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Theorem cphsqrcl3 21362
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 994 . . . . . . . . . 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 21355 . . . . . . . . . 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 18188 . . . . . . . . . 10  |-  CC  =  ( Base ` fld )
76subrgss 17206 . . . . . . . . 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 996 . . . . . . . 8  |-  ( ( ( W  e.  CPreHil  /\  _i  e.  K  /\  A  e.  K )  /\  -u A  e.  RR+ )  ->  A  e.  K )
108, 9sseldd 3498 . . . . . . 7  |-  ( ( ( W  e.  CPreHil  /\  _i  e.  K  /\  A  e.  K )  /\  -u A  e.  RR+ )  ->  A  e.  CC )
1110negnegd 9910 . . . . . 6  |-  ( ( ( W  e.  CPreHil  /\  _i  e.  K  /\  A  e.  K )  /\  -u A  e.  RR+ )  ->  -u -u A  =  A )
1211fveq2d 5861 . . . . 5  |-  ( ( ( W  e.  CPreHil  /\  _i  e.  K  /\  A  e.  K )  /\  -u A  e.  RR+ )  ->  ( sqr `  -u -u A )  =  ( sqr `  A
) )
13 rpre 11215 . . . . . . 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 11221 . . . . . . 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 13202 . . . . 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 2503 . . . 4  |-  ( ( ( W  e.  CPreHil  /\  _i  e.  K  /\  A  e.  K )  /\  -u A  e.  RR+ )  ->  ( sqr `  A )  =  ( _i  x.  ( sqr `  -u A ) ) )
19 simpl2 995 . . . . 5  |-  ( ( ( W  e.  CPreHil  /\  _i  e.  K  /\  A  e.  K )  /\  -u A  e.  RR+ )  ->  _i  e.  K )
20 cnfldneg 18208 . . . . . . . 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 17211 . . . . . . . . 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 2460 . . . . . . . . 9  |-  ( invg ` fld )  =  ( invg ` fld )
2524subginvcl 15998 . . . . . . . 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 2549 . . . . . 6  |-  ( ( ( W  e.  CPreHil  /\  _i  e.  K  /\  A  e.  K )  /\  -u A  e.  RR+ )  ->  -u A  e.  K )
282, 3cphsqrcl 21359 . . . . . 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 1225 . . . . 5  |-  ( ( ( W  e.  CPreHil  /\  _i  e.  K  /\  A  e.  K )  /\  -u A  e.  RR+ )  ->  ( sqr `  -u A )  e.  K )
30 cnfldmul 18190 . . . . . 6  |-  x.  =  ( .r ` fld )
3130subrgmcl 17217 . . . . 5  |-  ( ( K  e.  (SubRing ` fld )  /\  _i  e.  K  /\  ( sqr `  -u A
)  e.  K )  ->  ( _i  x.  ( sqr `  -u A
) )  e.  K
)
325, 19, 29, 31syl3anc 1223 . . . 4  |-  ( ( ( W  e.  CPreHil  /\  _i  e.  K  /\  A  e.  K )  /\  -u A  e.  RR+ )  ->  (
_i  x.  ( sqr `  -u A ) )  e.  K )
3318, 32eqeltrd 2548 . . 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 21361 . . . 4  |-  ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  ->  ( sqr `  A )  e.  K )
36353expia 1193 . . 3  |-  ( ( W  e.  CPreHil  /\  A  e.  K )  ->  ( -.  -u A  e.  RR+  ->  ( sqr `  A
)  e.  K ) )
37363adant2 1010 . 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 968    = wceq 1374    e. wcel 1762    C_ wss 3469   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   CCcc 9479   RRcr 9480   0cc0 9481   _ici 9483    x. cmul 9486    <_ cle 9618   -ucneg 9795   RR+crp 11209   sqrcsqr 13016   Basecbs 14479  Scalarcsca 14547   invgcminusg 15717  SubGrpcsubg 15983  SubRingcsubrg 17201  ℂfldccnfld 18184   CPreHilccph 21341
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-tpos 6945  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-map 7412  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-sup 7890  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-10 10591  df-n0 10785  df-z 10854  df-dec 10966  df-uz 11072  df-rp 11210  df-ico 11524  df-fz 11662  df-seq 12064  df-exp 12123  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-mulr 14558  df-starv 14559  df-tset 14563  df-ple 14564  df-ds 14566  df-unif 14567  df-0g 14686  df-mnd 15721  df-mhm 15770  df-grp 15851  df-minusg 15852  df-subg 15986  df-ghm 16053  df-cmn 16589  df-mgp 16925  df-ur 16937  df-rng 16981  df-cring 16982  df-oppr 17049  df-dvdsr 17067  df-unit 17068  df-invr 17098  df-dvr 17109  df-rnghom 17141  df-drng 17174  df-subrg 17203  df-staf 17270  df-srng 17271  df-lvec 17525  df-cnfld 18185  df-phl 18421  df-cph 21343
This theorem is referenced by: (None)
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