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Theorem ornglmulle 28407
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 28403 . . . . 5  |-  ( R  e. oRing  ->  R  e. oGrp )
31, 2syl 17 . . . 4  |-  ( ph  ->  R  e. oGrp )
4 isogrp 28303 . . . . 5  |-  ( R  e. oGrp 
<->  ( R  e.  Grp  /\  R  e. oMnd ) )
54simprbi 465 . . . 4  |-  ( R  e. oGrp  ->  R  e. oMnd )
63, 5syl 17 . . 3  |-  ( ph  ->  R  e. oMnd )
7 orngring 28402 . . . . . 6  |-  ( R  e. oRing  ->  R  e.  Ring )
81, 7syl 17 . . . . 5  |-  ( ph  ->  R  e.  Ring )
9 ringgrp 17720 . . . . 5  |-  ( R  e.  Ring  ->  R  e. 
Grp )
108, 9syl 17 . . . 4  |-  ( ph  ->  R  e.  Grp )
11 ornglmullt.b . . . . 5  |-  B  =  ( Base `  R
)
12 ornglmullt.0 . . . . 5  |-  .0.  =  ( 0g `  R )
1311, 12grpidcl 16645 . . . 4  |-  ( R  e.  Grp  ->  .0.  e.  B )
1410, 13syl 17 . . 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 17729 . . . . 5  |-  ( ( R  e.  Ring  /\  Z  e.  B  /\  Y  e.  B )  ->  ( Z  .x.  Y )  e.  B )
198, 15, 16, 18syl3anc 1264 . . . 4  |-  ( ph  ->  ( Z  .x.  Y
)  e.  B )
20 ornglmullt.2 . . . . 5  |-  ( ph  ->  X  e.  B )
2111, 17ringcl 17729 . . . . 5  |-  ( ( R  e.  Ring  /\  Z  e.  B  /\  X  e.  B )  ->  ( Z  .x.  X )  e.  B )
228, 15, 20, 21syl3anc 1264 . . . 4  |-  ( ph  ->  ( Z  .x.  X
)  e.  B )
23 eqid 2429 . . . . 5  |-  ( -g `  R )  =  (
-g `  R )
2411, 23grpsubcl 16685 . . . 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 1264 . . 3  |-  ( ph  ->  ( ( Z  .x.  Y ) ( -g `  R ) ( Z 
.x.  X ) )  e.  B )
26 orngmulle.6 . . . . 5  |-  ( ph  ->  .0.  .<_  Z )
2711, 23grpsubcl 16685 . . . . . 6  |-  ( ( R  e.  Grp  /\  Y  e.  B  /\  X  e.  B )  ->  ( Y ( -g `  R ) X )  e.  B )
2810, 16, 20, 27syl3anc 1264 . . . . 5  |-  ( ph  ->  ( Y ( -g `  R ) X )  e.  B )
2911, 12, 23grpsubid 16689 . . . . . . 7  |-  ( ( R  e.  Grp  /\  X  e.  B )  ->  ( X ( -g `  R ) X )  =  .0.  )
3010, 20, 29syl2anc 665 . . . . . 6  |-  ( ph  ->  ( X ( -g `  R ) X )  =  .0.  )
31 orngmulle.5 . . . . . . 7  |-  ( ph  ->  X  .<_  Y )
32 orngmulle.l . . . . . . . 8  |-  .<_  =  ( le `  R )
3311, 32, 23ogrpsub 28318 . . . . . . 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 1277 . . . . . 6  |-  ( ph  ->  ( X ( -g `  R ) X ) 
.<_  ( Y ( -g `  R ) X ) )
3530, 34eqbrtrrd 4448 . . . . 5  |-  ( ph  ->  .0.  .<_  ( Y (
-g `  R ) X ) )
3611, 32, 12, 17orngmul 28405 . . . . 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 1273 . . . 4  |-  ( ph  ->  .0.  .<_  ( Z  .x.  ( Y ( -g `  R
) X ) ) )
3811, 17, 23, 8, 15, 16, 20ringsubdi 17762 . . . 4  |-  ( ph  ->  ( Z  .x.  ( Y ( -g `  R
) X ) )  =  ( ( Z 
.x.  Y ) (
-g `  R )
( Z  .x.  X
) ) )
3937, 38breqtrd 4450 . . 3  |-  ( ph  ->  .0.  .<_  ( ( Z 
.x.  Y ) (
-g `  R )
( Z  .x.  X
) ) )
40 eqid 2429 . . . 4  |-  ( +g  `  R )  =  ( +g  `  R )
4111, 32, 40omndadd 28307 . . 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 1277 . 2  |-  ( ph  ->  (  .0.  ( +g  `  R ) ( Z 
.x.  X ) ) 
.<_  ( ( ( Z 
.x.  Y ) (
-g `  R )
( Z  .x.  X
) ) ( +g  `  R ) ( Z 
.x.  X ) ) )
4311, 40, 12grplid 16647 . . 3  |-  ( ( R  e.  Grp  /\  ( Z  .x.  X )  e.  B )  -> 
(  .0.  ( +g  `  R ) ( Z 
.x.  X ) )  =  ( Z  .x.  X ) )
4410, 22, 43syl2anc 665 . 2  |-  ( ph  ->  (  .0.  ( +g  `  R ) ( Z 
.x.  X ) )  =  ( Z  .x.  X ) )
4511, 40, 23grpnpcan 16697 . . 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 1264 . 2  |-  ( ph  ->  ( ( ( Z 
.x.  Y ) (
-g `  R )
( Z  .x.  X
) ) ( +g  `  R ) ( Z 
.x.  X ) )  =  ( Z  .x.  Y ) )
4742, 44, 463brtr3d 4455 1  |-  ( ph  ->  ( Z  .x.  X
)  .<_  ( Z  .x.  Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1870   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   Basecbs 15084   +g cplusg 15152   .rcmulr 15153   lecple 15159   0gc0g 15297   Grpcgrp 16620   -gcsg 16622   Ringcrg 17715  oMndcomnd 28298  oGrpcogrp 28299  oRingcorng 28397
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
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 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  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 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-plusg 15165  df-0g 15299  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-grp 16624  df-minusg 16625  df-sbg 16626  df-mgp 17659  df-ur 17671  df-ring 17717  df-omnd 28300  df-ogrp 28301  df-orng 28399
This theorem is referenced by:  ornglmullt  28409
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