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Theorem ofldsqr 24241
Description: In an ordered field, all squares are positive. (Contributed by Thierry Arnoux, 20-Jan-2018.)
Hypotheses
Ref Expression
ofldadd.0  |-  B  =  ( Base `  F
)
ofldadd.1  |-  .<_  =  ( le `  F )
ofldmul.2  |-  .0.  =  ( 0g `  F )
ofldmul.3  |-  .x.  =  ( .r `  F )
Assertion
Ref Expression
ofldsqr  |-  ( ( F  e. oField  /\  X  e.  B )  ->  .0.  .<_  ( X  .x.  X ) )

Proof of Theorem ofldsqr
StepHypRef Expression
1 simpll 732 . . 3  |-  ( ( ( F  e. oField  /\  X  e.  B )  /\  .0.  .<_  X )  ->  F  e. oField )
2 simplr 733 . . 3  |-  ( ( ( F  e. oField  /\  X  e.  B )  /\  .0.  .<_  X )  ->  X  e.  B )
3 simpr 449 . . 3  |-  ( ( ( F  e. oField  /\  X  e.  B )  /\  .0.  .<_  X )  ->  .0.  .<_  X )
4 ofldadd.0 . . . 4  |-  B  =  ( Base `  F
)
5 ofldadd.1 . . . 4  |-  .<_  =  ( le `  F )
6 ofldmul.2 . . . 4  |-  .0.  =  ( 0g `  F )
7 ofldmul.3 . . . 4  |-  .x.  =  ( .r `  F )
84, 5, 6, 7ofldmul 24240 . . 3  |-  ( ( F  e. oField  /\  ( X  e.  B  /\  .0.  .<_  X )  /\  ( X  e.  B  /\  .0.  .<_  X ) )  ->  .0.  .<_  ( X 
.x.  X ) )
91, 2, 3, 2, 3, 8syl122anc 1194 . 2  |-  ( ( ( F  e. oField  /\  X  e.  B )  /\  .0.  .<_  X )  ->  .0.  .<_  ( X  .x.  X ) )
10 simpll 732 . . . 4  |-  ( ( ( F  e. oField  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  F  e. oField )
11 ofldfld 24237 . . . . . . . 8  |-  ( F  e. oField  ->  F  e. Field )
12 isfld 15845 . . . . . . . . 9  |-  ( F  e. Field 
<->  ( F  e.  DivRing  /\  F  e.  CRing ) )
1312simplbi 448 . . . . . . . 8  |-  ( F  e. Field  ->  F  e.  DivRing )
14 drngrng 15843 . . . . . . . 8  |-  ( F  e.  DivRing  ->  F  e.  Ring )
1511, 13, 143syl 19 . . . . . . 7  |-  ( F  e. oField  ->  F  e.  Ring )
1615ad2antrr 708 . . . . . 6  |-  ( ( ( F  e. oField  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  F  e.  Ring )
17 rnggrp 15670 . . . . . 6  |-  ( F  e.  Ring  ->  F  e. 
Grp )
1816, 17syl 16 . . . . 5  |-  ( ( ( F  e. oField  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  F  e.  Grp )
19 simplr 733 . . . . 5  |-  ( ( ( F  e. oField  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  X  e.  B )
20 eqid 2437 . . . . . 6  |-  ( inv g `  F )  =  ( inv g `  F )
214, 20grpinvcl 14851 . . . . 5  |-  ( ( F  e.  Grp  /\  X  e.  B )  ->  ( ( inv g `  F ) `  X
)  e.  B )
2218, 19, 21syl2anc 644 . . . 4  |-  ( ( ( F  e. oField  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
( ( inv g `  F ) `  X
)  e.  B )
234, 6grpidcl 14834 . . . . . . 7  |-  ( F  e.  Grp  ->  .0.  e.  B )
2418, 23syl 16 . . . . . 6  |-  ( ( ( F  e. oField  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  .0.  e.  B )
25 simpl 445 . . . . . . . . . . 11  |-  ( ( F  e. oField  /\  X  e.  B )  ->  F  e. oField )
2615, 17, 233syl 19 . . . . . . . . . . . 12  |-  ( F  e. oField  ->  .0.  e.  B
)
2725, 26syl 16 . . . . . . . . . . 11  |-  ( ( F  e. oField  /\  X  e.  B )  ->  .0.  e.  B )
28 simpr 449 . . . . . . . . . . 11  |-  ( ( F  e. oField  /\  X  e.  B )  ->  X  e.  B )
2925, 27, 283jca 1135 . . . . . . . . . 10  |-  ( ( F  e. oField  /\  X  e.  B )  ->  ( F  e. oField  /\  .0.  e.  B  /\  X  e.  B
) )
30 eqid 2437 . . . . . . . . . . . 12  |-  ( lt
`  F )  =  ( lt `  F
)
315, 30pltle 14419 . . . . . . . . . . 11  |-  ( ( F  e. oField  /\  .0.  e.  B  /\  X  e.  B
)  ->  (  .0.  ( lt `  F ) X  ->  .0.  .<_  X ) )
3231con3and 430 . . . . . . . . . 10  |-  ( ( ( F  e. oField  /\  .0.  e.  B  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  -.  .0.  ( lt `  F ) X )
3329, 32sylan 459 . . . . . . . . 9  |-  ( ( ( F  e. oField  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  -.  .0.  ( lt `  F ) X )
34 ofldtos 24238 . . . . . . . . . . . 12  |-  ( F  e. oField  ->  F  e. Toset )
354, 5, 30tosso 14466 . . . . . . . . . . . . . 14  |-  ( F  e. Toset  ->  ( F  e. Toset  <->  ( ( lt `  F
)  Or  B  /\  (  _I  |`  B ) 
C_  .<_  ) ) )
3635ibi 234 . . . . . . . . . . . . 13  |-  ( F  e. Toset  ->  ( ( lt
`  F )  Or  B  /\  (  _I  |`  B )  C_  .<_  ) )
3736simpld 447 . . . . . . . . . . . 12  |-  ( F  e. Toset  ->  ( lt `  F )  Or  B
)
3810, 34, 373syl 19 . . . . . . . . . . 11  |-  ( ( ( F  e. oField  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
( lt `  F
)  Or  B )
39 solin 4527 . . . . . . . . . . 11  |-  ( ( ( lt `  F
)  Or  B  /\  (  .0.  e.  B  /\  X  e.  B )
)  ->  (  .0.  ( lt `  F ) X  \/  .0.  =  X  \/  X ( lt `  F )  .0.  ) )
4038, 24, 19, 39syl12anc 1183 . . . . . . . . . 10  |-  ( ( ( F  e. oField  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
(  .0.  ( lt
`  F ) X  \/  .0.  =  X  \/  X ( lt
`  F )  .0.  ) )
41 3orass 940 . . . . . . . . . 10  |-  ( (  .0.  ( lt `  F ) X  \/  .0.  =  X  \/  X
( lt `  F
)  .0.  )  <->  (  .0.  ( lt `  F ) X  \/  (  .0.  =  X  \/  X
( lt `  F
)  .0.  ) ) )
4240, 41sylib 190 . . . . . . . . 9  |-  ( ( ( F  e. oField  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
(  .0.  ( lt
`  F ) X  \/  (  .0.  =  X  \/  X ( lt `  F )  .0.  ) ) )
43 orel1 373 . . . . . . . . 9  |-  ( -.  .0.  ( lt `  F ) X  -> 
( (  .0.  ( lt `  F ) X  \/  (  .0.  =  X  \/  X ( lt `  F )  .0.  ) )  ->  (  .0.  =  X  \/  X
( lt `  F
)  .0.  ) ) )
4433, 42, 43sylc 59 . . . . . . . 8  |-  ( ( ( F  e. oField  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
(  .0.  =  X  \/  X ( lt
`  F )  .0.  ) )
45 orcom 378 . . . . . . . . 9  |-  ( (  .0.  =  X  \/  X ( lt `  F )  .0.  )  <->  ( X ( lt `  F )  .0.  \/  .0.  =  X ) )
46 eqcom 2439 . . . . . . . . . 10  |-  (  .0.  =  X  <->  X  =  .0.  )
4746orbi2i 507 . . . . . . . . 9  |-  ( ( X ( lt `  F )  .0.  \/  .0.  =  X )  <->  ( X
( lt `  F
)  .0.  \/  X  =  .0.  ) )
4845, 47bitri 242 . . . . . . . 8  |-  ( (  .0.  =  X  \/  X ( lt `  F )  .0.  )  <->  ( X ( lt `  F )  .0.  \/  X  =  .0.  )
)
4944, 48sylib 190 . . . . . . 7  |-  ( ( ( F  e. oField  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
( X ( lt
`  F )  .0. 
\/  X  =  .0.  ) )
50 tospos 24187 . . . . . . . . 9  |-  ( F  e. Toset  ->  F  e.  Poset )
5110, 34, 503syl 19 . . . . . . . 8  |-  ( ( ( F  e. oField  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  F  e.  Poset )
524, 5, 30pleval2 14423 . . . . . . . 8  |-  ( ( F  e.  Poset  /\  X  e.  B  /\  .0.  e.  B )  ->  ( X  .<_  .0.  <->  ( X ( lt `  F )  .0.  \/  X  =  .0.  ) ) )
5351, 19, 24, 52syl3anc 1185 . . . . . . 7  |-  ( ( ( F  e. oField  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
( X  .<_  .0.  <->  ( X
( lt `  F
)  .0.  \/  X  =  .0.  ) ) )
5449, 53mpbird 225 . . . . . 6  |-  ( ( ( F  e. oField  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  X  .<_  .0.  )
55 eqid 2437 . . . . . . 7  |-  ( +g  `  F )  =  ( +g  `  F )
564, 5, 55ofldadd 24239 . . . . . 6  |-  ( ( F  e. oField  /\  ( X  e.  B  /\  .0.  e.  B  /\  (
( inv g `  F ) `  X
)  e.  B )  /\  X  .<_  .0.  )  ->  ( X ( +g  `  F ) ( ( inv g `  F
) `  X )
)  .<_  (  .0.  ( +g  `  F ) ( ( inv g `  F ) `  X
) ) )
5710, 19, 24, 22, 54, 56syl131anc 1198 . . . . 5  |-  ( ( ( F  e. oField  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
( X ( +g  `  F ) ( ( inv g `  F
) `  X )
)  .<_  (  .0.  ( +g  `  F ) ( ( inv g `  F ) `  X
) ) )
584, 55, 6, 20grprinv 14853 . . . . . 6  |-  ( ( F  e.  Grp  /\  X  e.  B )  ->  ( X ( +g  `  F ) ( ( inv g `  F
) `  X )
)  =  .0.  )
5918, 19, 58syl2anc 644 . . . . 5  |-  ( ( ( F  e. oField  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
( X ( +g  `  F ) ( ( inv g `  F
) `  X )
)  =  .0.  )
604, 55, 6grplid 14836 . . . . . 6  |-  ( ( F  e.  Grp  /\  ( ( inv g `  F ) `  X
)  e.  B )  ->  (  .0.  ( +g  `  F ) ( ( inv g `  F ) `  X
) )  =  ( ( inv g `  F ) `  X
) )
6118, 22, 60syl2anc 644 . . . . 5  |-  ( ( ( F  e. oField  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
(  .0.  ( +g  `  F ) ( ( inv g `  F
) `  X )
)  =  ( ( inv g `  F
) `  X )
)
6257, 59, 613brtr3d 4242 . . . 4  |-  ( ( ( F  e. oField  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  .0.  .<_  ( ( inv g `  F ) `
 X ) )
634, 5, 6, 7ofldmul 24240 . . . 4  |-  ( ( F  e. oField  /\  (
( ( inv g `  F ) `  X
)  e.  B  /\  .0.  .<_  ( ( inv g `  F ) `
 X ) )  /\  ( ( ( inv g `  F
) `  X )  e.  B  /\  .0.  .<_  ( ( inv g `  F ) `  X
) ) )  ->  .0.  .<_  ( ( ( inv g `  F
) `  X )  .x.  ( ( inv g `  F ) `  X
) ) )
6410, 22, 62, 22, 62, 63syl122anc 1194 . . 3  |-  ( ( ( F  e. oField  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  .0.  .<_  ( ( ( inv g `  F
) `  X )  .x.  ( ( inv g `  F ) `  X
) ) )
654, 7, 20, 16, 19, 19rngm2neg 15708 . . 3  |-  ( ( ( F  e. oField  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
( ( ( inv g `  F ) `
 X )  .x.  ( ( inv g `  F ) `  X
) )  =  ( X  .x.  X ) )
6664, 65breqtrd 4237 . 2  |-  ( ( ( F  e. oField  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  .0.  .<_  ( X  .x.  X ) )
679, 66pm2.61dan 768 1  |-  ( ( F  e. oField  /\  X  e.  B )  ->  .0.  .<_  ( X  .x.  X ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    \/ w3o 936    /\ w3a 937    = wceq 1653    e. wcel 1726    C_ wss 3321   class class class wbr 4213    _I cid 4494    Or wor 4503    |` cres 4881   ` cfv 5455  (class class class)co 6082   Basecbs 13470   +g cplusg 13530   .rcmulr 13531   lecple 13537   0gc0g 13724   Posetcpo 14398   ltcplt 14399  Tosetctos 14463   Grpcgrp 14686   inv gcminusg 14687   Ringcrg 15661   CRingccrg 15662   DivRingcdr 15836  Fieldcfield 15837  oFieldcofld 24234
This theorem is referenced by:  ofld0le1  24243
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-riota 6550  df-recs 6634  df-rdg 6669  df-er 6906  df-en 7111  df-dom 7112  df-sdom 7113  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-nn 10002  df-2 10059  df-ndx 13473  df-slot 13474  df-base 13475  df-sets 13476  df-plusg 13543  df-0g 13728  df-poset 14404  df-plt 14416  df-toset 14464  df-mnd 14691  df-grp 14813  df-minusg 14814  df-mgp 15650  df-rng 15664  df-ur 15666  df-drng 15838  df-field 15839  df-ofld 24235
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