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Theorem cphsqrcl2 20704
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 12730 . . . . 5  |-  ( sqr `  0 )  =  0
2 fveq2 5690 . . . . 5  |-  ( A  =  0  ->  ( sqr `  A )  =  ( sqr `  0
) )
3 id 22 . . . . 5  |-  ( A  =  0  ->  A  =  0 )
41, 2, 33eqtr4a 2500 . . . 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 992 . . 3  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =  0 )  ->  A  e.  K )
75, 6eqeltrd 2516 . 2  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =  0 )  ->  ( sqr `  A )  e.  K
)
8 simpl1 991 . . . . . . 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 20698 . . . . . . 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 17821 . . . . . . 7  |-  CC  =  ( Base ` fld )
1413subrgss 16865 . . . . . 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 992 . . . . . . . 8  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  A  e.  K )
179, 10cphabscl 20703 . . . . . . . 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 3356 . . . . . . . 8  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  A  e.  CC )
2019abscld 12921 . . . . . . 7  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( abs `  A )  e.  RR )
2119absge0d 12929 . . . . . . 7  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  0  <_  ( abs `  A ) )
229, 10cphsqrcl 20702 . . . . . . 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 1220 . . . . . 6  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( sqr `  ( abs `  A
) )  e.  K
)
24 cnfldadd 17822 . . . . . . . . 9  |-  +  =  ( +g  ` fld )
2524subrgacl 16875 . . . . . . . 8  |-  ( ( K  e.  (SubRing ` fld )  /\  ( abs `  A )  e.  K  /\  A  e.  K )  ->  (
( abs `  A
)  +  A )  e.  K )
2612, 18, 16, 25syl3anc 1218 . . . . . . 7  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( ( abs `  A )  +  A )  e.  K
)
279, 10cphabscl 20703 . . . . . . . 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 3356 . . . . . . . 8  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( ( abs `  A )  +  A )  e.  CC )
30 simpl3 993 . . . . . . . . 9  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  -.  -u A  e.  RR+ )
3120recnd 9411 . . . . . . . . . . . . . 14  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( abs `  A )  e.  CC )
3231, 19subnegd 9725 . . . . . . . . . . . . 13  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( ( abs `  A )  -  -u A )  =  ( ( abs `  A
)  +  A ) )
3332eqeq1d 2450 . . . . . . . . . . . 12  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( (
( abs `  A
)  -  -u A
)  =  0  <->  (
( abs `  A
)  +  A )  =  0 ) )
3419negcld 9705 . . . . . . . . . . . . 13  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  -u A  e.  CC )
3531, 34subeq0ad 9728 . . . . . . . . . . . 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 12776 . . . . . . . . . . . . 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 2502 . . . . . . . . . . . 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 2644 . . . . . . . . 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 12932 . . . . . . 7  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( abs `  ( ( abs `  A
)  +  A ) )  =/=  0 )
459, 10cphdivcl 20700 . . . . . . 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 1220 . . . . . 6  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( (
( abs `  A
)  +  A )  /  ( abs `  (
( abs `  A
)  +  A ) ) )  e.  K
)
47 cnfldmul 17823 . . . . . . 7  |-  x.  =  ( .r ` fld )
4847subrgmcl 16876 . . . . . 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 1218 . . . . 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 3356 . . . 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 2442 . . . . . . 7  |-  ( ( sqr `  ( abs `  A ) )  x.  ( ( ( abs `  A )  +  A
)  /  ( abs `  ( ( abs `  A
)  +  A ) ) ) )  =  ( ( sqr `  ( abs `  A ) )  x.  ( ( ( abs `  A )  +  A )  / 
( abs `  (
( abs `  A
)  +  A ) ) ) )
5251sqreulem 12846 . . . . . 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 1000 . . . 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 1001 . . . 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 1002 . . . . 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 2608 . . . . 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 12854 . . 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 2517 . 2  |-  ( ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  /\  A  =/=  0
)  ->  ( sqr `  A )  e.  K
)
617, 60pm2.61dane 2688 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 965    = wceq 1369    e. wcel 1756    =/= wne 2605    e/ wnel 2606    C_ wss 3327   class class class wbr 4291   ` cfv 5417  (class class class)co 6090   CCcc 9279   RRcr 9280   0cc0 9281   _ici 9283    + caddc 9284    x. cmul 9286    <_ cle 9418    - cmin 9594   -ucneg 9595    / cdiv 9992   2c2 10370   RR+crp 10990   ^cexp 11864   Recre 12585   sqrcsqr 12721   abscabs 12722   Basecbs 14173  Scalarcsca 14240  SubRingcsubrg 16860  ℂfldccnfld 17817   CPreHilccph 20684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358  ax-pre-sup 9359  ax-addf 9360  ax-mulf 9361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-tpos 6744  df-recs 6831  df-rdg 6865  df-1o 6919  df-oadd 6923  df-er 7100  df-map 7215  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-sup 7690  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-div 9993  df-nn 10322  df-2 10379  df-3 10380  df-4 10381  df-5 10382  df-6 10383  df-7 10384  df-8 10385  df-9 10386  df-10 10387  df-n0 10579  df-z 10646  df-dec 10755  df-uz 10861  df-rp 10991  df-ico 11305  df-fz 11437  df-seq 11806  df-exp 11865  df-cj 12587  df-re 12588  df-im 12589  df-sqr 12723  df-abs 12724  df-struct 14175  df-ndx 14176  df-slot 14177  df-base 14178  df-sets 14179  df-ress 14180  df-plusg 14250  df-mulr 14251  df-starv 14252  df-tset 14256  df-ple 14257  df-ds 14259  df-unif 14260  df-0g 14379  df-mnd 15414  df-mhm 15463  df-grp 15544  df-minusg 15545  df-subg 15677  df-ghm 15744  df-cmn 16278  df-mgp 16591  df-ur 16603  df-rng 16646  df-cring 16647  df-oppr 16714  df-dvdsr 16732  df-unit 16733  df-invr 16763  df-dvr 16774  df-rnghom 16805  df-drng 16833  df-subrg 16862  df-staf 16929  df-srng 16930  df-lvec 17183  df-cnfld 17818  df-phl 18054  df-cph 20686
This theorem is referenced by:  cphsqrcl3  20705
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