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Theorem orngsqr 27619
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 754 . . . 4  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  .0.  .<_  X )  ->  X  e.  B )
3 simpr 461 . . . 4  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  .0.  .<_  X )  ->  .0.  .<_  X )
42, 3jca 532 . . 3  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  .0.  .<_  X )  ->  ( X  e.  B  /\  .0.  .<_  X ) )
5 orngmul.0 . . . 4  |-  B  =  ( Base `  R
)
6 orngmul.1 . . . 4  |-  .<_  =  ( le `  R )
7 orngmul.2 . . . 4  |-  .0.  =  ( 0g `  R )
8 orngmul.3 . . . 4  |-  .x.  =  ( .r `  R )
95, 6, 7, 8orngmul 27618 . . 3  |-  ( ( R  e. oRing  /\  ( X  e.  B  /\  .0.  .<_  X )  /\  ( X  e.  B  /\  .0.  .<_  X ) )  ->  .0.  .<_  ( X 
.x.  X ) )
101, 4, 4, 9syl3anc 1228 . 2  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  .0.  .<_  X )  ->  .0.  .<_  ( X  .x.  X ) )
11 simpll 753 . . . 4  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  R  e. oRing )
12 orngring 27615 . . . . . . . 8  |-  ( R  e. oRing  ->  R  e.  Ring )
1312ad2antrr 725 . . . . . . 7  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  R  e.  Ring )
14 ringgrp 17075 . . . . . . 7  |-  ( R  e.  Ring  ->  R  e. 
Grp )
1513, 14syl 16 . . . . . 6  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  R  e.  Grp )
16 simplr 754 . . . . . 6  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  X  e.  B )
17 eqid 2467 . . . . . . 7  |-  ( invg `  R )  =  ( invg `  R )
185, 17grpinvcl 15967 . . . . . 6  |-  ( ( R  e.  Grp  /\  X  e.  B )  ->  ( ( invg `  R ) `  X
)  e.  B )
1915, 16, 18syl2anc 661 . . . . 5  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
( ( invg `  R ) `  X
)  e.  B )
20 orngogrp 27616 . . . . . . . . 9  |-  ( R  e. oRing  ->  R  e. oGrp )
21 isogrp 27516 . . . . . . . . . 10  |-  ( R  e. oGrp 
<->  ( R  e.  Grp  /\  R  e. oMnd ) )
2221simprbi 464 . . . . . . . . 9  |-  ( R  e. oGrp  ->  R  e. oMnd )
2320, 22syl 16 . . . . . . . 8  |-  ( R  e. oRing  ->  R  e. oMnd )
2411, 23syl 16 . . . . . . 7  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  R  e. oMnd )
255, 7grpidcl 15950 . . . . . . . . 9  |-  ( R  e.  Grp  ->  .0.  e.  B )
2615, 25syl 16 . . . . . . . 8  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  .0.  e.  B )
2716, 26, 193jca 1176 . . . . . . 7  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
( X  e.  B  /\  .0.  e.  B  /\  ( ( invg `  R ) `  X
)  e.  B ) )
28 simpl 457 . . . . . . . . . . . 12  |-  ( ( R  e. oRing  /\  X  e.  B )  ->  R  e. oRing )
2912, 14, 253syl 20 . . . . . . . . . . . . 13  |-  ( R  e. oRing  ->  .0.  e.  B
)
3028, 29syl 16 . . . . . . . . . . . 12  |-  ( ( R  e. oRing  /\  X  e.  B )  ->  .0.  e.  B )
31 simpr 461 . . . . . . . . . . . 12  |-  ( ( R  e. oRing  /\  X  e.  B )  ->  X  e.  B )
3228, 30, 313jca 1176 . . . . . . . . . . 11  |-  ( ( R  e. oRing  /\  X  e.  B )  ->  ( R  e. oRing  /\  .0.  e.  B  /\  X  e.  B
) )
33 eqid 2467 . . . . . . . . . . . . . 14  |-  ( lt
`  R )  =  ( lt `  R
)
346, 33pltle 15465 . . . . . . . . . . . . 13  |-  ( ( R  e. oRing  /\  .0.  e.  B  /\  X  e.  B
)  ->  (  .0.  ( lt `  R ) X  ->  .0.  .<_  X ) )
3534con3d 133 . . . . . . . . . . . 12  |-  ( ( R  e. oRing  /\  .0.  e.  B  /\  X  e.  B
)  ->  ( -.  .0.  .<_  X  ->  -.  .0.  ( lt `  R
) X ) )
3635imp 429 . . . . . . . . . . 11  |-  ( ( ( R  e. oRing  /\  .0.  e.  B  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  -.  .0.  ( lt `  R ) X )
3732, 36sylan 471 . . . . . . . . . 10  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  -.  .0.  ( lt `  R ) X )
38 omndtos 27519 . . . . . . . . . . . . . 14  |-  ( R  e. oMnd  ->  R  e. Toset )
3923, 38syl 16 . . . . . . . . . . . . 13  |-  ( R  e. oRing  ->  R  e. Toset )
405, 6, 33tosso 15540 . . . . . . . . . . . . . . 15  |-  ( R  e. Toset  ->  ( R  e. Toset  <->  ( ( lt `  R
)  Or  B  /\  (  _I  |`  B ) 
C_  .<_  ) ) )
4140ibi 241 . . . . . . . . . . . . . 14  |-  ( R  e. Toset  ->  ( ( lt
`  R )  Or  B  /\  (  _I  |`  B )  C_  .<_  ) )
4241simpld 459 . . . . . . . . . . . . 13  |-  ( R  e. Toset  ->  ( lt `  R )  Or  B
)
4311, 39, 423syl 20 . . . . . . . . . . . 12  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
( lt `  R
)  Or  B )
44 solin 4829 . . . . . . . . . . . 12  |-  ( ( ( lt `  R
)  Or  B  /\  (  .0.  e.  B  /\  X  e.  B )
)  ->  (  .0.  ( lt `  R ) X  \/  .0.  =  X  \/  X ( lt `  R )  .0.  ) )
4543, 26, 16, 44syl12anc 1226 . . . . . . . . . . 11  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
(  .0.  ( lt
`  R ) X  \/  .0.  =  X  \/  X ( lt
`  R )  .0.  ) )
46 3orass 976 . . . . . . . . . . 11  |-  ( (  .0.  ( lt `  R ) X  \/  .0.  =  X  \/  X
( lt `  R
)  .0.  )  <->  (  .0.  ( lt `  R ) X  \/  (  .0.  =  X  \/  X
( lt `  R
)  .0.  ) ) )
4745, 46sylib 196 . . . . . . . . . 10  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
(  .0.  ( lt
`  R ) X  \/  (  .0.  =  X  \/  X ( lt `  R )  .0.  ) ) )
48 orel1 382 . . . . . . . . . 10  |-  ( -.  .0.  ( lt `  R ) X  -> 
( (  .0.  ( lt `  R ) X  \/  (  .0.  =  X  \/  X ( lt `  R )  .0.  ) )  ->  (  .0.  =  X  \/  X
( lt `  R
)  .0.  ) ) )
4937, 47, 48sylc 60 . . . . . . . . 9  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
(  .0.  =  X  \/  X ( lt
`  R )  .0.  ) )
50 orcom 387 . . . . . . . . . . 11  |-  ( (  .0.  =  X  \/  X ( lt `  R )  .0.  )  <->  ( X ( lt `  R )  .0.  \/  .0.  =  X ) )
51 eqcom 2476 . . . . . . . . . . . 12  |-  (  .0.  =  X  <->  X  =  .0.  )
5251orbi2i 519 . . . . . . . . . . 11  |-  ( ( X ( lt `  R )  .0.  \/  .0.  =  X )  <->  ( X
( lt `  R
)  .0.  \/  X  =  .0.  ) )
5350, 52bitri 249 . . . . . . . . . 10  |-  ( (  .0.  =  X  \/  X ( lt `  R )  .0.  )  <->  ( X ( lt `  R )  .0.  \/  X  =  .0.  )
)
5453imbi2i 312 . . . . . . . . 9  |-  ( ( ( ( R  e. oRing  /\  X  e.  B
)  /\  -.  .0.  .<_  X )  ->  (  .0.  =  X  \/  X
( lt `  R
)  .0.  ) )  <-> 
( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
( X ( lt
`  R )  .0. 
\/  X  =  .0.  ) ) )
5549, 54mpbi 208 . . . . . . . 8  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
( X ( lt
`  R )  .0. 
\/  X  =  .0.  ) )
56 tospos 27470 . . . . . . . . . 10  |-  ( R  e. Toset  ->  R  e.  Poset )
5711, 39, 563syl 20 . . . . . . . . 9  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  R  e.  Poset )
585, 6, 33pleval2 15469 . . . . . . . . 9  |-  ( ( R  e.  Poset  /\  X  e.  B  /\  .0.  e.  B )  ->  ( X  .<_  .0.  <->  ( X ( lt `  R )  .0.  \/  X  =  .0.  ) ) )
5957, 16, 26, 58syl3anc 1228 . . . . . . . 8  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
( X  .<_  .0.  <->  ( X
( lt `  R
)  .0.  \/  X  =  .0.  ) ) )
6055, 59mpbird 232 . . . . . . 7  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  X  .<_  .0.  )
61 eqid 2467 . . . . . . . 8  |-  ( +g  `  R )  =  ( +g  `  R )
625, 6, 61omndadd 27520 . . . . . . 7  |-  ( ( 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
) ) )
6324, 27, 60, 62syl3anc 1228 . . . . . 6  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
( X ( +g  `  R ) ( ( invg `  R
) `  X )
)  .<_  (  .0.  ( +g  `  R ) ( ( invg `  R ) `  X
) ) )
645, 61, 7, 17grprinv 15969 . . . . . . 7  |-  ( ( R  e.  Grp  /\  X  e.  B )  ->  ( X ( +g  `  R ) ( ( invg `  R
) `  X )
)  =  .0.  )
6515, 16, 64syl2anc 661 . . . . . 6  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
( X ( +g  `  R ) ( ( invg `  R
) `  X )
)  =  .0.  )
665, 61, 7grplid 15952 . . . . . . 7  |-  ( ( R  e.  Grp  /\  ( ( invg `  R ) `  X
)  e.  B )  ->  (  .0.  ( +g  `  R ) ( ( invg `  R ) `  X
) )  =  ( ( invg `  R ) `  X
) )
6715, 19, 66syl2anc 661 . . . . . 6  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
(  .0.  ( +g  `  R ) ( ( invg `  R
) `  X )
)  =  ( ( invg `  R
) `  X )
)
6863, 65, 673brtr3d 4482 . . . . 5  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  .0.  .<_  ( ( invg `  R ) `
 X ) )
6919, 68jca 532 . . . 4  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
( ( ( invg `  R ) `
 X )  e.  B  /\  .0.  .<_  ( ( invg `  R ) `  X
) ) )
705, 6, 7, 8orngmul 27618 . . . 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
) ) )
7111, 69, 69, 70syl3anc 1228 . . 3  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  .0.  .<_  ( ( ( invg `  R
) `  X )  .x.  ( ( invg `  R ) `  X
) ) )
725, 8, 17, 13, 16, 16ringm2neg 17116 . . 3  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
( ( ( invg `  R ) `
 X )  .x.  ( ( invg `  R ) `  X
) )  =  ( X  .x.  X ) )
7371, 72breqtrd 4477 . 2  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  .0.  .<_  ( X  .x.  X ) )
7410, 73pm2.61dan 789 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 1379    e. wcel 1767    C_ wss 3481   class class class wbr 4453    _I cid 4796    Or wor 4805    |` cres 5007   ` cfv 5594  (class class class)co 6295   Basecbs 14507   +g cplusg 14572   .rcmulr 14573   lecple 14579   0gc0g 14712   Posetcpo 15444   ltcplt 15445  Tosetctos 15537   Grpcgrp 15925   invgcminusg 15926   Ringcrg 17070  oMndcomnd 27511  oGrpcogrp 27512  oRingcorng 27610
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-plusg 14585  df-0g 14714  df-poset 15450  df-plt 15462  df-toset 15538  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15929  df-minusg 15930  df-mgp 17014  df-ur 17026  df-ring 17072  df-omnd 27513  df-ogrp 27514  df-orng 27612
This theorem is referenced by:  orng0le1  27627
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