 29, 31syl3anc 1219 . . . 4  |-  ( ( ( W  e.  CPreHil  /\  _i  e.  K  /\  A  e.  K )  /\  -u A  e.  RR+ )  ->  (
_i  x.  ( sqr `  -u A ) )  e.  K )
3318, 32eqeltrd 2539 . . 3  |-  ( ( ( W  e.  CPreHil  /\  _i  e.  K  /\  A  e.  K )  /\  -u A  e.  RR+ )  ->  ( sqr `  A )  e.  K )
3433ex 434 . 2  |-  ( ( W  e.  CPreHil  /\  _i  e.  K  /\  A  e.  K )  ->  ( -u A  e.  RR+  ->  ( sqr `  A )  e.  K ) )
352, 3cphsqrcl2 20823 . . . 4  |-  ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  ->  ( sqr `  A )  e.  K )
36353expia 1190 . . 3  |-  ( ( W  e.  CPreHil  /\  A  e.  K )  ->  ( -.  -u A  e.  RR+  ->  ( sqr `  A
)  e.  K ) )
37363adant2 1007 . 2  |-  ( ( W  e.  CPreHil  /\  _i  e.  K  /\  A  e.  K )  ->  ( -.  -u A  e.  RR+  ->  ( sqr `  A
)  e.  K ) )
3834, 37pm2.61d 158 1  |-  ( ( W  e.  CPreHil  /\  _i  e.  K  /\  A  e.  K )  ->  ( sqr `  A )  e.  K )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    C_ wss 3428   class class class wbr 4392   ` cfv 5518  (class class class)co 6192   CCcc 9383   RRcr 9384   0cc0 9385   _ici 9387    x. cmul 9390    <_ cle 9522   -ucneg 9699   RR+crp 11094   sqrcsqr 12826   Basecbs 14278  Scalarcsca 14345   invgcminusg 15515  SubGrpcsubg 15779  SubRingcsubrg 16969  ℂfldccnfld 17929   CPreHilccph 20803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462  ax-pre-sup 9463  ax-addf 9464  ax-mulf 9465
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1st 6679  df-2nd 6680  df-tpos 6847  df-recs 6934  df-rdg 6968  df-1o 7022  df-oadd 7026  df-er 7203  df-map 7318  df-en 7413  df-dom 7414  df-sdom 7415  df-fin 7416  df-sup 7794  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-div 10097  df-nn 10426  df-2 10483  df-3 10484  df-4 10485  df-5 10486  df-6 10487  df-7 10488  df-8 10489  df-9 10490  df-10 10491  df-n0 10683  df-z 10750  df-dec 10859  df-uz 10965  df-rp 11095  df-ico 11409  df-fz 11541  df-seq 11910  df-exp 11969  df-cj 12692  df-re 12693  df-im 12694  df-sqr 12828  df-abs 12829  df-struct 14280  df-ndx 14281  df-slot 14282  df-base 14283  df-sets 14284  df-ress 14285  df-plusg 14355  df-mulr 14356  df-starv 14357  df-tset 14361  df-ple 14362  df-ds 14364  df-unif 14365  df-0g 14484  df-mnd 15519  df-mhm 15568  df-grp 15649  df-minusg 15650  df-subg 15782  df-ghm 15849  df-cmn 16385  df-mgp 16699  df-ur 16711  df-rng 16755  df-cring 16756  df-oppr 16823  df-dvdsr 16841  df-unit 16842  df-invr 16872  df-dvr 16883  df-rnghom 16914  df-drng 16942  df-subrg 16971  df-staf 17038  df-srng 17039  df-lvec 17292  df-cnfld 17930  df-phl 18166  df-cph 20805
This theorem is referenced by: (None)
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