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Theorem orngsqr 26406
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 26405 . . 3  |-  ( ( R  e. oRing  /\  ( X  e.  B  /\  .0.  .<_  X )  /\  ( X  e.  B  /\  .0.  .<_  X ) )  ->  .0.  .<_  ( X 
.x.  X ) )
101, 4, 4, 9syl3anc 1219 . 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 orngrng 26402 . . . . . . . 8  |-  ( R  e. oRing  ->  R  e.  Ring )
1312ad2antrr 725 . . . . . . 7  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  R  e.  Ring )
14 rnggrp 16756 . . . . . . 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 2451 . . . . . . 7  |-  ( invg `  R )  =  ( invg `  R )
185, 17grpinvcl 15685 . . . . . 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 26403 . . . . . . . . 9  |-  ( R  e. oRing  ->  R  e. oGrp )
21 isogrp 26299 . . . . . . . . . 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 15668 . . . . . . . . 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 1168 . . . . . . 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 1168 . . . . . . . . . . 11  |-  ( ( R  e. oRing  /\  X  e.  B )  ->  ( R  e. oRing  /\  .0.  e.  B  /\  X  e.  B
) )
33 eqid 2451 . . . . . . . . . . . . . 14  |-  ( lt
`  R )  =  ( lt `  R
)
346, 33pltle 15233 . . . . . . . . . . . . 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 26302 . . . . . . . . . . . . . 14  |-  ( R  e. oMnd  ->  R  e. Toset )
3923, 38syl 16 . . . . . . . . . . . . 13  |-  ( R  e. oRing  ->  R  e. Toset )
405, 6, 33tosso 15308 . . . . . . . . . . . . . . 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 4762 . . . . . . . . . . . 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 1217 . . . . . . . . . . 11  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
(  .0.  ( lt
`  R ) X  \/  .0.  =  X  \/  X ( lt
`  R )  .0.  ) )
46 3orass 968 . . . . . . . . . . 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 2460 . . . . . . . . . . . 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 26253 . . . . . . . . . 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 15237 . . . . . . . . 9  |-  ( ( R  e.  Poset  /\  X  e.  B  /\  .0.  e.  B )  ->  ( X  .<_  .0.  <->  ( X ( lt `  R )  .0.  \/  X  =  .0.  ) ) )
5957, 16, 26, 58syl3anc 1219 . . . . . . . 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 2451 . . . . . . . 8  |-  ( +g  `  R )  =  ( +g  `  R )
625, 6, 61omndadd 26303 . . . . . . 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 1219 . . . . . 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 15687 . . . . . . 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 15670 . . . . . . 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 4419 . . . . 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 26405 . . . 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 1219 . . 3  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  ->  .0.  .<_  ( ( ( invg `  R
) `  X )  .x.  ( ( invg `  R ) `  X
) ) )
725, 8, 17, 13, 16, 16rngm2neg 16795 . . 3  |-  ( ( ( R  e. oRing  /\  X  e.  B )  /\  -.  .0.  .<_  X )  -> 
( ( ( invg `  R ) `
 X )  .x.  ( ( invg `  R ) `  X
) )  =  ( X  .x.  X ) )
7371, 72breqtrd 4414 . 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 964    /\ w3a 965    = wceq 1370    e. wcel 1758    C_ wss 3426   class class class wbr 4390    _I cid 4729    Or wor 4738    |` cres 4940   ` cfv 5516  (class class class)co 6190   Basecbs 14276   +g cplusg 14340   .rcmulr 14341   lecple 14347   0gc0g 14480   Posetcpo 15212   ltcplt 15213  Tosetctos 15305   Grpcgrp 15512   invgcminusg 15513   Ringcrg 16751  oMndcomnd 26294  oGrpcogrp 26295  oRingcorng 26397
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-recs 6932  df-rdg 6966  df-er 7201  df-en 7411  df-dom 7412  df-sdom 7413  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-nn 10424  df-2 10481  df-ndx 14279  df-slot 14280  df-base 14281  df-sets 14282  df-plusg 14353  df-0g 14482  df-poset 15218  df-plt 15230  df-toset 15306  df-mnd 15517  df-grp 15647  df-minusg 15648  df-mgp 16697  df-ur 16709  df-rng 16753  df-omnd 26296  df-ogrp 26297  df-orng 26399
This theorem is referenced by:  orng0le1  26414
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