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Theorem ornglmullt 28644
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
ornglmullt  |-  ( ph  ->  ( Z  .x.  X
)  .<  ( Z  .x.  Y ) )

Proof of Theorem ornglmullt
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, 22ornglmulle 28642 . 2  |-  ( ph  ->  ( Z  .x.  X
) ( le `  R ) ( Z 
.x.  Y ) )
24 simpr 468 . . . . . 6  |-  ( (
ph  /\  ( Z  .x.  X )  =  ( Z  .x.  Y ) )  ->  ( Z  .x.  X )  =  ( Z  .x.  Y ) )
2524oveq2d 6324 . . . . 5  |-  ( (
ph  /\  ( Z  .x.  X )  =  ( Z  .x.  Y ) )  ->  ( (
( invr `  R ) `  Z )  .x.  ( Z  .x.  X ) )  =  ( ( (
invr `  R ) `  Z )  .x.  ( Z  .x.  Y ) ) )
26 ornglmullt.d . . . . . . . . . 10  |-  ( ph  ->  R  e.  DivRing )
2710pltne 16286 . . . . . . . . . . . . 13  |-  ( ( R  e. oRing  /\  .0.  e.  B  /\  Z  e.  B
)  ->  (  .0.  .<  Z  ->  .0.  =/=  Z
) )
2827imp 436 . . . . . . . . . . . 12  |-  ( ( ( R  e. oRing  /\  .0.  e.  B  /\  Z  e.  B )  /\  .0.  .<  Z )  ->  .0.  =/=  Z )
294, 18, 7, 19, 28syl31anc 1295 . . . . . . . . . . 11  |-  ( ph  ->  .0.  =/=  Z )
3029necomd 2698 . . . . . . . . . 10  |-  ( ph  ->  Z  =/=  .0.  )
31 eqid 2471 . . . . . . . . . . . 12  |-  (Unit `  R )  =  (Unit `  R )
321, 31, 3drngunit 18058 . . . . . . . . . . 11  |-  ( R  e.  DivRing  ->  ( Z  e.  (Unit `  R )  <->  ( Z  e.  B  /\  Z  =/=  .0.  ) ) )
3332biimpar 493 . . . . . . . . . 10  |-  ( ( R  e.  DivRing  /\  ( Z  e.  B  /\  Z  =/=  .0.  ) )  ->  Z  e.  (Unit `  R ) )
3426, 7, 30, 33syl12anc 1290 . . . . . . . . 9  |-  ( ph  ->  Z  e.  (Unit `  R ) )
35 eqid 2471 . . . . . . . . . 10  |-  ( invr `  R )  =  (
invr `  R )
36 eqid 2471 . . . . . . . . . 10  |-  ( 1r
`  R )  =  ( 1r `  R
)
3731, 35, 2, 36unitlinv 17983 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  Z  e.  (Unit `  R )
)  ->  ( (
( invr `  R ) `  Z )  .x.  Z
)  =  ( 1r
`  R ) )
3815, 34, 37syl2anc 673 . . . . . . . 8  |-  ( ph  ->  ( ( ( invr `  R ) `  Z
)  .x.  Z )  =  ( 1r `  R ) )
3938oveq1d 6323 . . . . . . 7  |-  ( ph  ->  ( ( ( (
invr `  R ) `  Z )  .x.  Z
)  .x.  X )  =  ( ( 1r
`  R )  .x.  X ) )
4031, 35, 1ringinvcl 17982 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  Z  e.  (Unit `  R )
)  ->  ( ( invr `  R ) `  Z )  e.  B
)
4115, 34, 40syl2anc 673 . . . . . . . 8  |-  ( ph  ->  ( ( invr `  R
) `  Z )  e.  B )
421, 2ringass 17875 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  (
( ( invr `  R
) `  Z )  e.  B  /\  Z  e.  B  /\  X  e.  B ) )  -> 
( ( ( (
invr `  R ) `  Z )  .x.  Z
)  .x.  X )  =  ( ( (
invr `  R ) `  Z )  .x.  ( Z  .x.  X ) ) )
4315, 41, 7, 5, 42syl13anc 1294 . . . . . . 7  |-  ( ph  ->  ( ( ( (
invr `  R ) `  Z )  .x.  Z
)  .x.  X )  =  ( ( (
invr `  R ) `  Z )  .x.  ( Z  .x.  X ) ) )
441, 2, 36ringlidm 17882 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (
( 1r `  R
)  .x.  X )  =  X )
4515, 5, 44syl2anc 673 . . . . . . 7  |-  ( ph  ->  ( ( 1r `  R )  .x.  X
)  =  X )
4639, 43, 453eqtr3d 2513 . . . . . 6  |-  ( ph  ->  ( ( ( invr `  R ) `  Z
)  .x.  ( Z  .x.  X ) )  =  X )
4746adantr 472 . . . . 5  |-  ( (
ph  /\  ( Z  .x.  X )  =  ( Z  .x.  Y ) )  ->  ( (
( invr `  R ) `  Z )  .x.  ( Z  .x.  X ) )  =  X )
4838oveq1d 6323 . . . . . . 7  |-  ( ph  ->  ( ( ( (
invr `  R ) `  Z )  .x.  Z
)  .x.  Y )  =  ( ( 1r
`  R )  .x.  Y ) )
491, 2ringass 17875 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  (
( ( invr `  R
) `  Z )  e.  B  /\  Z  e.  B  /\  Y  e.  B ) )  -> 
( ( ( (
invr `  R ) `  Z )  .x.  Z
)  .x.  Y )  =  ( ( (
invr `  R ) `  Z )  .x.  ( Z  .x.  Y ) ) )
5015, 41, 7, 6, 49syl13anc 1294 . . . . . . 7  |-  ( ph  ->  ( ( ( (
invr `  R ) `  Z )  .x.  Z
)  .x.  Y )  =  ( ( (
invr `  R ) `  Z )  .x.  ( Z  .x.  Y ) ) )
511, 2, 36ringlidm 17882 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  Y  e.  B )  ->  (
( 1r `  R
)  .x.  Y )  =  Y )
5215, 6, 51syl2anc 673 . . . . . . 7  |-  ( ph  ->  ( ( 1r `  R )  .x.  Y
)  =  Y )
5348, 50, 523eqtr3d 2513 . . . . . 6  |-  ( ph  ->  ( ( ( invr `  R ) `  Z
)  .x.  ( Z  .x.  Y ) )  =  Y )
5453adantr 472 . . . . 5  |-  ( (
ph  /\  ( Z  .x.  X )  =  ( Z  .x.  Y ) )  ->  ( (
( invr `  R ) `  Z )  .x.  ( Z  .x.  Y ) )  =  Y )
5525, 47, 543eqtr3d 2513 . . . 4  |-  ( (
ph  /\  ( Z  .x.  X )  =  ( Z  .x.  Y ) )  ->  X  =  Y )
5610pltne 16286 . . . . . . . 8  |-  ( ( R  e. oRing  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  ->  X  =/=  Y ) )
5756imp 436 . . . . . . 7  |-  ( ( ( R  e. oRing  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  ->  X  =/=  Y )
584, 5, 6, 9, 57syl31anc 1295 . . . . . 6  |-  ( ph  ->  X  =/=  Y )
5958adantr 472 . . . . 5  |-  ( (
ph  /\  ( Z  .x.  X )  =  ( Z  .x.  Y ) )  ->  X  =/=  Y )
6059neneqd 2648 . . . 4  |-  ( (
ph  /\  ( Z  .x.  X )  =  ( Z  .x.  Y ) )  ->  -.  X  =  Y )
6155, 60pm2.65da 586 . . 3  |-  ( ph  ->  -.  ( Z  .x.  X )  =  ( Z  .x.  Y ) )
6261neqned 2650 . 2  |-  ( ph  ->  ( Z  .x.  X
)  =/=  ( Z 
.x.  Y ) )
631, 2ringcl 17872 . . . 4  |-  ( ( R  e.  Ring  /\  Z  e.  B  /\  X  e.  B )  ->  ( Z  .x.  X )  e.  B )
6415, 7, 5, 63syl3anc 1292 . . 3  |-  ( ph  ->  ( Z  .x.  X
)  e.  B )
651, 2ringcl 17872 . . . 4  |-  ( ( R  e.  Ring  /\  Z  e.  B  /\  Y  e.  B )  ->  ( Z  .x.  Y )  e.  B )
6615, 7, 6, 65syl3anc 1292 . . 3  |-  ( ph  ->  ( Z  .x.  Y
)  e.  B )
678, 10pltval 16284 . . 3  |-  ( ( R  e. oRing  /\  ( Z  .x.  X )  e.  B  /\  ( Z 
.x.  Y )  e.  B )  ->  (
( Z  .x.  X
)  .<  ( Z  .x.  Y )  <->  ( ( Z  .x.  X ) ( le `  R ) ( Z  .x.  Y
)  /\  ( Z  .x.  X )  =/=  ( Z  .x.  Y ) ) ) )
684, 64, 66, 67syl3anc 1292 . 2  |-  ( ph  ->  ( ( Z  .x.  X )  .<  ( Z  .x.  Y )  <->  ( ( Z  .x.  X ) ( le `  R ) ( Z  .x.  Y
)  /\  ( Z  .x.  X )  =/=  ( Z  .x.  Y ) ) ) )
6923, 62, 68mpbir2and 936 1  |-  ( ph  ->  ( Z  .x.  X
)  .<  ( Z  .x.  Y ) )
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   1rcur 17813   Ringcrg 17858  Unitcui 17945   invrcinvr 17977   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-drng 18055  df-omnd 28536  df-ogrp 28537  df-orng 28634
This theorem is referenced by: (None)
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