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Theorem orngrmullt 26276
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 2443 . . 3  |-  ( le
`  R )  =  ( le `  R
)
9 ornglmullt.5 . . . 4  |-  ( ph  ->  X  .<  Y )
10 ornglmullt.l . . . . . 6  |-  .<  =  ( lt `  R )
118, 10pltle 15131 . . . . 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 1221 . . 3  |-  ( ph  ->  X ( le `  R ) Y )
14 orngrng 26268 . . . . . 6  |-  ( R  e. oRing  ->  R  e.  Ring )
154, 14syl 16 . . . . 5  |-  ( ph  ->  R  e.  Ring )
16 rnggrp 16650 . . . . 5  |-  ( R  e.  Ring  ->  R  e. 
Grp )
171, 3grpidcl 15566 . . . . 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 15131 . . . . 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 1221 . . 3  |-  ( ph  ->  .0.  ( le `  R ) Z )
231, 2, 3, 4, 5, 6, 7, 8, 13, 22orngrmulle 26274 . 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 6106 . . . . 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 15132 . . . . . . . . . . 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 1221 . . . . . . . . 9  |-  ( ph  ->  .0.  =/=  Z )
3029necomd 2695 . . . . . . . 8  |-  ( ph  ->  Z  =/=  .0.  )
31 eqid 2443 . . . . . . . . . 10  |-  (Unit `  R )  =  (Unit `  R )
321, 31, 3drngunit 16837 . . . . . . . . 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 1216 . . . . . . 7  |-  ( ph  ->  Z  e.  (Unit `  R ) )
35 eqid 2443 . . . . . . . 8  |-  (/r `  R
)  =  (/r `  R
)
361, 31, 35, 2dvrcan3 16784 . . . . . . 7  |-  ( ( R  e.  Ring  /\  X  e.  B  /\  Z  e.  (Unit `  R )
)  ->  ( ( X  .x.  Z ) (/r `  R ) Z )  =  X )
3715, 5, 34, 36syl3anc 1218 . . . . . 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 16784 . . . . . . 7  |-  ( ( R  e.  Ring  /\  Y  e.  B  /\  Z  e.  (Unit `  R )
)  ->  ( ( Y  .x.  Z ) (/r `  R ) Z )  =  Y )
4015, 6, 34, 39syl3anc 1218 . . . . . 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 2483 . . . 4  |-  ( (
ph  /\  ( X  .x.  Z )  =  ( Y  .x.  Z ) )  ->  X  =  Y )
4310pltne 15132 . . . . . . . 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 1221 . . . . . 6  |-  ( ph  ->  X  =/=  Y )
4645adantr 465 . . . . 5  |-  ( (
ph  /\  ( X  .x.  Z )  =  ( Y  .x.  Z ) )  ->  X  =/=  Y )
4746neneqd 2624 . . . 4  |-  ( (
ph  /\  ( X  .x.  Z )  =  ( Y  .x.  Z ) )  ->  -.  X  =  Y )
4842, 47pm2.65da 576 . . 3  |-  ( ph  ->  -.  ( X  .x.  Z )  =  ( Y  .x.  Z ) )
4948neneqad 2681 . 2  |-  ( ph  ->  ( X  .x.  Z
)  =/=  ( Y 
.x.  Z ) )
501, 2rngcl 16658 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  .x.  Z )  e.  B )
5115, 5, 7, 50syl3anc 1218 . . 3  |-  ( ph  ->  ( X  .x.  Z
)  e.  B )
521, 2rngcl 16658 . . . 4  |-  ( ( R  e.  Ring  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  .x.  Z )  e.  B )
5315, 6, 7, 52syl3anc 1218 . . 3  |-  ( ph  ->  ( Y  .x.  Z
)  e.  B )
548, 10pltval 15130 . . 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 1218 . 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 913 1  |-  ( ph  ->  ( X  .x.  Z
)  .<  ( Y  .x.  Z ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2606   class class class wbr 4292   ` cfv 5418  (class class class)co 6091   Basecbs 14174   .rcmulr 14239   lecple 14245   0gc0g 14378   ltcplt 15111   Grpcgrp 15410   Ringcrg 16645  Unitcui 16731  /rcdvr 16774   DivRingcdr 16832  oRingcorng 26263
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-tpos 6745  df-recs 6832  df-rdg 6866  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-3 10381  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-mulr 14252  df-0g 14380  df-plt 15128  df-mnd 15415  df-grp 15545  df-minusg 15546  df-sbg 15547  df-mgp 16592  df-ur 16604  df-rng 16647  df-oppr 16715  df-dvdsr 16733  df-unit 16734  df-invr 16764  df-dvr 16775  df-drng 16834  df-omnd 26162  df-ogrp 26163  df-orng 26265
This theorem is referenced by:  isarchiofld  26285
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