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Theorem orngrmullt 27447
Description: In an ordered ring, multiplication with a positive does not change strict comparison. (Contributed by Thierry Arnoux, 9-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 )
ornglmullt.l  |-  .<  =  ( lt `  R )
ornglmullt.d  |-  ( ph  ->  R  e.  DivRing )
ornglmullt.5  |-  ( ph  ->  X  .<  Y )
ornglmullt.6  |-  ( ph  ->  .0.  .<  Z )
Assertion
Ref Expression
orngrmullt  |-  ( ph  ->  ( X  .x.  Z
)  .<  ( Y  .x.  Z ) )

Proof of Theorem orngrmullt
StepHypRef Expression
1 ornglmullt.b . . 3  |-  B  =  ( Base `  R
)
2 ornglmullt.t . . 3  |-  .x.  =  ( .r `  R )
3 ornglmullt.0 . . 3  |-  .0.  =  ( 0g `  R )
4 ornglmullt.1 . . 3  |-  ( ph  ->  R  e. oRing )
5 ornglmullt.2 . . 3  |-  ( ph  ->  X  e.  B )
6 ornglmullt.3 . . 3  |-  ( ph  ->  Y  e.  B )
7 ornglmullt.4 . . 3  |-  ( ph  ->  Z  e.  B )
8 eqid 2460 . . 3  |-  ( le
`  R )  =  ( le `  R
)
9 ornglmullt.5 . . . 4  |-  ( ph  ->  X  .<  Y )
10 ornglmullt.l . . . . . 6  |-  .<  =  ( lt `  R )
118, 10pltle 15437 . . . . 5  |-  ( ( R  e. oRing  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  ->  X
( le `  R
) Y ) )
1211imp 429 . . . 4  |-  ( ( ( R  e. oRing  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  ->  X
( le `  R
) Y )
134, 5, 6, 9, 12syl31anc 1226 . . 3  |-  ( ph  ->  X ( le `  R ) Y )
14 orngrng 27439 . . . . . 6  |-  ( R  e. oRing  ->  R  e.  Ring )
154, 14syl 16 . . . . 5  |-  ( ph  ->  R  e.  Ring )
16 rnggrp 16984 . . . . 5  |-  ( R  e.  Ring  ->  R  e. 
Grp )
171, 3grpidcl 15872 . . . . 5  |-  ( R  e.  Grp  ->  .0.  e.  B )
1815, 16, 173syl 20 . . . 4  |-  ( ph  ->  .0.  e.  B )
19 ornglmullt.6 . . . 4  |-  ( ph  ->  .0.  .<  Z )
208, 10pltle 15437 . . . . 5  |-  ( ( R  e. oRing  /\  .0.  e.  B  /\  Z  e.  B
)  ->  (  .0.  .<  Z  ->  .0.  ( le `  R ) Z ) )
2120imp 429 . . . 4  |-  ( ( ( R  e. oRing  /\  .0.  e.  B  /\  Z  e.  B )  /\  .0.  .<  Z )  ->  .0.  ( le `  R ) Z )
224, 18, 7, 19, 21syl31anc 1226 . . 3  |-  ( ph  ->  .0.  ( le `  R ) Z )
231, 2, 3, 4, 5, 6, 7, 8, 13, 22orngrmulle 27445 . 2  |-  ( ph  ->  ( X  .x.  Z
) ( le `  R ) ( Y 
.x.  Z ) )
24 simpr 461 . . . . . 6  |-  ( (
ph  /\  ( X  .x.  Z )  =  ( Y  .x.  Z ) )  ->  ( X  .x.  Z )  =  ( Y  .x.  Z ) )
2524oveq1d 6290 . . . . 5  |-  ( (
ph  /\  ( X  .x.  Z )  =  ( Y  .x.  Z ) )  ->  ( ( X  .x.  Z ) (/r `  R ) Z )  =  ( ( Y 
.x.  Z ) (/r `  R ) Z ) )
26 ornglmullt.d . . . . . . . 8  |-  ( ph  ->  R  e.  DivRing )
2710pltne 15438 . . . . . . . . . . 11  |-  ( ( R  e. oRing  /\  .0.  e.  B  /\  Z  e.  B
)  ->  (  .0.  .<  Z  ->  .0.  =/=  Z
) )
2827imp 429 . . . . . . . . . 10  |-  ( ( ( R  e. oRing  /\  .0.  e.  B  /\  Z  e.  B )  /\  .0.  .<  Z )  ->  .0.  =/=  Z )
294, 18, 7, 19, 28syl31anc 1226 . . . . . . . . 9  |-  ( ph  ->  .0.  =/=  Z )
3029necomd 2731 . . . . . . . 8  |-  ( ph  ->  Z  =/=  .0.  )
31 eqid 2460 . . . . . . . . . 10  |-  (Unit `  R )  =  (Unit `  R )
321, 31, 3drngunit 17177 . . . . . . . . 9  |-  ( R  e.  DivRing  ->  ( Z  e.  (Unit `  R )  <->  ( Z  e.  B  /\  Z  =/=  .0.  ) ) )
3332biimpar 485 . . . . . . . 8  |-  ( ( R  e.  DivRing  /\  ( Z  e.  B  /\  Z  =/=  .0.  ) )  ->  Z  e.  (Unit `  R ) )
3426, 7, 30, 33syl12anc 1221 . . . . . . 7  |-  ( ph  ->  Z  e.  (Unit `  R ) )
35 eqid 2460 . . . . . . . 8  |-  (/r `  R
)  =  (/r `  R
)
361, 31, 35, 2dvrcan3 17118 . . . . . . 7  |-  ( ( R  e.  Ring  /\  X  e.  B  /\  Z  e.  (Unit `  R )
)  ->  ( ( X  .x.  Z ) (/r `  R ) Z )  =  X )
3715, 5, 34, 36syl3anc 1223 . . . . . 6  |-  ( ph  ->  ( ( X  .x.  Z ) (/r `  R
) Z )  =  X )
3837adantr 465 . . . . 5  |-  ( (
ph  /\  ( X  .x.  Z )  =  ( Y  .x.  Z ) )  ->  ( ( X  .x.  Z ) (/r `  R ) Z )  =  X )
391, 31, 35, 2dvrcan3 17118 . . . . . . 7  |-  ( ( R  e.  Ring  /\  Y  e.  B  /\  Z  e.  (Unit `  R )
)  ->  ( ( Y  .x.  Z ) (/r `  R ) Z )  =  Y )
4015, 6, 34, 39syl3anc 1223 . . . . . 6  |-  ( ph  ->  ( ( Y  .x.  Z ) (/r `  R
) Z )  =  Y )
4140adantr 465 . . . . 5  |-  ( (
ph  /\  ( X  .x.  Z )  =  ( Y  .x.  Z ) )  ->  ( ( Y  .x.  Z ) (/r `  R ) Z )  =  Y )
4225, 38, 413eqtr3d 2509 . . . 4  |-  ( (
ph  /\  ( X  .x.  Z )  =  ( Y  .x.  Z ) )  ->  X  =  Y )
4310pltne 15438 . . . . . . . 8  |-  ( ( R  e. oRing  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  ->  X  =/=  Y ) )
4443imp 429 . . . . . . 7  |-  ( ( ( R  e. oRing  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  ->  X  =/=  Y )
454, 5, 6, 9, 44syl31anc 1226 . . . . . 6  |-  ( ph  ->  X  =/=  Y )
4645adantr 465 . . . . 5  |-  ( (
ph  /\  ( X  .x.  Z )  =  ( Y  .x.  Z ) )  ->  X  =/=  Y )
4746neneqd 2662 . . . 4  |-  ( (
ph  /\  ( X  .x.  Z )  =  ( Y  .x.  Z ) )  ->  -.  X  =  Y )
4842, 47pm2.65da 576 . . 3  |-  ( ph  ->  -.  ( X  .x.  Z )  =  ( Y  .x.  Z ) )
4948neqned 2663 . 2  |-  ( ph  ->  ( X  .x.  Z
)  =/=  ( Y 
.x.  Z ) )
501, 2rngcl 16992 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  .x.  Z )  e.  B )
5115, 5, 7, 50syl3anc 1223 . . 3  |-  ( ph  ->  ( X  .x.  Z
)  e.  B )
521, 2rngcl 16992 . . . 4  |-  ( ( R  e.  Ring  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  .x.  Z )  e.  B )
5315, 6, 7, 52syl3anc 1223 . . 3  |-  ( ph  ->  ( Y  .x.  Z
)  e.  B )
548, 10pltval 15436 . . 3  |-  ( ( R  e. oRing  /\  ( X  .x.  Z )  e.  B  /\  ( Y 
.x.  Z )  e.  B )  ->  (
( X  .x.  Z
)  .<  ( Y  .x.  Z )  <->  ( ( X  .x.  Z ) ( le `  R ) ( Y  .x.  Z
)  /\  ( X  .x.  Z )  =/=  ( Y  .x.  Z ) ) ) )
554, 51, 53, 54syl3anc 1223 . 2  |-  ( ph  ->  ( ( X  .x.  Z )  .<  ( Y  .x.  Z )  <->  ( ( X  .x.  Z ) ( le `  R ) ( Y  .x.  Z
)  /\  ( X  .x.  Z )  =/=  ( Y  .x.  Z ) ) ) )
5623, 49, 55mpbir2and 915 1  |-  ( ph  ->  ( X  .x.  Z
)  .<  ( Y  .x.  Z ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2655   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   Basecbs 14479   .rcmulr 14545   lecple 14551   0gc0g 14684   ltcplt 15417   Grpcgrp 15716   Ringcrg 16979  Unitcui 17065  /rcdvr 17108   DivRingcdr 17172  oRingcorng 27434
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 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-tpos 6945  df-recs 7032  df-rdg 7066  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-3 10584  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-mulr 14558  df-0g 14686  df-plt 15434  df-mnd 15721  df-grp 15851  df-minusg 15852  df-sbg 15853  df-mgp 16925  df-ur 16937  df-rng 16981  df-oppr 17049  df-dvdsr 17067  df-unit 17068  df-invr 17098  df-dvr 17109  df-drng 17174  df-omnd 27337  df-ogrp 27338  df-orng 27436
This theorem is referenced by:  isarchiofld  27456
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