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Theorem orngsqr 26207
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 748 . . 3  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  .0.  .<_  X )  ->  R  e. oRing )
2 simplr 749 . . . 4  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  .0.  .<_  X )  ->  X  e.  B )
3 simpr 458 . . . 4  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  .0.  .<_  X )  ->  .0.  .<_  X )
42, 3jca 529 . . 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 26206 . . 3  |-  ( ( R  e. oRing  /\  ( X  e.  B  /\  .0.  .<_  X )  /\  ( X  e.  B  /\  .0.  .<_  X ) )  ->  .0.  .<_  ( X 
.x.  X ) )
101, 4, 4, 9syl3anc 1213 . 2  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  .0.  .<_  X )  ->  .0.  .<_  ( X  .x.  X ) )
11 simpll 748 . . . 4  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  R  e. oRing )
12 orngrng 26203 . . . . . . . 8  |-  ( R  e. oRing  ->  R  e.  Ring )
1312ad2antrr 720 . . . . . . 7  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  R  e.  Ring )
14 rnggrp 16640 . . . . . . 7  |-  ( R  e.  Ring  ->  R  e. 
Grp )
1513, 14syl 16 . . . . . 6  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  R  e.  Grp )
16 simplr 749 . . . . . 6  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  X  e.  B )
17 eqid 2441 . . . . . . 7  |-  ( invg `  R )  =  ( invg `  R )
185, 17grpinvcl 15576 . . . . . 6  |-  ( ( R  e.  Grp  /\  X  e.  B )  ->  ( ( invg `  R ) `  X
)  e.  B )
1915, 16, 18syl2anc 656 . . . . 5  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
( ( invg `  R ) `  X
)  e.  B )
20 orngogrp 26204 . . . . . . . . 9  |-  ( R  e. oRing  ->  R  e. oGrp )
21 isogrp 26098 . . . . . . . . . 10  |-  ( R  e. oGrp 
<->  ( R  e.  Grp  /\  R  e. oMnd ) )
2221simprbi 461 . . . . . . . . 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 15559 . . . . . . . . 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 1163 . . . . . . 7  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
( X  e.  B  /\  .0.  e.  B  /\  ( ( invg `  R ) `  X
)  e.  B ) )
28 simpl 454 . . . . . . . . . . . 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 458 . . . . . . . . . . . 12  |-  ( ( R  e. oRing  /\  X  e.  B )  ->  X  e.  B )
3228, 30, 313jca 1163 . . . . . . . . . . 11  |-  ( ( R  e. oRing  /\  X  e.  B )  ->  ( R  e. oRing  /\  .0.  e.  B  /\  X  e.  B
) )
33 eqid 2441 . . . . . . . . . . . . . 14  |-  ( lt
`  R )  =  ( lt `  R
)
346, 33pltle 15127 . . . . . . . . . . . . 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 468 . . . . . . . . . 10  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  -.  .0.  ( lt `  R ) X )
38 omndtos 26101 . . . . . . . . . . . . . 14  |-  ( R  e. oMnd  ->  R  e. Toset )
3923, 38syl 16 . . . . . . . . . . . . 13  |-  ( R  e. oRing  ->  R  e. Toset )
405, 6, 33tosso 15202 . . . . . . . . . . . . . . 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 456 . . . . . . . . . . . . 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 4660 . . . . . . . . . . . 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 1211 . . . . . . . . . . 11  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
(  .0.  ( lt
`  R ) X  \/  .0.  =  X  \/  X ( lt
`  R )  .0.  ) )
46 3orass 963 . . . . . . . . . . 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 2443 . . . . . . . . . . . 12  |-  (  .0.  =  X  <->  X  =  .0.  )
5251orbi2i 516 . . . . . . . . . . 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 26052 . . . . . . . . . 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 15131 . . . . . . . . 9  |-  ( ( R  e.  Poset  /\  X  e.  B  /\  .0.  e.  B )  ->  ( X  .<_  .0.  <->  ( X ( lt `  R )  .0.  \/  X  =  .0.  ) ) )
5957, 16, 26, 58syl3anc 1213 . . . . . . . 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 2441 . . . . . . . 8  |-  ( +g  `  R )  =  ( +g  `  R )
625, 6, 61omndadd 26102 . . . . . . 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 1213 . . . . . 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 15578 . . . . . . 7  |-  ( ( R  e.  Grp  /\  X  e.  B )  ->  ( X ( +g  `  R ) ( ( invg `  R
) `  X )
)  =  .0.  )
6515, 16, 64syl2anc 656 . . . . . 6  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
( X ( +g  `  R ) ( ( invg `  R
) `  X )
)  =  .0.  )
665, 61, 7grplid 15561 . . . . . . 7  |-  ( ( R  e.  Grp  /\  ( ( invg `  R ) `  X
)  e.  B )  ->  (  .0.  ( +g  `  R ) ( ( invg `  R ) `  X
) )  =  ( ( invg `  R ) `  X
) )
6715, 19, 66syl2anc 656 . . . . . 6  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
(  .0.  ( +g  `  R ) ( ( invg `  R
) `  X )
)  =  ( ( invg `  R
) `  X )
)
6863, 65, 673brtr3d 4318 . . . . 5  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  .0.  .<_  ( ( invg `  R ) `
 X ) )
6919, 68jca 529 . . . 4  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
( ( ( invg `  R ) `
 X )  e.  B  /\  .0.  .<_  ( ( invg `  R ) `  X
) ) )
705, 6, 7, 8orngmul 26206 . . . 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 1213 . . 3  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  .0.  .<_  ( ( ( invg `  R
) `  X )  .x.  ( ( invg `  R ) `  X
) ) )
725, 8, 17, 13, 16, 16rngm2neg 16679 . . 3  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
( ( ( invg `  R ) `
 X )  .x.  ( ( invg `  R ) `  X
) )  =  ( X  .x.  X ) )
7371, 72breqtrd 4313 . 2  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  .0.  .<_  ( X  .x.  X ) )
7410, 73pm2.61dan 784 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 959    /\ w3a 960    = wceq 1364    e. wcel 1761    C_ wss 3325   class class class wbr 4289    _I cid 4627    Or wor 4636    |` cres 4838   ` cfv 5415  (class class class)co 6090   Basecbs 14170   +g cplusg 14234   .rcmulr 14235   lecple 14241   0gc0g 14374   Posetcpo 15106   ltcplt 15107  Tosetctos 15199   Grpcgrp 15406   invgcminusg 15407   Ringcrg 16635  oMndcomnd 26093  oGrpcogrp 26094  oRingcorng 26198
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-recs 6828  df-rdg 6862  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-plusg 14247  df-0g 14376  df-poset 15112  df-plt 15124  df-toset 15200  df-mnd 15411  df-grp 15538  df-minusg 15539  df-mgp 16582  df-ur 16594  df-rng 16637  df-omnd 26095  df-ogrp 26096  df-orng 26200
This theorem is referenced by:  orng0le1  26215
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