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Theorem orngrmullt 28645
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 2471 . . 3  |-  ( le
`  R )  =  ( le `  R
)
9 ornglmullt.5 . . . 4  |-  ( ph  ->  X  .<  Y )
10 ornglmullt.l . . . . . 6  |-  .<  =  ( lt `  R )
118, 10pltle 16285 . . . . 5  |-  ( ( R  e. oRing  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  ->  X
( le `  R
) Y ) )
1211imp 436 . . . 4  |-  ( ( ( R  e. oRing  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  ->  X
( le `  R
) Y )
134, 5, 6, 9, 12syl31anc 1295 . . 3  |-  ( ph  ->  X ( le `  R ) Y )
14 orngring 28637 . . . . . 6  |-  ( R  e. oRing  ->  R  e.  Ring )
154, 14syl 17 . . . . 5  |-  ( ph  ->  R  e.  Ring )
16 ringgrp 17863 . . . . 5  |-  ( R  e.  Ring  ->  R  e. 
Grp )
171, 3grpidcl 16772 . . . . 5  |-  ( R  e.  Grp  ->  .0.  e.  B )
1815, 16, 173syl 18 . . . 4  |-  ( ph  ->  .0.  e.  B )
19 ornglmullt.6 . . . 4  |-  ( ph  ->  .0.  .<  Z )
208, 10pltle 16285 . . . . 5  |-  ( ( R  e. oRing  /\  .0.  e.  B  /\  Z  e.  B
)  ->  (  .0.  .<  Z  ->  .0.  ( le `  R ) Z ) )
2120imp 436 . . . 4  |-  ( ( ( R  e. oRing  /\  .0.  e.  B  /\  Z  e.  B )  /\  .0.  .<  Z )  ->  .0.  ( le `  R ) Z )
224, 18, 7, 19, 21syl31anc 1295 . . 3  |-  ( ph  ->  .0.  ( le `  R ) Z )
231, 2, 3, 4, 5, 6, 7, 8, 13, 22orngrmulle 28643 . 2  |-  ( ph  ->  ( X  .x.  Z
) ( le `  R ) ( Y 
.x.  Z ) )
24 simpr 468 . . . . . 6  |-  ( (
ph  /\  ( X  .x.  Z )  =  ( Y  .x.  Z ) )  ->  ( X  .x.  Z )  =  ( Y  .x.  Z ) )
2524oveq1d 6323 . . . . 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 16286 . . . . . . . . . . 11  |-  ( ( R  e. oRing  /\  .0.  e.  B  /\  Z  e.  B
)  ->  (  .0.  .<  Z  ->  .0.  =/=  Z
) )
2827imp 436 . . . . . . . . . 10  |-  ( ( ( R  e. oRing  /\  .0.  e.  B  /\  Z  e.  B )  /\  .0.  .<  Z )  ->  .0.  =/=  Z )
294, 18, 7, 19, 28syl31anc 1295 . . . . . . . . 9  |-  ( ph  ->  .0.  =/=  Z )
3029necomd 2698 . . . . . . . 8  |-  ( ph  ->  Z  =/=  .0.  )
31 eqid 2471 . . . . . . . . . 10  |-  (Unit `  R )  =  (Unit `  R )
321, 31, 3drngunit 18058 . . . . . . . . 9  |-  ( R  e.  DivRing  ->  ( Z  e.  (Unit `  R )  <->  ( Z  e.  B  /\  Z  =/=  .0.  ) ) )
3332biimpar 493 . . . . . . . 8  |-  ( ( R  e.  DivRing  /\  ( Z  e.  B  /\  Z  =/=  .0.  ) )  ->  Z  e.  (Unit `  R ) )
3426, 7, 30, 33syl12anc 1290 . . . . . . 7  |-  ( ph  ->  Z  e.  (Unit `  R ) )
35 eqid 2471 . . . . . . . 8  |-  (/r `  R
)  =  (/r `  R
)
361, 31, 35, 2dvrcan3 17998 . . . . . . 7  |-  ( ( R  e.  Ring  /\  X  e.  B  /\  Z  e.  (Unit `  R )
)  ->  ( ( X  .x.  Z ) (/r `  R ) Z )  =  X )
3715, 5, 34, 36syl3anc 1292 . . . . . 6  |-  ( ph  ->  ( ( X  .x.  Z ) (/r `  R
) Z )  =  X )
3837adantr 472 . . . . 5  |-  ( (
ph  /\  ( X  .x.  Z )  =  ( Y  .x.  Z ) )  ->  ( ( X  .x.  Z ) (/r `  R ) Z )  =  X )
391, 31, 35, 2dvrcan3 17998 . . . . . . 7  |-  ( ( R  e.  Ring  /\  Y  e.  B  /\  Z  e.  (Unit `  R )
)  ->  ( ( Y  .x.  Z ) (/r `  R ) Z )  =  Y )
4015, 6, 34, 39syl3anc 1292 . . . . . 6  |-  ( ph  ->  ( ( Y  .x.  Z ) (/r `  R
) Z )  =  Y )
4140adantr 472 . . . . 5  |-  ( (
ph  /\  ( X  .x.  Z )  =  ( Y  .x.  Z ) )  ->  ( ( Y  .x.  Z ) (/r `  R ) Z )  =  Y )
4225, 38, 413eqtr3d 2513 . . . 4  |-  ( (
ph  /\  ( X  .x.  Z )  =  ( Y  .x.  Z ) )  ->  X  =  Y )
4310pltne 16286 . . . . . . . 8  |-  ( ( R  e. oRing  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  ->  X  =/=  Y ) )
4443imp 436 . . . . . . 7  |-  ( ( ( R  e. oRing  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  ->  X  =/=  Y )
454, 5, 6, 9, 44syl31anc 1295 . . . . . 6  |-  ( ph  ->  X  =/=  Y )
4645adantr 472 . . . . 5  |-  ( (
ph  /\  ( X  .x.  Z )  =  ( Y  .x.  Z ) )  ->  X  =/=  Y )
4746neneqd 2648 . . . 4  |-  ( (
ph  /\  ( X  .x.  Z )  =  ( Y  .x.  Z ) )  ->  -.  X  =  Y )
4842, 47pm2.65da 586 . . 3  |-  ( ph  ->  -.  ( X  .x.  Z )  =  ( Y  .x.  Z ) )
4948neqned 2650 . 2  |-  ( ph  ->  ( X  .x.  Z
)  =/=  ( Y 
.x.  Z ) )
501, 2ringcl 17872 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  .x.  Z )  e.  B )
5115, 5, 7, 50syl3anc 1292 . . 3  |-  ( ph  ->  ( X  .x.  Z
)  e.  B )
521, 2ringcl 17872 . . . 4  |-  ( ( R  e.  Ring  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  .x.  Z )  e.  B )
5315, 6, 7, 52syl3anc 1292 . . 3  |-  ( ph  ->  ( Y  .x.  Z
)  e.  B )
548, 10pltval 16284 . . 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 1292 . 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 936 1  |-  ( ph  ->  ( X  .x.  Z
)  .<  ( Y  .x.  Z ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   class class class wbr 4395   ` cfv 5589  (class class class)co 6308   Basecbs 15199   .rcmulr 15269   lecple 15275   0gc0g 15416   ltcplt 16264   Grpcgrp 16747   Ringcrg 17858  Unitcui 17945  /rcdvr 17988   DivRingcdr 18053  oRingcorng 28632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-tpos 6991  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-0g 15418  df-plt 16282  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-grp 16751  df-minusg 16752  df-sbg 16753  df-mgp 17802  df-ur 17814  df-ring 17860  df-oppr 17929  df-dvdsr 17947  df-unit 17948  df-invr 17978  df-dvr 17989  df-drng 18055  df-omnd 28536  df-ogrp 28537  df-orng 28634
This theorem is referenced by:  isarchiofld  28654
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