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Theorem orngsqr 28576
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 758 . . 3  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  .0.  .<_  X )  ->  R  e. oRing )
2 simplr 760 . . 3  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  .0.  .<_  X )  ->  X  e.  B )
3 simpr 462 . . 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 28575 . . 3  |-  ( ( R  e. oRing  /\  ( X  e.  B  /\  .0.  .<_  X )  /\  ( X  e.  B  /\  .0.  .<_  X ) )  ->  .0.  .<_  ( X 
.x.  X ) )
91, 2, 3, 2, 3, 8syl122anc 1273 . 2  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  .0.  .<_  X )  ->  .0.  .<_  ( X  .x.  X ) )
10 simpll 758 . . . 4  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  R  e. oRing )
11 orngring 28572 . . . . . . 7  |-  ( R  e. oRing  ->  R  e.  Ring )
1211ad2antrr 730 . . . . . 6  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  R  e.  Ring )
13 ringgrp 17785 . . . . . 6  |-  ( R  e.  Ring  ->  R  e. 
Grp )
1412, 13syl 17 . . . . 5  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  R  e.  Grp )
15 simplr 760 . . . . 5  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  X  e.  B )
16 eqid 2422 . . . . . 6  |-  ( invg `  R )  =  ( invg `  R )
174, 16grpinvcl 16711 . . . . 5  |-  ( ( R  e.  Grp  /\  X  e.  B )  ->  ( ( invg `  R ) `  X
)  e.  B )
1814, 15, 17syl2anc 665 . . . 4  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
( ( invg `  R ) `  X
)  e.  B )
19 orngogrp 28573 . . . . . . . 8  |-  ( R  e. oRing  ->  R  e. oGrp )
20 isogrp 28473 . . . . . . . . 9  |-  ( R  e. oGrp 
<->  ( R  e.  Grp  /\  R  e. oMnd ) )
2120simprbi 465 . . . . . . . 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 16694 . . . . . . 7  |-  ( R  e.  Grp  ->  .0.  e.  B )
2514, 24syl 17 . . . . . 6  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  .0.  e.  B )
26 simpl 458 . . . . . . . . . . 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 462 . . . . . . . . . . 11  |-  ( ( R  e. oRing  /\  X  e.  B )  ->  X  e.  B )
3026, 28, 293jca 1185 . . . . . . . . . 10  |-  ( ( R  e. oRing  /\  X  e.  B )  ->  ( R  e. oRing  /\  .0.  e.  B  /\  X  e.  B
) )
31 eqid 2422 . . . . . . . . . . . 12  |-  ( lt
`  R )  =  ( lt `  R
)
325, 31pltle 16207 . . . . . . . . . . 11  |-  ( ( R  e. oRing  /\  .0.  e.  B  /\  X  e.  B
)  ->  (  .0.  ( lt `  R ) X  ->  .0.  .<_  X ) )
3332con3dimp 442 . . . . . . . . . 10  |-  ( ( ( R  e. oRing  /\  .0.  e.  B  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  -.  .0.  ( lt `  R ) X )
3430, 33sylan 473 . . . . . . . . 9  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  -.  .0.  ( lt `  R ) X )
35 omndtos 28476 . . . . . . . . . . . . 13  |-  ( R  e. oMnd  ->  R  e. Toset )
3622, 35syl 17 . . . . . . . . . . . 12  |-  ( R  e. oRing  ->  R  e. Toset )
374, 5, 31tosso 16282 . . . . . . . . . . . . . 14  |-  ( R  e. Toset  ->  ( R  e. Toset  <->  ( ( lt `  R
)  Or  B  /\  (  _I  |`  B ) 
C_  .<_  ) ) )
3837ibi 244 . . . . . . . . . . . . 13  |-  ( R  e. Toset  ->  ( ( lt
`  R )  Or  B  /\  (  _I  |`  B )  C_  .<_  ) )
3938simpld 460 . . . . . . . . . . . 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 4797 . . . . . . . . . . 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 1262 . . . . . . . . . 10  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
(  .0.  ( lt
`  R ) X  \/  .0.  =  X  \/  X ( lt
`  R )  .0.  ) )
43 3orass 985 . . . . . . . . . 10  |-  ( (  .0.  ( lt `  R ) X  \/  .0.  =  X  \/  X
( lt `  R
)  .0.  )  <->  (  .0.  ( lt `  R ) X  \/  (  .0.  =  X  \/  X
( lt `  R
)  .0.  ) ) )
4442, 43sylib 199 . . . . . . . . 9  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
(  .0.  ( lt
`  R ) X  \/  (  .0.  =  X  \/  X ( lt `  R )  .0.  ) ) )
45 orel1 383 . . . . . . . . 9  |-  ( -.  .0.  ( lt `  R ) X  -> 
( (  .0.  ( lt `  R ) X  \/  (  .0.  =  X  \/  X ( lt `  R )  .0.  ) )  ->  (  .0.  =  X  \/  X
( lt `  R
)  .0.  ) ) )
4634, 44, 45sylc 62 . . . . . . . 8  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
(  .0.  =  X  \/  X ( lt
`  R )  .0.  ) )
47 orcom 388 . . . . . . . . 9  |-  ( (  .0.  =  X  \/  X ( lt `  R )  .0.  )  <->  ( X ( lt `  R )  .0.  \/  .0.  =  X ) )
48 eqcom 2431 . . . . . . . . . 10  |-  (  .0.  =  X  <->  X  =  .0.  )
4948orbi2i 521 . . . . . . . . 9  |-  ( ( X ( lt `  R )  .0.  \/  .0.  =  X )  <->  ( X
( lt `  R
)  .0.  \/  X  =  .0.  ) )
5047, 49bitri 252 . . . . . . . 8  |-  ( (  .0.  =  X  \/  X ( lt `  R )  .0.  )  <->  ( X ( lt `  R )  .0.  \/  X  =  .0.  )
)
5146, 50sylib 199 . . . . . . 7  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
( X ( lt
`  R )  .0. 
\/  X  =  .0.  ) )
52 tospos 28427 . . . . . . . . 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 16211 . . . . . . . 8  |-  ( ( R  e.  Poset  /\  X  e.  B  /\  .0.  e.  B )  ->  ( X  .<_  .0.  <->  ( X ( lt `  R )  .0.  \/  X  =  .0.  ) ) )
5553, 15, 25, 54syl3anc 1264 . . . . . . 7  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
( X  .<_  .0.  <->  ( X
( lt `  R
)  .0.  \/  X  =  .0.  ) ) )
5651, 55mpbird 235 . . . . . 6  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  X  .<_  .0.  )
57 eqid 2422 . . . . . . 7  |-  ( +g  `  R )  =  ( +g  `  R )
584, 5, 57omndadd 28477 . . . . . 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 1277 . . . . 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 16713 . . . . . 6  |-  ( ( R  e.  Grp  /\  X  e.  B )  ->  ( X ( +g  `  R ) ( ( invg `  R
) `  X )
)  =  .0.  )
6114, 15, 60syl2anc 665 . . . . 5  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
( X ( +g  `  R ) ( ( invg `  R
) `  X )
)  =  .0.  )
624, 57, 6grplid 16696 . . . . . 6  |-  ( ( R  e.  Grp  /\  ( ( invg `  R ) `  X
)  e.  B )  ->  (  .0.  ( +g  `  R ) ( ( invg `  R ) `  X
) )  =  ( ( invg `  R ) `  X
) )
6314, 18, 62syl2anc 665 . . . . 5  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
(  .0.  ( +g  `  R ) ( ( invg `  R
) `  X )
)  =  ( ( invg `  R
) `  X )
)
6459, 61, 633brtr3d 4453 . . . 4  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  .0.  .<_  ( ( invg `  R ) `
 X ) )
654, 5, 6, 7orngmul 28575 . . . 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 1273 . . 3  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  .0.  .<_  ( ( ( invg `  R
) `  X )  .x.  ( ( invg `  R ) `  X
) ) )
674, 7, 16, 12, 15, 15ringm2neg 17826 . . 3  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
( ( ( invg `  R ) `
 X )  .x.  ( ( invg `  R ) `  X
) )  =  ( X  .x.  X ) )
6866, 67breqtrd 4448 . 2  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  .0.  .<_  ( X  .x.  X ) )
699, 68pm2.61dan 798 1  |-  ( ( R  e. oRing  /\  X  e.  B )  ->  .0.  .<_  ( X  .x.  X ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    \/ w3o 981    /\ w3a 982    = wceq 1437    e. wcel 1872    C_ wss 3436   class class class wbr 4423    _I cid 4763    Or wor 4773    |` cres 4855   ` cfv 5601  (class class class)co 6306   Basecbs 15121   +g cplusg 15190   .rcmulr 15191   lecple 15197   0gc0g 15338   Posetcpo 16185   ltcplt 16186  Tosetctos 16279   Grpcgrp 16669   invgcminusg 16670   Ringcrg 17780  oMndcomnd 28468  oGrpcogrp 28469  oRingcorng 28567
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6598  ax-cnex 9603  ax-resscn 9604  ax-1cn 9605  ax-icn 9606  ax-addcl 9607  ax-addrcl 9608  ax-mulcl 9609  ax-mulrcl 9610  ax-mulcom 9611  ax-addass 9612  ax-mulass 9613  ax-distr 9614  ax-i2m1 9615  ax-1ne0 9616  ax-1rid 9617  ax-rnegex 9618  ax-rrecex 9619  ax-cnre 9620  ax-pre-lttri 9621  ax-pre-lttrn 9622  ax-pre-ltadd 9623  ax-pre-mulgt0 9624
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6268  df-ov 6309  df-oprab 6310  df-mpt2 6311  df-om 6708  df-wrecs 7040  df-recs 7102  df-rdg 7140  df-er 7375  df-en 7582  df-dom 7583  df-sdom 7584  df-pnf 9685  df-mnf 9686  df-xr 9687  df-ltxr 9688  df-le 9689  df-sub 9870  df-neg 9871  df-nn 10618  df-2 10676  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-plusg 15203  df-0g 15340  df-preset 16173  df-poset 16191  df-plt 16204  df-toset 16280  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-grp 16673  df-minusg 16674  df-mgp 17724  df-ur 17736  df-ring 17782  df-omnd 28470  df-ogrp 28471  df-orng 28569
This theorem is referenced by:  orng0le1  28584
  Copyright terms: Public domain W3C validator