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Theorem orngsqr 27947
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 753 . . 3  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  .0.  .<_  X )  ->  R  e. oRing )
2 simplr 755 . . 3  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  .0.  .<_  X )  ->  X  e.  B )
3 simpr 461 . . 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 27946 . . 3  |-  ( ( R  e. oRing  /\  ( X  e.  B  /\  .0.  .<_  X )  /\  ( X  e.  B  /\  .0.  .<_  X ) )  ->  .0.  .<_  ( X 
.x.  X ) )
91, 2, 3, 2, 3, 8syl122anc 1237 . 2  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  .0.  .<_  X )  ->  .0.  .<_  ( X  .x.  X ) )
10 simpll 753 . . . 4  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  R  e. oRing )
11 orngring 27943 . . . . . . 7  |-  ( R  e. oRing  ->  R  e.  Ring )
1211ad2antrr 725 . . . . . 6  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  R  e.  Ring )
13 ringgrp 17329 . . . . . 6  |-  ( R  e.  Ring  ->  R  e. 
Grp )
1412, 13syl 16 . . . . 5  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  R  e.  Grp )
15 simplr 755 . . . . 5  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  X  e.  B )
16 eqid 2457 . . . . . 6  |-  ( invg `  R )  =  ( invg `  R )
174, 16grpinvcl 16221 . . . . 5  |-  ( ( R  e.  Grp  /\  X  e.  B )  ->  ( ( invg `  R ) `  X
)  e.  B )
1814, 15, 17syl2anc 661 . . . 4  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
( ( invg `  R ) `  X
)  e.  B )
19 orngogrp 27944 . . . . . . . 8  |-  ( R  e. oRing  ->  R  e. oGrp )
20 isogrp 27844 . . . . . . . . 9  |-  ( R  e. oGrp 
<->  ( R  e.  Grp  /\  R  e. oMnd ) )
2120simprbi 464 . . . . . . . 8  |-  ( R  e. oGrp  ->  R  e. oMnd )
2219, 21syl 16 . . . . . . 7  |-  ( R  e. oRing  ->  R  e. oMnd )
2310, 22syl 16 . . . . . 6  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  R  e. oMnd )
244, 6grpidcl 16204 . . . . . . 7  |-  ( R  e.  Grp  ->  .0.  e.  B )
2514, 24syl 16 . . . . . 6  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  .0.  e.  B )
26 simpl 457 . . . . . . . . . . 11  |-  ( ( R  e. oRing  /\  X  e.  B )  ->  R  e. oRing )
2711, 13, 243syl 20 . . . . . . . . . . . 12  |-  ( R  e. oRing  ->  .0.  e.  B
)
2826, 27syl 16 . . . . . . . . . . 11  |-  ( ( R  e. oRing  /\  X  e.  B )  ->  .0.  e.  B )
29 simpr 461 . . . . . . . . . . 11  |-  ( ( R  e. oRing  /\  X  e.  B )  ->  X  e.  B )
3026, 28, 293jca 1176 . . . . . . . . . 10  |-  ( ( R  e. oRing  /\  X  e.  B )  ->  ( R  e. oRing  /\  .0.  e.  B  /\  X  e.  B
) )
31 eqid 2457 . . . . . . . . . . . 12  |-  ( lt
`  R )  =  ( lt `  R
)
325, 31pltle 15717 . . . . . . . . . . 11  |-  ( ( R  e. oRing  /\  .0.  e.  B  /\  X  e.  B
)  ->  (  .0.  ( lt `  R ) X  ->  .0.  .<_  X ) )
3332con3dimp 441 . . . . . . . . . 10  |-  ( ( ( R  e. oRing  /\  .0.  e.  B  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  -.  .0.  ( lt `  R ) X )
3430, 33sylan 471 . . . . . . . . 9  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  -.  .0.  ( lt `  R ) X )
35 omndtos 27847 . . . . . . . . . . . . 13  |-  ( R  e. oMnd  ->  R  e. Toset )
3622, 35syl 16 . . . . . . . . . . . 12  |-  ( R  e. oRing  ->  R  e. Toset )
374, 5, 31tosso 15792 . . . . . . . . . . . . . 14  |-  ( R  e. Toset  ->  ( R  e. Toset  <->  ( ( lt `  R
)  Or  B  /\  (  _I  |`  B ) 
C_  .<_  ) ) )
3837ibi 241 . . . . . . . . . . . . 13  |-  ( R  e. Toset  ->  ( ( lt
`  R )  Or  B  /\  (  _I  |`  B )  C_  .<_  ) )
3938simpld 459 . . . . . . . . . . . 12  |-  ( R  e. Toset  ->  ( lt `  R )  Or  B
)
4010, 36, 393syl 20 . . . . . . . . . . 11  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
( lt `  R
)  Or  B )
41 solin 4832 . . . . . . . . . . 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 1226 . . . . . . . . . 10  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
(  .0.  ( lt
`  R ) X  \/  .0.  =  X  \/  X ( lt
`  R )  .0.  ) )
43 3orass 976 . . . . . . . . . 10  |-  ( (  .0.  ( lt `  R ) X  \/  .0.  =  X  \/  X
( lt `  R
)  .0.  )  <->  (  .0.  ( lt `  R ) X  \/  (  .0.  =  X  \/  X
( lt `  R
)  .0.  ) ) )
4442, 43sylib 196 . . . . . . . . 9  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
(  .0.  ( lt
`  R ) X  \/  (  .0.  =  X  \/  X ( lt `  R )  .0.  ) ) )
45 orel1 382 . . . . . . . . 9  |-  ( -.  .0.  ( lt `  R ) X  -> 
( (  .0.  ( lt `  R ) X  \/  (  .0.  =  X  \/  X ( lt `  R )  .0.  ) )  ->  (  .0.  =  X  \/  X
( lt `  R
)  .0.  ) ) )
4634, 44, 45sylc 60 . . . . . . . 8  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
(  .0.  =  X  \/  X ( lt
`  R )  .0.  ) )
47 orcom 387 . . . . . . . . 9  |-  ( (  .0.  =  X  \/  X ( lt `  R )  .0.  )  <->  ( X ( lt `  R )  .0.  \/  .0.  =  X ) )
48 eqcom 2466 . . . . . . . . . 10  |-  (  .0.  =  X  <->  X  =  .0.  )
4948orbi2i 519 . . . . . . . . 9  |-  ( ( X ( lt `  R )  .0.  \/  .0.  =  X )  <->  ( X
( lt `  R
)  .0.  \/  X  =  .0.  ) )
5047, 49bitri 249 . . . . . . . 8  |-  ( (  .0.  =  X  \/  X ( lt `  R )  .0.  )  <->  ( X ( lt `  R )  .0.  \/  X  =  .0.  )
)
5146, 50sylib 196 . . . . . . 7  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
( X ( lt
`  R )  .0. 
\/  X  =  .0.  ) )
52 tospos 27798 . . . . . . . . 9  |-  ( R  e. Toset  ->  R  e.  Poset )
5310, 36, 523syl 20 . . . . . . . 8  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  R  e.  Poset )
544, 5, 31pleval2 15721 . . . . . . . 8  |-  ( ( R  e.  Poset  /\  X  e.  B  /\  .0.  e.  B )  ->  ( X  .<_  .0.  <->  ( X ( lt `  R )  .0.  \/  X  =  .0.  ) ) )
5553, 15, 25, 54syl3anc 1228 . . . . . . 7  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
( X  .<_  .0.  <->  ( X
( lt `  R
)  .0.  \/  X  =  .0.  ) ) )
5651, 55mpbird 232 . . . . . 6  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  X  .<_  .0.  )
57 eqid 2457 . . . . . . 7  |-  ( +g  `  R )  =  ( +g  `  R )
584, 5, 57omndadd 27848 . . . . . 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 1241 . . . . 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 16223 . . . . . 6  |-  ( ( R  e.  Grp  /\  X  e.  B )  ->  ( X ( +g  `  R ) ( ( invg `  R
) `  X )
)  =  .0.  )
6114, 15, 60syl2anc 661 . . . . 5  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
( X ( +g  `  R ) ( ( invg `  R
) `  X )
)  =  .0.  )
624, 57, 6grplid 16206 . . . . . 6  |-  ( ( R  e.  Grp  /\  ( ( invg `  R ) `  X
)  e.  B )  ->  (  .0.  ( +g  `  R ) ( ( invg `  R ) `  X
) )  =  ( ( invg `  R ) `  X
) )
6314, 18, 62syl2anc 661 . . . . 5  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
(  .0.  ( +g  `  R ) ( ( invg `  R
) `  X )
)  =  ( ( invg `  R
) `  X )
)
6459, 61, 633brtr3d 4485 . . . 4  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  .0.  .<_  ( ( invg `  R ) `
 X ) )
654, 5, 6, 7orngmul 27946 . . . 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 1237 . . 3  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  .0.  .<_  ( ( ( invg `  R
) `  X )  .x.  ( ( invg `  R ) `  X
) ) )
674, 7, 16, 12, 15, 15ringm2neg 17370 . . 3  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
( ( ( invg `  R ) `
 X )  .x.  ( ( invg `  R ) `  X
) )  =  ( X  .x.  X ) )
6866, 67breqtrd 4480 . 2  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  .0.  .<_  ( X  .x.  X ) )
699, 68pm2.61dan 791 1  |-  ( ( R  e. oRing  /\  X  e.  B )  ->  .0.  .<_  ( X  .x.  X ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    \/ w3o 972    /\ w3a 973    = wceq 1395    e. wcel 1819    C_ wss 3471   class class class wbr 4456    _I cid 4799    Or wor 4808    |` cres 5010   ` cfv 5594  (class class class)co 6296   Basecbs 14643   +g cplusg 14711   .rcmulr 14712   lecple 14718   0gc0g 14856   Posetcpo 15695   ltcplt 15696  Tosetctos 15789   Grpcgrp 16179   invgcminusg 16180   Ringcrg 17324  oMndcomnd 27839  oGrpcogrp 27840  oRingcorng 27938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-ndx 14646  df-slot 14647  df-base 14648  df-sets 14649  df-plusg 14724  df-0g 14858  df-preset 15683  df-poset 15701  df-plt 15714  df-toset 15790  df-mgm 15998  df-sgrp 16037  df-mnd 16047  df-grp 16183  df-minusg 16184  df-mgp 17268  df-ur 17280  df-ring 17326  df-omnd 27841  df-ogrp 27842  df-orng 27940
This theorem is referenced by:  orng0le1  27955
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