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Theorem ornglmulle 27611
Description: In an ordered ring, multiplication with a positive does not change comparison. (Contributed by Thierry Arnoux, 10-Apr-2018.)
Hypotheses
Ref Expression
ornglmullt.b  |-  B  =  ( Base `  R
)
ornglmullt.t  |-  .x.  =  ( .r `  R )
ornglmullt.0  |-  .0.  =  ( 0g `  R )
ornglmullt.1  |-  ( ph  ->  R  e. oRing )
ornglmullt.2  |-  ( ph  ->  X  e.  B )
ornglmullt.3  |-  ( ph  ->  Y  e.  B )
ornglmullt.4  |-  ( ph  ->  Z  e.  B )
orngmulle.l  |-  .<_  =  ( le `  R )
orngmulle.5  |-  ( ph  ->  X  .<_  Y )
orngmulle.6  |-  ( ph  ->  .0.  .<_  Z )
Assertion
Ref Expression
ornglmulle  |-  ( ph  ->  ( Z  .x.  X
)  .<_  ( Z  .x.  Y ) )

Proof of Theorem ornglmulle
StepHypRef Expression
1 ornglmullt.1 . . . . 5  |-  ( ph  ->  R  e. oRing )
2 orngogrp 27607 . . . . 5  |-  ( R  e. oRing  ->  R  e. oGrp )
31, 2syl 16 . . . 4  |-  ( ph  ->  R  e. oGrp )
4 isogrp 27507 . . . . 5  |-  ( R  e. oGrp 
<->  ( R  e.  Grp  /\  R  e. oMnd ) )
54simprbi 464 . . . 4  |-  ( R  e. oGrp  ->  R  e. oMnd )
63, 5syl 16 . . 3  |-  ( ph  ->  R  e. oMnd )
7 orngring 27606 . . . . . 6  |-  ( R  e. oRing  ->  R  e.  Ring )
81, 7syl 16 . . . . 5  |-  ( ph  ->  R  e.  Ring )
9 ringgrp 17075 . . . . 5  |-  ( R  e.  Ring  ->  R  e. 
Grp )
108, 9syl 16 . . . 4  |-  ( ph  ->  R  e.  Grp )
11 ornglmullt.b . . . . 5  |-  B  =  ( Base `  R
)
12 ornglmullt.0 . . . . 5  |-  .0.  =  ( 0g `  R )
1311, 12grpidcl 15950 . . . 4  |-  ( R  e.  Grp  ->  .0.  e.  B )
1410, 13syl 16 . . 3  |-  ( ph  ->  .0.  e.  B )
15 ornglmullt.4 . . . . 5  |-  ( ph  ->  Z  e.  B )
16 ornglmullt.3 . . . . 5  |-  ( ph  ->  Y  e.  B )
17 ornglmullt.t . . . . . 6  |-  .x.  =  ( .r `  R )
1811, 17ringcl 17084 . . . . 5  |-  ( ( R  e.  Ring  /\  Z  e.  B  /\  Y  e.  B )  ->  ( Z  .x.  Y )  e.  B )
198, 15, 16, 18syl3anc 1228 . . . 4  |-  ( ph  ->  ( Z  .x.  Y
)  e.  B )
20 ornglmullt.2 . . . . 5  |-  ( ph  ->  X  e.  B )
2111, 17ringcl 17084 . . . . 5  |-  ( ( R  e.  Ring  /\  Z  e.  B  /\  X  e.  B )  ->  ( Z  .x.  X )  e.  B )
228, 15, 20, 21syl3anc 1228 . . . 4  |-  ( ph  ->  ( Z  .x.  X
)  e.  B )
23 eqid 2467 . . . . 5  |-  ( -g `  R )  =  (
-g `  R )
2411, 23grpsubcl 15990 . . . 4  |-  ( ( R  e.  Grp  /\  ( Z  .x.  Y )  e.  B  /\  ( Z  .x.  X )  e.  B )  ->  (
( Z  .x.  Y
) ( -g `  R
) ( Z  .x.  X ) )  e.  B )
2510, 19, 22, 24syl3anc 1228 . . 3  |-  ( ph  ->  ( ( Z  .x.  Y ) ( -g `  R ) ( Z 
.x.  X ) )  e.  B )
26 orngmulle.6 . . . . 5  |-  ( ph  ->  .0.  .<_  Z )
2711, 23grpsubcl 15990 . . . . . 6  |-  ( ( R  e.  Grp  /\  Y  e.  B  /\  X  e.  B )  ->  ( Y ( -g `  R ) X )  e.  B )
2810, 16, 20, 27syl3anc 1228 . . . . 5  |-  ( ph  ->  ( Y ( -g `  R ) X )  e.  B )
2911, 12, 23grpsubid 15994 . . . . . . 7  |-  ( ( R  e.  Grp  /\  X  e.  B )  ->  ( X ( -g `  R ) X )  =  .0.  )
3010, 20, 29syl2anc 661 . . . . . 6  |-  ( ph  ->  ( X ( -g `  R ) X )  =  .0.  )
31 orngmulle.5 . . . . . . 7  |-  ( ph  ->  X  .<_  Y )
32 orngmulle.l . . . . . . . 8  |-  .<_  =  ( le `  R )
3311, 32, 23ogrpsub 27522 . . . . . . 7  |-  ( ( R  e. oGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  X  e.  B )  /\  X  .<_  Y )  ->  ( X (
-g `  R ) X )  .<_  ( Y ( -g `  R
) X ) )
343, 20, 16, 20, 31, 33syl131anc 1241 . . . . . 6  |-  ( ph  ->  ( X ( -g `  R ) X ) 
.<_  ( Y ( -g `  R ) X ) )
3530, 34eqbrtrrd 4475 . . . . 5  |-  ( ph  ->  .0.  .<_  ( Y (
-g `  R ) X ) )
3611, 32, 12, 17orngmul 27609 . . . . 5  |-  ( ( R  e. oRing  /\  ( Z  e.  B  /\  .0.  .<_  Z )  /\  ( ( Y (
-g `  R ) X )  e.  B  /\  .0.  .<_  ( Y (
-g `  R ) X ) ) )  ->  .0.  .<_  ( Z 
.x.  ( Y (
-g `  R ) X ) ) )
371, 15, 26, 28, 35, 36syl122anc 1237 . . . 4  |-  ( ph  ->  .0.  .<_  ( Z  .x.  ( Y ( -g `  R
) X ) ) )
3811, 17, 23, 8, 15, 16, 20ringsubdi 17117 . . . 4  |-  ( ph  ->  ( Z  .x.  ( Y ( -g `  R
) X ) )  =  ( ( Z 
.x.  Y ) (
-g `  R )
( Z  .x.  X
) ) )
3937, 38breqtrd 4477 . . 3  |-  ( ph  ->  .0.  .<_  ( ( Z 
.x.  Y ) (
-g `  R )
( Z  .x.  X
) ) )
40 eqid 2467 . . . 4  |-  ( +g  `  R )  =  ( +g  `  R )
4111, 32, 40omndadd 27511 . . 3  |-  ( ( R  e. oMnd  /\  (  .0.  e.  B  /\  (
( Z  .x.  Y
) ( -g `  R
) ( Z  .x.  X ) )  e.  B  /\  ( Z 
.x.  X )  e.  B )  /\  .0.  .<_  ( ( Z  .x.  Y ) ( -g `  R ) ( Z 
.x.  X ) ) )  ->  (  .0.  ( +g  `  R ) ( Z  .x.  X
) )  .<_  ( ( ( Z  .x.  Y
) ( -g `  R
) ( Z  .x.  X ) ) ( +g  `  R ) ( Z  .x.  X
) ) )
426, 14, 25, 22, 39, 41syl131anc 1241 . 2  |-  ( ph  ->  (  .0.  ( +g  `  R ) ( Z 
.x.  X ) ) 
.<_  ( ( ( Z 
.x.  Y ) (
-g `  R )
( Z  .x.  X
) ) ( +g  `  R ) ( Z 
.x.  X ) ) )
4311, 40, 12grplid 15952 . . 3  |-  ( ( R  e.  Grp  /\  ( Z  .x.  X )  e.  B )  -> 
(  .0.  ( +g  `  R ) ( Z 
.x.  X ) )  =  ( Z  .x.  X ) )
4410, 22, 43syl2anc 661 . 2  |-  ( ph  ->  (  .0.  ( +g  `  R ) ( Z 
.x.  X ) )  =  ( Z  .x.  X ) )
4511, 40, 23grpnpcan 16002 . . 3  |-  ( ( R  e.  Grp  /\  ( Z  .x.  Y )  e.  B  /\  ( Z  .x.  X )  e.  B )  ->  (
( ( Z  .x.  Y ) ( -g `  R ) ( Z 
.x.  X ) ) ( +g  `  R
) ( Z  .x.  X ) )  =  ( Z  .x.  Y
) )
4610, 19, 22, 45syl3anc 1228 . 2  |-  ( ph  ->  ( ( ( Z 
.x.  Y ) (
-g `  R )
( Z  .x.  X
) ) ( +g  `  R ) ( Z 
.x.  X ) )  =  ( Z  .x.  Y ) )
4742, 44, 463brtr3d 4482 1  |-  ( ph  ->  ( Z  .x.  X
)  .<_  ( Z  .x.  Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   Basecbs 14507   +g cplusg 14572   .rcmulr 14573   lecple 14579   0gc0g 14712   Grpcgrp 15925   -gcsg 15927   Ringcrg 17070  oMndcomnd 27502  oGrpcogrp 27503  oRingcorng 27601
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-1st 6795  df-2nd 6796  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-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15929  df-minusg 15930  df-sbg 15931  df-mgp 17014  df-ur 17026  df-ring 17072  df-omnd 27504  df-ogrp 27505  df-orng 27603
This theorem is referenced by:  ornglmullt  27613
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