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Theorem orngrmulle 26442
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
orngrmulle  |-  ( ph  ->  ( X  .x.  Z
)  .<_  ( Y  .x.  Z ) )

Proof of Theorem orngrmulle
StepHypRef Expression
1 ornglmullt.1 . . . . 5  |-  ( ph  ->  R  e. oRing )
2 orngogrp 26437 . . . . 5  |-  ( R  e. oRing  ->  R  e. oGrp )
31, 2syl 16 . . . 4  |-  ( ph  ->  R  e. oGrp )
4 isogrp 26337 . . . . 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 orngrng 26436 . . . . . 6  |-  ( R  e. oRing  ->  R  e.  Ring )
81, 7syl 16 . . . . 5  |-  ( ph  ->  R  e.  Ring )
9 rnggrp 16783 . . . . 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 15689 . . . 4  |-  ( R  e.  Grp  ->  .0.  e.  B )
1410, 13syl 16 . . 3  |-  ( ph  ->  .0.  e.  B )
15 ornglmullt.3 . . . . 5  |-  ( ph  ->  Y  e.  B )
16 ornglmullt.4 . . . . 5  |-  ( ph  ->  Z  e.  B )
17 ornglmullt.t . . . . . 6  |-  .x.  =  ( .r `  R )
1811, 17rngcl 16791 . . . . 5  |-  ( ( R  e.  Ring  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  .x.  Z )  e.  B )
198, 15, 16, 18syl3anc 1219 . . . 4  |-  ( ph  ->  ( Y  .x.  Z
)  e.  B )
20 ornglmullt.2 . . . . 5  |-  ( ph  ->  X  e.  B )
2111, 17rngcl 16791 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  .x.  Z )  e.  B )
228, 20, 16, 21syl3anc 1219 . . . 4  |-  ( ph  ->  ( X  .x.  Z
)  e.  B )
23 eqid 2454 . . . . 5  |-  ( -g `  R )  =  (
-g `  R )
2411, 23grpsubcl 15729 . . . 4  |-  ( ( R  e.  Grp  /\  ( Y  .x.  Z )  e.  B  /\  ( X  .x.  Z )  e.  B )  ->  (
( Y  .x.  Z
) ( -g `  R
) ( X  .x.  Z ) )  e.  B )
2510, 19, 22, 24syl3anc 1219 . . 3  |-  ( ph  ->  ( ( Y  .x.  Z ) ( -g `  R ) ( X 
.x.  Z ) )  e.  B )
2611, 23grpsubcl 15729 . . . . . 6  |-  ( ( R  e.  Grp  /\  Y  e.  B  /\  X  e.  B )  ->  ( Y ( -g `  R ) X )  e.  B )
2710, 15, 20, 26syl3anc 1219 . . . . 5  |-  ( ph  ->  ( Y ( -g `  R ) X )  e.  B )
2811, 12, 23grpsubid 15733 . . . . . . 7  |-  ( ( R  e.  Grp  /\  X  e.  B )  ->  ( X ( -g `  R ) X )  =  .0.  )
2910, 20, 28syl2anc 661 . . . . . 6  |-  ( ph  ->  ( X ( -g `  R ) X )  =  .0.  )
30 orngmulle.5 . . . . . . 7  |-  ( ph  ->  X  .<_  Y )
31 orngmulle.l . . . . . . . 8  |-  .<_  =  ( le `  R )
3211, 31, 23ogrpsub 26352 . . . . . . 7  |-  ( ( R  e. oGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  X  e.  B )  /\  X  .<_  Y )  ->  ( X (
-g `  R ) X )  .<_  ( Y ( -g `  R
) X ) )
333, 20, 15, 20, 30, 32syl131anc 1232 . . . . . 6  |-  ( ph  ->  ( X ( -g `  R ) X ) 
.<_  ( Y ( -g `  R ) X ) )
3429, 33eqbrtrrd 4425 . . . . 5  |-  ( ph  ->  .0.  .<_  ( Y (
-g `  R ) X ) )
35 orngmulle.6 . . . . 5  |-  ( ph  ->  .0.  .<_  Z )
3611, 31, 12, 17orngmul 26439 . . . . 5  |-  ( ( R  e. oRing  /\  (
( Y ( -g `  R ) X )  e.  B  /\  .0.  .<_  ( Y ( -g `  R
) X ) )  /\  ( Z  e.  B  /\  .0.  .<_  Z ) )  ->  .0.  .<_  ( ( Y (
-g `  R ) X )  .x.  Z
) )
371, 27, 34, 16, 35, 36syl122anc 1228 . . . 4  |-  ( ph  ->  .0.  .<_  ( ( Y ( -g `  R
) X )  .x.  Z ) )
3811, 17, 23, 8, 15, 20, 16rngsubdir 16824 . . . 4  |-  ( ph  ->  ( ( Y (
-g `  R ) X )  .x.  Z
)  =  ( ( Y  .x.  Z ) ( -g `  R
) ( X  .x.  Z ) ) )
3937, 38breqtrd 4427 . . 3  |-  ( ph  ->  .0.  .<_  ( ( Y 
.x.  Z ) (
-g `  R )
( X  .x.  Z
) ) )
40 eqid 2454 . . . 4  |-  ( +g  `  R )  =  ( +g  `  R )
4111, 31, 40omndadd 26341 . . 3  |-  ( ( R  e. oMnd  /\  (  .0.  e.  B  /\  (
( Y  .x.  Z
) ( -g `  R
) ( X  .x.  Z ) )  e.  B  /\  ( X 
.x.  Z )  e.  B )  /\  .0.  .<_  ( ( Y  .x.  Z ) ( -g `  R ) ( X 
.x.  Z ) ) )  ->  (  .0.  ( +g  `  R ) ( X  .x.  Z
) )  .<_  ( ( ( Y  .x.  Z
) ( -g `  R
) ( X  .x.  Z ) ) ( +g  `  R ) ( X  .x.  Z
) ) )
426, 14, 25, 22, 39, 41syl131anc 1232 . 2  |-  ( ph  ->  (  .0.  ( +g  `  R ) ( X 
.x.  Z ) ) 
.<_  ( ( ( Y 
.x.  Z ) (
-g `  R )
( X  .x.  Z
) ) ( +g  `  R ) ( X 
.x.  Z ) ) )
4311, 40, 12grplid 15691 . . 3  |-  ( ( R  e.  Grp  /\  ( X  .x.  Z )  e.  B )  -> 
(  .0.  ( +g  `  R ) ( X 
.x.  Z ) )  =  ( X  .x.  Z ) )
4410, 22, 43syl2anc 661 . 2  |-  ( ph  ->  (  .0.  ( +g  `  R ) ( X 
.x.  Z ) )  =  ( X  .x.  Z ) )
4511, 40, 23grpnpcan 15740 . . 3  |-  ( ( R  e.  Grp  /\  ( Y  .x.  Z )  e.  B  /\  ( X  .x.  Z )  e.  B )  ->  (
( ( Y  .x.  Z ) ( -g `  R ) ( X 
.x.  Z ) ) ( +g  `  R
) ( X  .x.  Z ) )  =  ( Y  .x.  Z
) )
4610, 19, 22, 45syl3anc 1219 . 2  |-  ( ph  ->  ( ( ( Y 
.x.  Z ) (
-g `  R )
( X  .x.  Z
) ) ( +g  `  R ) ( X 
.x.  Z ) )  =  ( Y  .x.  Z ) )
4742, 44, 463brtr3d 4432 1  |-  ( ph  ->  ( X  .x.  Z
)  .<_  ( Y  .x.  Z ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   class class class wbr 4403   ` cfv 5529  (class class class)co 6203   Basecbs 14296   +g cplusg 14361   .rcmulr 14362   lecple 14368   0gc0g 14501   Grpcgrp 15533   -gcsg 15536   Ringcrg 16778  oMndcomnd 26332  oGrpcogrp 26333  oRingcorng 26431
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 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-nn 10438  df-2 10495  df-ndx 14299  df-slot 14300  df-base 14301  df-sets 14302  df-plusg 14374  df-0g 14503  df-mnd 15538  df-grp 15668  df-minusg 15669  df-sbg 15670  df-mgp 16724  df-ur 16736  df-rng 16780  df-omnd 26334  df-ogrp 26335  df-orng 26433
This theorem is referenced by:  orngrmullt  26444
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