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Theorem cphsqrcl2 21363
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 13027 . . . . 5  |-  ( sqr `  0 )  =  0
2 fveq2 5859 . . . . 5  |-  ( A  =  0  ->  ( sqr `  A )  =  ( sqr `  0
) )
3 id 22 . . . . 5  |-  ( A  =  0  ->  A  =  0 )
41, 2, 33eqtr4a 2529 . . . 4  |-  ( A  =  0  ->  ( sqr `  A )  =  A )
54adantl 466 . . 3  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =  0 )  ->  ( sqr `  A )  =  A )
6 simpl2 995 . . 3  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =  0 )  ->  A  e.  K )
75, 6eqeltrd 2550 . 2  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =  0 )  ->  ( sqr `  A )  e.  K
)
8 simpl1 994 . . . . . . 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 21357 . . . . . . 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 18190 . . . . . . 7  |-  CC  =  ( Base ` fld )
1413subrgss 17208 . . . . . 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 995 . . . . . . . 8  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  A  e.  K )
179, 10cphabscl 21362 . . . . . . . 8  |-  ( ( W  e.  CPreHil  /\  A  e.  K )  ->  ( abs `  A )  e.  K )
188, 16, 17syl2anc 661 . . . . . . 7  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( abs `  A )  e.  K
)
1915, 16sseldd 3500 . . . . . . . 8  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  A  e.  CC )
2019abscld 13218 . . . . . . 7  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( abs `  A )  e.  RR )
2119absge0d 13226 . . . . . . 7  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  0  <_  ( abs `  A ) )
229, 10cphsqrcl 21361 . . . . . . 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 1225 . . . . . 6  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( sqr `  ( abs `  A
) )  e.  K
)
24 cnfldadd 18191 . . . . . . . . 9  |-  +  =  ( +g  ` fld )
2524subrgacl 17218 . . . . . . . 8  |-  ( ( K  e.  (SubRing ` fld )  /\  ( abs `  A )  e.  K  /\  A  e.  K )  ->  (
( abs `  A
)  +  A )  e.  K )
2612, 18, 16, 25syl3anc 1223 . . . . . . 7  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( ( abs `  A )  +  A )  e.  K
)
279, 10cphabscl 21362 . . . . . . . 8  |-  ( ( W  e.  CPreHil  /\  (
( abs `  A
)  +  A )  e.  K )  -> 
( abs `  (
( abs `  A
)  +  A ) )  e.  K )
288, 26, 27syl2anc 661 . . . . . . 7  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( abs `  ( ( abs `  A
)  +  A ) )  e.  K )
2915, 26sseldd 3500 . . . . . . . 8  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( ( abs `  A )  +  A )  e.  CC )
30 simpl3 996 . . . . . . . . 9  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  -.  -u A  e.  RR+ )
3120recnd 9613 . . . . . . . . . . . . . 14  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( abs `  A )  e.  CC )
3231, 19subnegd 9928 . . . . . . . . . . . . 13  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( ( abs `  A )  -  -u A )  =  ( ( abs `  A
)  +  A ) )
3332eqeq1d 2464 . . . . . . . . . . . 12  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( (
( abs `  A
)  -  -u A
)  =  0  <->  (
( abs `  A
)  +  A )  =  0 ) )
3419negcld 9908 . . . . . . . . . . . . 13  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  -u A  e.  CC )
3531, 34subeq0ad 9931 . . . . . . . . . . . 12  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( (
( abs `  A
)  -  -u A
)  =  0  <->  ( abs `  A )  = 
-u A ) )
3633, 35bitr3d 255 . . . . . . . . . . 11  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( (
( abs `  A
)  +  A )  =  0  <->  ( abs `  A )  =  -u A ) )
37 absrpcl 13073 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( abs `  A
)  e.  RR+ )
3819, 37sylancom 667 . . . . . . . . . . . 12  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( abs `  A )  e.  RR+ )
39 eleq1 2534 . . . . . . . . . . . 12  |-  ( ( abs `  A )  =  -u A  ->  (
( abs `  A
)  e.  RR+  <->  -u A  e.  RR+ ) )
4038, 39syl5ibcom 220 . . . . . . . . . . 11  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( ( abs `  A )  = 
-u A  ->  -u A  e.  RR+ ) )
4136, 40sylbid 215 . . . . . . . . . 10  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( (
( abs `  A
)  +  A )  =  0  ->  -u A  e.  RR+ ) )
4241necon3bd 2674 . . . . . . . . 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 13229 . . . . . . 7  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( abs `  ( ( abs `  A
)  +  A ) )  =/=  0 )
459, 10cphdivcl 21359 . . . . . . 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 1225 . . . . . 6  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( (
( abs `  A
)  +  A )  /  ( abs `  (
( abs `  A
)  +  A ) ) )  e.  K
)
47 cnfldmul 18192 . . . . . . 7  |-  x.  =  ( .r ` fld )
4847subrgmcl 17219 . . . . . 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 1223 . . . . 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 3500 . . . 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 2462 . . . . . . 7  |-  ( ( sqr `  ( abs `  A ) )  x.  ( ( ( abs `  A )  +  A
)  /  ( abs `  ( ( abs `  A
)  +  A ) ) ) )  =  ( ( sqr `  ( abs `  A ) )  x.  ( ( ( abs `  A )  +  A )  / 
( abs `  (
( abs `  A
)  +  A ) ) ) )
5251sqreulem 13143 . . . . . 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 661 . . . . 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 1003 . . . 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 1004 . . . 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 1005 . . . . 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 2660 . . . . 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 196 . . . 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 13151 . . 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 2551 . 2  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( sqr `  A )  e.  K
)
617, 60pm2.61dane 2780 1  |-  ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  ->  ( 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    =/= wne 2657    e/ wnel 2658    C_ wss 3471   class class class wbr 4442   ` cfv 5581  (class class class)co 6277   CCcc 9481   RRcr 9482   0cc0 9483   _ici 9485    + caddc 9486    x. cmul 9488    <_ cle 9620    - cmin 9796   -ucneg 9797    / cdiv 10197   2c2 10576   RR+crp 11211   ^cexp 12124   Recre 12882   sqrcsqr 13018   abscabs 13019   Basecbs 14481  Scalarcsca 14549  SubRingcsubrg 17203  ℂfldccnfld 18186   CPreHilccph 21343
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 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560  ax-pre-sup 9561  ax-addf 9562  ax-mulf 9563
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 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-tpos 6947  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-sup 7892  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-div 10198  df-nn 10528  df-2 10585  df-3 10586  df-4 10587  df-5 10588  df-6 10589  df-7 10590  df-8 10591  df-9 10592  df-10 10593  df-n0 10787  df-z 10856  df-dec 10968  df-uz 11074  df-rp 11212  df-ico 11526  df-fz 11664  df-seq 12066  df-exp 12125  df-cj 12884  df-re 12885  df-im 12886  df-sqr 13020  df-abs 13021  df-struct 14483  df-ndx 14484  df-slot 14485  df-base 14486  df-sets 14487  df-ress 14488  df-plusg 14559  df-mulr 14560  df-starv 14561  df-tset 14565  df-ple 14566  df-ds 14568  df-unif 14569  df-0g 14688  df-mnd 15723  df-mhm 15772  df-grp 15853  df-minusg 15854  df-subg 15988  df-ghm 16055  df-cmn 16591  df-mgp 16927  df-ur 16939  df-rng 16983  df-cring 16984  df-oppr 17051  df-dvdsr 17069  df-unit 17070  df-invr 17100  df-dvr 17111  df-rnghom 17143  df-drng 17176  df-subrg 17205  df-staf 17272  df-srng 17273  df-lvec 17527  df-cnfld 18187  df-phl 18423  df-cph 21345
This theorem is referenced by:  cphsqrcl3  21364
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