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Theorem cphsqrcl2 18638
Description: The scalar field of a complex pre-Hilbert space is closed under square roots of all numbers except possibly the negative reals. (Contributed by Mario Carneiro, 8-Oct-2015.)
Hypotheses
Ref Expression
cphsca.f  |-  F  =  (Scalar `  W )
cphsca.k  |-  K  =  ( Base `  F
)
Assertion
Ref Expression
cphsqrcl2  |-  ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  ->  ( sqr `  A )  e.  K )

Proof of Theorem cphsqrcl2
StepHypRef Expression
1 sqr0 11743 . . . . 5  |-  ( sqr `  0 )  =  0
2 fveq2 5541 . . . . 5  |-  ( A  =  0  ->  ( sqr `  A )  =  ( sqr `  0
) )
3 id 19 . . . . 5  |-  ( A  =  0  ->  A  =  0 )
41, 2, 33eqtr4a 2354 . . . 4  |-  ( A  =  0  ->  ( sqr `  A )  =  A )
54adantl 452 . . 3  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =  0 )  ->  ( sqr `  A )  =  A )
6 simpl2 959 . . 3  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =  0 )  ->  A  e.  K )
75, 6eqeltrd 2370 . 2  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =  0 )  ->  ( sqr `  A )  e.  K
)
8 simpl1 958 . . . . . . 7  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  W  e.  CPreHil )
9 cphsca.f . . . . . . . 8  |-  F  =  (Scalar `  W )
10 cphsca.k . . . . . . . 8  |-  K  =  ( Base `  F
)
119, 10cphsubrg 18632 . . . . . . 7  |-  ( W  e.  CPreHil  ->  K  e.  (SubRing ` fld ) )
128, 11syl 15 . . . . . 6  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  K  e.  (SubRing ` fld ) )
13 cnfldbas 16399 . . . . . . 7  |-  CC  =  ( Base ` fld )
1413subrgss 15562 . . . . . 6  |-  ( K  e.  (SubRing ` fld )  ->  K  C_  CC )
1512, 14syl 15 . . . . 5  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  K  C_  CC )
16 simpl2 959 . . . . . . . 8  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  A  e.  K )
179, 10cphabscl 18637 . . . . . . . 8  |-  ( ( W  e.  CPreHil  /\  A  e.  K )  ->  ( abs `  A )  e.  K )
188, 16, 17syl2anc 642 . . . . . . 7  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( abs `  A )  e.  K
)
1915, 16sseldd 3194 . . . . . . . 8  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  A  e.  CC )
2019abscld 11934 . . . . . . 7  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( abs `  A )  e.  RR )
2119absge0d 11942 . . . . . . 7  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  0  <_  ( abs `  A ) )
229, 10cphsqrcl 18636 . . . . . . 7  |-  ( ( W  e.  CPreHil  /\  (
( abs `  A
)  e.  K  /\  ( abs `  A )  e.  RR  /\  0  <_  ( abs `  A
) ) )  -> 
( sqr `  ( abs `  A ) )  e.  K )
238, 18, 20, 21, 22syl13anc 1184 . . . . . 6  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( sqr `  ( abs `  A
) )  e.  K
)
24 cnfldadd 16400 . . . . . . . . 9  |-  +  =  ( +g  ` fld )
2524subrgacl 15572 . . . . . . . 8  |-  ( ( K  e.  (SubRing ` fld )  /\  ( abs `  A )  e.  K  /\  A  e.  K )  ->  (
( abs `  A
)  +  A )  e.  K )
2612, 18, 16, 25syl3anc 1182 . . . . . . 7  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( ( abs `  A )  +  A )  e.  K
)
279, 10cphabscl 18637 . . . . . . . 8  |-  ( ( W  e.  CPreHil  /\  (
( abs `  A
)  +  A )  e.  K )  -> 
( abs `  (
( abs `  A
)  +  A ) )  e.  K )
288, 26, 27syl2anc 642 . . . . . . 7  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( abs `  ( ( abs `  A
)  +  A ) )  e.  K )
2915, 26sseldd 3194 . . . . . . . 8  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( ( abs `  A )  +  A )  e.  CC )
30 simpl3 960 . . . . . . . . 9  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  -.  -u A  e.  RR+ )
3120recnd 8877 . . . . . . . . . . . . . 14  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( abs `  A )  e.  CC )
3231, 19subnegd 9180 . . . . . . . . . . . . 13  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( ( abs `  A )  -  -u A )  =  ( ( abs `  A
)  +  A ) )
3332eqeq1d 2304 . . . . . . . . . . . 12  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( (
( abs `  A
)  -  -u A
)  =  0  <->  (
( abs `  A
)  +  A )  =  0 ) )
3419negcld 9160 . . . . . . . . . . . . 13  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  -u A  e.  CC )
35 subeq0 9089 . . . . . . . . . . . . 13  |-  ( ( ( abs `  A
)  e.  CC  /\  -u A  e.  CC )  ->  ( ( ( abs `  A )  -  -u A )  =  0  <->  ( abs `  A
)  =  -u A
) )
3631, 34, 35syl2anc 642 . . . . . . . . . . . 12  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( (
( abs `  A
)  -  -u A
)  =  0  <->  ( abs `  A )  = 
-u A ) )
3733, 36bitr3d 246 . . . . . . . . . . 11  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( (
( abs `  A
)  +  A )  =  0  <->  ( abs `  A )  =  -u A ) )
38 absrpcl 11789 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( abs `  A
)  e.  RR+ )
3919, 38sylancom 648 . . . . . . . . . . . 12  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( abs `  A )  e.  RR+ )
40 eleq1 2356 . . . . . . . . . . . 12  |-  ( ( abs `  A )  =  -u A  ->  (
( abs `  A
)  e.  RR+  <->  -u A  e.  RR+ ) )
4139, 40syl5ibcom 211 . . . . . . . . . . 11  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( ( abs `  A )  = 
-u A  ->  -u A  e.  RR+ ) )
4237, 41sylbid 206 . . . . . . . . . 10  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( (
( abs `  A
)  +  A )  =  0  ->  -u A  e.  RR+ ) )
4342necon3bd 2496 . . . . . . . . 9  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( -.  -u A  e.  RR+  ->  ( ( abs `  A
)  +  A )  =/=  0 ) )
4430, 43mpd 14 . . . . . . . 8  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( ( abs `  A )  +  A )  =/=  0
)
4529, 44absne0d 11945 . . . . . . 7  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( abs `  ( ( abs `  A
)  +  A ) )  =/=  0 )
469, 10cphdivcl 18634 . . . . . . 7  |-  ( ( W  e.  CPreHil  /\  (
( ( abs `  A
)  +  A )  e.  K  /\  ( abs `  ( ( abs `  A )  +  A
) )  e.  K  /\  ( abs `  (
( abs `  A
)  +  A ) )  =/=  0 ) )  ->  ( (
( abs `  A
)  +  A )  /  ( abs `  (
( abs `  A
)  +  A ) ) )  e.  K
)
478, 26, 28, 45, 46syl13anc 1184 . . . . . 6  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( (
( abs `  A
)  +  A )  /  ( abs `  (
( abs `  A
)  +  A ) ) )  e.  K
)
48 cnfldmul 16401 . . . . . . 7  |-  x.  =  ( .r ` fld )
4948subrgmcl 15573 . . . . . 6  |-  ( ( K  e.  (SubRing ` fld )  /\  ( sqr `  ( abs `  A
) )  e.  K  /\  ( ( ( abs `  A )  +  A
)  /  ( abs `  ( ( abs `  A
)  +  A ) ) )  e.  K
)  ->  ( ( sqr `  ( abs `  A
) )  x.  (
( ( abs `  A
)  +  A )  /  ( abs `  (
( abs `  A
)  +  A ) ) ) )  e.  K )
5012, 23, 47, 49syl3anc 1182 . . . . 5  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( ( sqr `  ( abs `  A
) )  x.  (
( ( abs `  A
)  +  A )  /  ( abs `  (
( abs `  A
)  +  A ) ) ) )  e.  K )
5115, 50sseldd 3194 . . . 4  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( ( sqr `  ( abs `  A
) )  x.  (
( ( abs `  A
)  +  A )  /  ( abs `  (
( abs `  A
)  +  A ) ) ) )  e.  CC )
52 eqid 2296 . . . . . . 7  |-  ( ( sqr `  ( abs `  A ) )  x.  ( ( ( abs `  A )  +  A
)  /  ( abs `  ( ( abs `  A
)  +  A ) ) ) )  =  ( ( sqr `  ( abs `  A ) )  x.  ( ( ( abs `  A )  +  A )  / 
( abs `  (
( abs `  A
)  +  A ) ) ) )
5352sqreulem 11859 . . . . . 6  |-  ( ( A  e.  CC  /\  ( ( abs `  A
)  +  A )  =/=  0 )  -> 
( ( ( ( sqr `  ( abs `  A ) )  x.  ( ( ( abs `  A )  +  A
)  /  ( abs `  ( ( abs `  A
)  +  A ) ) ) ) ^
2 )  =  A  /\  0  <_  (
Re `  ( ( sqr `  ( abs `  A
) )  x.  (
( ( abs `  A
)  +  A )  /  ( abs `  (
( abs `  A
)  +  A ) ) ) ) )  /\  ( _i  x.  ( ( sqr `  ( abs `  A ) )  x.  ( ( ( abs `  A )  +  A )  / 
( abs `  (
( abs `  A
)  +  A ) ) ) ) )  e/  RR+ ) )
5419, 44, 53syl2anc 642 . . . . 5  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( (
( ( sqr `  ( abs `  A ) )  x.  ( ( ( abs `  A )  +  A )  / 
( abs `  (
( abs `  A
)  +  A ) ) ) ) ^
2 )  =  A  /\  0  <_  (
Re `  ( ( sqr `  ( abs `  A
) )  x.  (
( ( abs `  A
)  +  A )  /  ( abs `  (
( abs `  A
)  +  A ) ) ) ) )  /\  ( _i  x.  ( ( sqr `  ( abs `  A ) )  x.  ( ( ( abs `  A )  +  A )  / 
( abs `  (
( abs `  A
)  +  A ) ) ) ) )  e/  RR+ ) )
5554simp1d 967 . . . 4  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( (
( sqr `  ( abs `  A ) )  x.  ( ( ( abs `  A )  +  A )  / 
( abs `  (
( abs `  A
)  +  A ) ) ) ) ^
2 )  =  A )
5654simp2d 968 . . . 4  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  0  <_  ( Re `  ( ( sqr `  ( abs `  A ) )  x.  ( ( ( abs `  A )  +  A
)  /  ( abs `  ( ( abs `  A
)  +  A ) ) ) ) ) )
5754simp3d 969 . . . . 5  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( _i  x.  ( ( sqr `  ( abs `  A ) )  x.  ( ( ( abs `  A )  +  A )  / 
( abs `  (
( abs `  A
)  +  A ) ) ) ) )  e/  RR+ )
58 df-nel 2462 . . . . 5  |-  ( ( _i  x.  ( ( sqr `  ( abs `  A ) )  x.  ( ( ( abs `  A )  +  A
)  /  ( abs `  ( ( abs `  A
)  +  A ) ) ) ) )  e/  RR+  <->  -.  ( _i  x.  ( ( sqr `  ( abs `  A ) )  x.  ( ( ( abs `  A )  +  A )  / 
( abs `  (
( abs `  A
)  +  A ) ) ) ) )  e.  RR+ )
5957, 58sylib 188 . . . 4  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  -.  (
_i  x.  ( ( sqr `  ( abs `  A
) )  x.  (
( ( abs `  A
)  +  A )  /  ( abs `  (
( abs `  A
)  +  A ) ) ) ) )  e.  RR+ )
6051, 19, 55, 56, 59eqsqrd 11867 . . 3  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( ( sqr `  ( abs `  A
) )  x.  (
( ( abs `  A
)  +  A )  /  ( abs `  (
( abs `  A
)  +  A ) ) ) )  =  ( sqr `  A
) )
6160, 50eqeltrrd 2371 . 2  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( sqr `  A )  e.  K
)
627, 61pm2.61dane 2537 1  |-  ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  ->  ( sqr `  A )  e.  K )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459    e/ wnel 2460    C_ wss 3165   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   _ici 8755    + caddc 8756    x. cmul 8758    <_ cle 8884    - cmin 9053   -ucneg 9054    / cdiv 9439   2c2 9811   RR+crp 10370   ^cexp 11120   Recre 11598   sqrcsqr 11734   abscabs 11735   Basecbs 13164  Scalarcsca 13227  SubRingcsubrg 15557  ℂfldccnfld 16393   CPreHilccph 18618
This theorem is referenced by:  cphsqrcl3  18639
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-rp 10371  df-ico 10678  df-fz 10799  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-tset 13243  df-ple 13244  df-ds 13246  df-0g 13420  df-mnd 14383  df-mhm 14431  df-grp 14505  df-minusg 14506  df-subg 14634  df-ghm 14697  df-cmn 15107  df-mgp 15342  df-rng 15356  df-cring 15357  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440  df-invr 15470  df-dvr 15481  df-rnghom 15512  df-drng 15530  df-subrg 15559  df-staf 15626  df-srng 15627  df-lvec 15872  df-cnfld 16394  df-phl 16546  df-cph 18620
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