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Theorem cphsqrcl2 19141
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 12039 . . . . 5  |-  ( sqr `  0 )  =  0
2 fveq2 5720 . . . . 5  |-  ( A  =  0  ->  ( sqr `  A )  =  ( sqr `  0
) )
3 id 20 . . . . 5  |-  ( A  =  0  ->  A  =  0 )
41, 2, 33eqtr4a 2493 . . . 4  |-  ( A  =  0  ->  ( sqr `  A )  =  A )
54adantl 453 . . 3  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =  0 )  ->  ( sqr `  A )  =  A )
6 simpl2 961 . . 3  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =  0 )  ->  A  e.  K )
75, 6eqeltrd 2509 . 2  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =  0 )  ->  ( sqr `  A )  e.  K
)
8 simpl1 960 . . . . . . 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 19135 . . . . . . 7  |-  ( W  e.  CPreHil  ->  K  e.  (SubRing ` fld ) )
128, 11syl 16 . . . . . 6  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  K  e.  (SubRing ` fld ) )
13 cnfldbas 16699 . . . . . . 7  |-  CC  =  ( Base ` fld )
1413subrgss 15861 . . . . . 6  |-  ( K  e.  (SubRing ` fld )  ->  K  C_  CC )
1512, 14syl 16 . . . . 5  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  K  C_  CC )
16 simpl2 961 . . . . . . . 8  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  A  e.  K )
179, 10cphabscl 19140 . . . . . . . 8  |-  ( ( W  e.  CPreHil  /\  A  e.  K )  ->  ( abs `  A )  e.  K )
188, 16, 17syl2anc 643 . . . . . . 7  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( abs `  A )  e.  K
)
1915, 16sseldd 3341 . . . . . . . 8  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  A  e.  CC )
2019abscld 12230 . . . . . . 7  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( abs `  A )  e.  RR )
2119absge0d 12238 . . . . . . 7  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  0  <_  ( abs `  A ) )
229, 10cphsqrcl 19139 . . . . . . 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 1186 . . . . . 6  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( sqr `  ( abs `  A
) )  e.  K
)
24 cnfldadd 16700 . . . . . . . . 9  |-  +  =  ( +g  ` fld )
2524subrgacl 15871 . . . . . . . 8  |-  ( ( K  e.  (SubRing ` fld )  /\  ( abs `  A )  e.  K  /\  A  e.  K )  ->  (
( abs `  A
)  +  A )  e.  K )
2612, 18, 16, 25syl3anc 1184 . . . . . . 7  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( ( abs `  A )  +  A )  e.  K
)
279, 10cphabscl 19140 . . . . . . . 8  |-  ( ( W  e.  CPreHil  /\  (
( abs `  A
)  +  A )  e.  K )  -> 
( abs `  (
( abs `  A
)  +  A ) )  e.  K )
288, 26, 27syl2anc 643 . . . . . . 7  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( abs `  ( ( abs `  A
)  +  A ) )  e.  K )
2915, 26sseldd 3341 . . . . . . . 8  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( ( abs `  A )  +  A )  e.  CC )
30 simpl3 962 . . . . . . . . 9  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  -.  -u A  e.  RR+ )
3120recnd 9106 . . . . . . . . . . . . . 14  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( abs `  A )  e.  CC )
3231, 19subnegd 9410 . . . . . . . . . . . . 13  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( ( abs `  A )  -  -u A )  =  ( ( abs `  A
)  +  A ) )
3332eqeq1d 2443 . . . . . . . . . . . 12  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( (
( abs `  A
)  -  -u A
)  =  0  <->  (
( abs `  A
)  +  A )  =  0 ) )
3419negcld 9390 . . . . . . . . . . . . 13  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  -u A  e.  CC )
3531, 34subeq0ad 9413 . . . . . . . . . . . 12  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( (
( abs `  A
)  -  -u A
)  =  0  <->  ( abs `  A )  = 
-u A ) )
3633, 35bitr3d 247 . . . . . . . . . . 11  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( (
( abs `  A
)  +  A )  =  0  <->  ( abs `  A )  =  -u A ) )
37 absrpcl 12085 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( abs `  A
)  e.  RR+ )
3819, 37sylancom 649 . . . . . . . . . . . 12  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( abs `  A )  e.  RR+ )
39 eleq1 2495 . . . . . . . . . . . 12  |-  ( ( abs `  A )  =  -u A  ->  (
( abs `  A
)  e.  RR+  <->  -u A  e.  RR+ ) )
4038, 39syl5ibcom 212 . . . . . . . . . . 11  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( ( abs `  A )  = 
-u A  ->  -u A  e.  RR+ ) )
4136, 40sylbid 207 . . . . . . . . . 10  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( (
( abs `  A
)  +  A )  =  0  ->  -u A  e.  RR+ ) )
4241necon3bd 2635 . . . . . . . . 9  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( -.  -u A  e.  RR+  ->  ( ( abs `  A
)  +  A )  =/=  0 ) )
4330, 42mpd 15 . . . . . . . 8  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( ( abs `  A )  +  A )  =/=  0
)
4429, 43absne0d 12241 . . . . . . 7  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( abs `  ( ( abs `  A
)  +  A ) )  =/=  0 )
459, 10cphdivcl 19137 . . . . . . 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
)
468, 26, 28, 44, 45syl13anc 1186 . . . . . 6  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( (
( abs `  A
)  +  A )  /  ( abs `  (
( abs `  A
)  +  A ) ) )  e.  K
)
47 cnfldmul 16701 . . . . . . 7  |-  x.  =  ( .r ` fld )
4847subrgmcl 15872 . . . . . 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 )
4912, 23, 46, 48syl3anc 1184 . . . . 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 )
5015, 49sseldd 3341 . . . 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 )
51 eqid 2435 . . . . . . 7  |-  ( ( sqr `  ( abs `  A ) )  x.  ( ( ( abs `  A )  +  A
)  /  ( abs `  ( ( abs `  A
)  +  A ) ) ) )  =  ( ( sqr `  ( abs `  A ) )  x.  ( ( ( abs `  A )  +  A )  / 
( abs `  (
( abs `  A
)  +  A ) ) ) )
5251sqreulem 12155 . . . . . 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+ ) )
5319, 43, 52syl2anc 643 . . . . 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+ ) )
5453simp1d 969 . . . 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 )
5553simp2d 970 . . . 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 ) ) ) ) ) )
5653simp3d 971 . . . . 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+ )
57 df-nel 2601 . . . . 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+ )
5856, 57sylib 189 . . . 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+ )
5950, 19, 54, 55, 58eqsqrd 12163 . . 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
) )
6059, 49eqeltrrd 2510 . 2  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( sqr `  A )  e.  K
)
617, 60pm2.61dane 2676 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    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598    e/ wnel 2599    C_ wss 3312   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   CCcc 8980   RRcr 8981   0cc0 8982   _ici 8984    + caddc 8985    x. cmul 8987    <_ cle 9113    - cmin 9283   -ucneg 9284    / cdiv 9669   2c2 10041   RR+crp 10604   ^cexp 11374   Recre 11894   sqrcsqr 12030   abscabs 12031   Basecbs 13461  Scalarcsca 13524  SubRingcsubrg 15856  ℂfldccnfld 16695   CPreHilccph 19121
This theorem is referenced by:  cphsqrcl3  19142
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-addf 9061  ax-mulf 9062
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-tpos 6471  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-rp 10605  df-ico 10914  df-fz 11036  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-starv 13536  df-tset 13540  df-ple 13541  df-ds 13543  df-unif 13544  df-0g 13719  df-mnd 14682  df-mhm 14730  df-grp 14804  df-minusg 14805  df-subg 14933  df-ghm 14996  df-cmn 15406  df-mgp 15641  df-rng 15655  df-cring 15656  df-ur 15657  df-oppr 15720  df-dvdsr 15738  df-unit 15739  df-invr 15769  df-dvr 15780  df-rnghom 15811  df-drng 15829  df-subrg 15858  df-staf 15925  df-srng 15926  df-lvec 16167  df-cnfld 16696  df-phl 16849  df-cph 19123
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