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Theorem orngsqr 28641
Description: In an ordered ring, all squares are positive. (Contributed by Thierry Arnoux, 20-Jan-2018.)
Hypotheses
Ref Expression
orngmul.0  |-  B  =  ( Base `  R
)
orngmul.1  |-  .<_  =  ( le `  R )
orngmul.2  |-  .0.  =  ( 0g `  R )
orngmul.3  |-  .x.  =  ( .r `  R )
Assertion
Ref Expression
orngsqr  |-  ( ( R  e. oRing  /\  X  e.  B )  ->  .0.  .<_  ( X  .x.  X ) )

Proof of Theorem orngsqr
StepHypRef Expression
1 simpll 768 . . 3  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  .0.  .<_  X )  ->  R  e. oRing )
2 simplr 770 . . 3  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  .0.  .<_  X )  ->  X  e.  B )
3 simpr 468 . . 3  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  .0.  .<_  X )  ->  .0.  .<_  X )
4 orngmul.0 . . . 4  |-  B  =  ( Base `  R
)
5 orngmul.1 . . . 4  |-  .<_  =  ( le `  R )
6 orngmul.2 . . . 4  |-  .0.  =  ( 0g `  R )
7 orngmul.3 . . . 4  |-  .x.  =  ( .r `  R )
84, 5, 6, 7orngmul 28640 . . 3  |-  ( ( R  e. oRing  /\  ( X  e.  B  /\  .0.  .<_  X )  /\  ( X  e.  B  /\  .0.  .<_  X ) )  ->  .0.  .<_  ( X 
.x.  X ) )
91, 2, 3, 2, 3, 8syl122anc 1301 . 2  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  .0.  .<_  X )  ->  .0.  .<_  ( X  .x.  X ) )
10 simpll 768 . . . 4  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  R  e. oRing )
11 orngring 28637 . . . . . . 7  |-  ( R  e. oRing  ->  R  e.  Ring )
1211ad2antrr 740 . . . . . 6  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  R  e.  Ring )
13 ringgrp 17863 . . . . . 6  |-  ( R  e.  Ring  ->  R  e. 
Grp )
1412, 13syl 17 . . . . 5  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  R  e.  Grp )
15 simplr 770 . . . . 5  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  X  e.  B )
16 eqid 2471 . . . . . 6  |-  ( invg `  R )  =  ( invg `  R )
174, 16grpinvcl 16789 . . . . 5  |-  ( ( R  e.  Grp  /\  X  e.  B )  ->  ( ( invg `  R ) `  X
)  e.  B )
1814, 15, 17syl2anc 673 . . . 4  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
( ( invg `  R ) `  X
)  e.  B )
19 orngogrp 28638 . . . . . . . 8  |-  ( R  e. oRing  ->  R  e. oGrp )
20 isogrp 28539 . . . . . . . . 9  |-  ( R  e. oGrp 
<->  ( R  e.  Grp  /\  R  e. oMnd ) )
2120simprbi 471 . . . . . . . 8  |-  ( R  e. oGrp  ->  R  e. oMnd )
2219, 21syl 17 . . . . . . 7  |-  ( R  e. oRing  ->  R  e. oMnd )
2310, 22syl 17 . . . . . 6  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  R  e. oMnd )
244, 6grpidcl 16772 . . . . . . 7  |-  ( R  e.  Grp  ->  .0.  e.  B )
2514, 24syl 17 . . . . . 6  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  .0.  e.  B )
26 simpl 464 . . . . . . . . . . 11  |-  ( ( R  e. oRing  /\  X  e.  B )  ->  R  e. oRing )
2711, 13, 243syl 18 . . . . . . . . . . . 12  |-  ( R  e. oRing  ->  .0.  e.  B
)
2826, 27syl 17 . . . . . . . . . . 11  |-  ( ( R  e. oRing  /\  X  e.  B )  ->  .0.  e.  B )
29 simpr 468 . . . . . . . . . . 11  |-  ( ( R  e. oRing  /\  X  e.  B )  ->  X  e.  B )
3026, 28, 293jca 1210 . . . . . . . . . 10  |-  ( ( R  e. oRing  /\  X  e.  B )  ->  ( R  e. oRing  /\  .0.  e.  B  /\  X  e.  B
) )
31 eqid 2471 . . . . . . . . . . . 12  |-  ( lt
`  R )  =  ( lt `  R
)
325, 31pltle 16285 . . . . . . . . . . 11  |-  ( ( R  e. oRing  /\  .0.  e.  B  /\  X  e.  B
)  ->  (  .0.  ( lt `  R ) X  ->  .0.  .<_  X ) )
3332con3dimp 448 . . . . . . . . . 10  |-  ( ( ( R  e. oRing  /\  .0.  e.  B  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  -.  .0.  ( lt `  R ) X )
3430, 33sylan 479 . . . . . . . . 9  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  -.  .0.  ( lt `  R ) X )
35 omndtos 28542 . . . . . . . . . . . . 13  |-  ( R  e. oMnd  ->  R  e. Toset )
3622, 35syl 17 . . . . . . . . . . . 12  |-  ( R  e. oRing  ->  R  e. Toset )
374, 5, 31tosso 16360 . . . . . . . . . . . . . 14  |-  ( R  e. Toset  ->  ( R  e. Toset  <->  ( ( lt `  R
)  Or  B  /\  (  _I  |`  B ) 
C_  .<_  ) ) )
3837ibi 249 . . . . . . . . . . . . 13  |-  ( R  e. Toset  ->  ( ( lt
`  R )  Or  B  /\  (  _I  |`  B )  C_  .<_  ) )
3938simpld 466 . . . . . . . . . . . 12  |-  ( R  e. Toset  ->  ( lt `  R )  Or  B
)
4010, 36, 393syl 18 . . . . . . . . . . 11  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
( lt `  R
)  Or  B )
41 solin 4783 . . . . . . . . . . 11  |-  ( ( ( lt `  R
)  Or  B  /\  (  .0.  e.  B  /\  X  e.  B )
)  ->  (  .0.  ( lt `  R ) X  \/  .0.  =  X  \/  X ( lt `  R )  .0.  ) )
4240, 25, 15, 41syl12anc 1290 . . . . . . . . . 10  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
(  .0.  ( lt
`  R ) X  \/  .0.  =  X  \/  X ( lt
`  R )  .0.  ) )
43 3orass 1010 . . . . . . . . . 10  |-  ( (  .0.  ( lt `  R ) X  \/  .0.  =  X  \/  X
( lt `  R
)  .0.  )  <->  (  .0.  ( lt `  R ) X  \/  (  .0.  =  X  \/  X
( lt `  R
)  .0.  ) ) )
4442, 43sylib 201 . . . . . . . . 9  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
(  .0.  ( lt
`  R ) X  \/  (  .0.  =  X  \/  X ( lt `  R )  .0.  ) ) )
45 orel1 389 . . . . . . . . 9  |-  ( -.  .0.  ( lt `  R ) X  -> 
( (  .0.  ( lt `  R ) X  \/  (  .0.  =  X  \/  X ( lt `  R )  .0.  ) )  ->  (  .0.  =  X  \/  X
( lt `  R
)  .0.  ) ) )
4634, 44, 45sylc 61 . . . . . . . 8  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
(  .0.  =  X  \/  X ( lt
`  R )  .0.  ) )
47 orcom 394 . . . . . . . . 9  |-  ( (  .0.  =  X  \/  X ( lt `  R )  .0.  )  <->  ( X ( lt `  R )  .0.  \/  .0.  =  X ) )
48 eqcom 2478 . . . . . . . . . 10  |-  (  .0.  =  X  <->  X  =  .0.  )
4948orbi2i 528 . . . . . . . . 9  |-  ( ( X ( lt `  R )  .0.  \/  .0.  =  X )  <->  ( X
( lt `  R
)  .0.  \/  X  =  .0.  ) )
5047, 49bitri 257 . . . . . . . 8  |-  ( (  .0.  =  X  \/  X ( lt `  R )  .0.  )  <->  ( X ( lt `  R )  .0.  \/  X  =  .0.  )
)
5146, 50sylib 201 . . . . . . 7  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
( X ( lt
`  R )  .0. 
\/  X  =  .0.  ) )
52 tospos 28494 . . . . . . . . 9  |-  ( R  e. Toset  ->  R  e.  Poset )
5310, 36, 523syl 18 . . . . . . . 8  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  R  e.  Poset )
544, 5, 31pleval2 16289 . . . . . . . 8  |-  ( ( R  e.  Poset  /\  X  e.  B  /\  .0.  e.  B )  ->  ( X  .<_  .0.  <->  ( X ( lt `  R )  .0.  \/  X  =  .0.  ) ) )
5553, 15, 25, 54syl3anc 1292 . . . . . . 7  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
( X  .<_  .0.  <->  ( X
( lt `  R
)  .0.  \/  X  =  .0.  ) ) )
5651, 55mpbird 240 . . . . . 6  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  X  .<_  .0.  )
57 eqid 2471 . . . . . . 7  |-  ( +g  `  R )  =  ( +g  `  R )
584, 5, 57omndadd 28543 . . . . . 6  |-  ( ( R  e. oMnd  /\  ( X  e.  B  /\  .0.  e.  B  /\  (
( invg `  R ) `  X
)  e.  B )  /\  X  .<_  .0.  )  ->  ( X ( +g  `  R ) ( ( invg `  R
) `  X )
)  .<_  (  .0.  ( +g  `  R ) ( ( invg `  R ) `  X
) ) )
5923, 15, 25, 18, 56, 58syl131anc 1305 . . . . 5  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
( X ( +g  `  R ) ( ( invg `  R
) `  X )
)  .<_  (  .0.  ( +g  `  R ) ( ( invg `  R ) `  X
) ) )
604, 57, 6, 16grprinv 16791 . . . . . 6  |-  ( ( R  e.  Grp  /\  X  e.  B )  ->  ( X ( +g  `  R ) ( ( invg `  R
) `  X )
)  =  .0.  )
6114, 15, 60syl2anc 673 . . . . 5  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
( X ( +g  `  R ) ( ( invg `  R
) `  X )
)  =  .0.  )
624, 57, 6grplid 16774 . . . . . 6  |-  ( ( R  e.  Grp  /\  ( ( invg `  R ) `  X
)  e.  B )  ->  (  .0.  ( +g  `  R ) ( ( invg `  R ) `  X
) )  =  ( ( invg `  R ) `  X
) )
6314, 18, 62syl2anc 673 . . . . 5  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
(  .0.  ( +g  `  R ) ( ( invg `  R
) `  X )
)  =  ( ( invg `  R
) `  X )
)
6459, 61, 633brtr3d 4425 . . . 4  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  .0.  .<_  ( ( invg `  R ) `
 X ) )
654, 5, 6, 7orngmul 28640 . . . 4  |-  ( ( R  e. oRing  /\  (
( ( invg `  R ) `  X
)  e.  B  /\  .0.  .<_  ( ( invg `  R ) `
 X ) )  /\  ( ( ( invg `  R
) `  X )  e.  B  /\  .0.  .<_  ( ( invg `  R ) `  X
) ) )  ->  .0.  .<_  ( ( ( invg `  R
) `  X )  .x.  ( ( invg `  R ) `  X
) ) )
6610, 18, 64, 18, 64, 65syl122anc 1301 . . 3  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  .0.  .<_  ( ( ( invg `  R
) `  X )  .x.  ( ( invg `  R ) `  X
) ) )
674, 7, 16, 12, 15, 15ringm2neg 17904 . . 3  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
( ( ( invg `  R ) `
 X )  .x.  ( ( invg `  R ) `  X
) )  =  ( X  .x.  X ) )
6866, 67breqtrd 4420 . 2  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  .0.  .<_  ( X  .x.  X ) )
699, 68pm2.61dan 808 1  |-  ( ( R  e. oRing  /\  X  e.  B )  ->  .0.  .<_  ( X  .x.  X ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    \/ wo 375    /\ wa 376    \/ w3o 1006    /\ w3a 1007    = wceq 1452    e. wcel 1904    C_ wss 3390   class class class wbr 4395    _I cid 4749    Or wor 4759    |` cres 4841   ` cfv 5589  (class class class)co 6308   Basecbs 15199   +g cplusg 15268   .rcmulr 15269   lecple 15275   0gc0g 15416   Posetcpo 16263   ltcplt 16264  Tosetctos 16357   Grpcgrp 16747   invgcminusg 16748   Ringcrg 17858  oMndcomnd 28534  oGrpcogrp 28535  oRingcorng 28632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  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-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-plusg 15281  df-0g 15418  df-preset 16251  df-poset 16269  df-plt 16282  df-toset 16358  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-grp 16751  df-minusg 16752  df-mgp 17802  df-ur 17814  df-ring 17860  df-omnd 28536  df-ogrp 28537  df-orng 28634
This theorem is referenced by:  orng0le1  28649
  Copyright terms: Public domain W3C validator