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Theorem ornglmullt 27667
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 2441 . . 3  |-  ( le
`  R )  =  ( le `  R
)
9 ornglmullt.5 . . . 4  |-  ( ph  ->  X  .<  Y )
10 ornglmullt.l . . . . . 6  |-  .<  =  ( lt `  R )
118, 10pltle 15462 . . . . 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 1230 . . 3  |-  ( ph  ->  X ( le `  R ) Y )
14 orngring 27660 . . . . . 6  |-  ( R  e. oRing  ->  R  e.  Ring )
154, 14syl 16 . . . . 5  |-  ( ph  ->  R  e.  Ring )
16 ringgrp 17074 . . . . 5  |-  ( R  e.  Ring  ->  R  e. 
Grp )
171, 3grpidcl 15949 . . . . 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 15462 . . . . 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 1230 . . 3  |-  ( ph  ->  .0.  ( le `  R ) Z )
231, 2, 3, 4, 5, 6, 7, 8, 13, 22ornglmulle 27665 . 2  |-  ( ph  ->  ( Z  .x.  X
) ( le `  R ) ( Z 
.x.  Y ) )
24 simpr 461 . . . . . 6  |-  ( (
ph  /\  ( Z  .x.  X )  =  ( Z  .x.  Y ) )  ->  ( Z  .x.  X )  =  ( Z  .x.  Y ) )
2524oveq2d 6294 . . . . 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 15463 . . . . . . . . . . . . 13  |-  ( ( R  e. oRing  /\  .0.  e.  B  /\  Z  e.  B
)  ->  (  .0.  .<  Z  ->  .0.  =/=  Z
) )
2827imp 429 . . . . . . . . . . . 12  |-  ( ( ( R  e. oRing  /\  .0.  e.  B  /\  Z  e.  B )  /\  .0.  .<  Z )  ->  .0.  =/=  Z )
294, 18, 7, 19, 28syl31anc 1230 . . . . . . . . . . 11  |-  ( ph  ->  .0.  =/=  Z )
3029necomd 2712 . . . . . . . . . 10  |-  ( ph  ->  Z  =/=  .0.  )
31 eqid 2441 . . . . . . . . . . . 12  |-  (Unit `  R )  =  (Unit `  R )
321, 31, 3drngunit 17272 . . . . . . . . . . 11  |-  ( R  e.  DivRing  ->  ( Z  e.  (Unit `  R )  <->  ( Z  e.  B  /\  Z  =/=  .0.  ) ) )
3332biimpar 485 . . . . . . . . . 10  |-  ( ( R  e.  DivRing  /\  ( Z  e.  B  /\  Z  =/=  .0.  ) )  ->  Z  e.  (Unit `  R ) )
3426, 7, 30, 33syl12anc 1225 . . . . . . . . 9  |-  ( ph  ->  Z  e.  (Unit `  R ) )
35 eqid 2441 . . . . . . . . . 10  |-  ( invr `  R )  =  (
invr `  R )
36 eqid 2441 . . . . . . . . . 10  |-  ( 1r
`  R )  =  ( 1r `  R
)
3731, 35, 2, 36unitlinv 17197 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  Z  e.  (Unit `  R )
)  ->  ( (
( invr `  R ) `  Z )  .x.  Z
)  =  ( 1r
`  R ) )
3815, 34, 37syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( ( ( invr `  R ) `  Z
)  .x.  Z )  =  ( 1r `  R ) )
3938oveq1d 6293 . . . . . . 7  |-  ( ph  ->  ( ( ( (
invr `  R ) `  Z )  .x.  Z
)  .x.  X )  =  ( ( 1r
`  R )  .x.  X ) )
4031, 35, 1ringinvcl 17196 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  Z  e.  (Unit `  R )
)  ->  ( ( invr `  R ) `  Z )  e.  B
)
4115, 34, 40syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( ( invr `  R
) `  Z )  e.  B )
421, 2ringass 17086 . . . . . . . 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 1229 . . . . . . 7  |-  ( ph  ->  ( ( ( (
invr `  R ) `  Z )  .x.  Z
)  .x.  X )  =  ( ( (
invr `  R ) `  Z )  .x.  ( Z  .x.  X ) ) )
441, 2, 36ringlidm 17093 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (
( 1r `  R
)  .x.  X )  =  X )
4515, 5, 44syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( ( 1r `  R )  .x.  X
)  =  X )
4639, 43, 453eqtr3d 2490 . . . . . 6  |-  ( ph  ->  ( ( ( invr `  R ) `  Z
)  .x.  ( Z  .x.  X ) )  =  X )
4746adantr 465 . . . . 5  |-  ( (
ph  /\  ( Z  .x.  X )  =  ( Z  .x.  Y ) )  ->  ( (
( invr `  R ) `  Z )  .x.  ( Z  .x.  X ) )  =  X )
4838oveq1d 6293 . . . . . . 7  |-  ( ph  ->  ( ( ( (
invr `  R ) `  Z )  .x.  Z
)  .x.  Y )  =  ( ( 1r
`  R )  .x.  Y ) )
491, 2ringass 17086 . . . . . . . 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 1229 . . . . . . 7  |-  ( ph  ->  ( ( ( (
invr `  R ) `  Z )  .x.  Z
)  .x.  Y )  =  ( ( (
invr `  R ) `  Z )  .x.  ( Z  .x.  Y ) ) )
511, 2, 36ringlidm 17093 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  Y  e.  B )  ->  (
( 1r `  R
)  .x.  Y )  =  Y )
5215, 6, 51syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( ( 1r `  R )  .x.  Y
)  =  Y )
5348, 50, 523eqtr3d 2490 . . . . . 6  |-  ( ph  ->  ( ( ( invr `  R ) `  Z
)  .x.  ( Z  .x.  Y ) )  =  Y )
5453adantr 465 . . . . 5  |-  ( (
ph  /\  ( Z  .x.  X )  =  ( Z  .x.  Y ) )  ->  ( (
( invr `  R ) `  Z )  .x.  ( Z  .x.  Y ) )  =  Y )
5525, 47, 543eqtr3d 2490 . . . 4  |-  ( (
ph  /\  ( Z  .x.  X )  =  ( Z  .x.  Y ) )  ->  X  =  Y )
5610pltne 15463 . . . . . . . 8  |-  ( ( R  e. oRing  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  ->  X  =/=  Y ) )
5756imp 429 . . . . . . 7  |-  ( ( ( R  e. oRing  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  ->  X  =/=  Y )
584, 5, 6, 9, 57syl31anc 1230 . . . . . 6  |-  ( ph  ->  X  =/=  Y )
5958adantr 465 . . . . 5  |-  ( (
ph  /\  ( Z  .x.  X )  =  ( Z  .x.  Y ) )  ->  X  =/=  Y )
6059neneqd 2643 . . . 4  |-  ( (
ph  /\  ( Z  .x.  X )  =  ( Z  .x.  Y ) )  ->  -.  X  =  Y )
6155, 60pm2.65da 576 . . 3  |-  ( ph  ->  -.  ( Z  .x.  X )  =  ( Z  .x.  Y ) )
6261neqned 2644 . 2  |-  ( ph  ->  ( Z  .x.  X
)  =/=  ( Z 
.x.  Y ) )
631, 2ringcl 17083 . . . 4  |-  ( ( R  e.  Ring  /\  Z  e.  B  /\  X  e.  B )  ->  ( Z  .x.  X )  e.  B )
6415, 7, 5, 63syl3anc 1227 . . 3  |-  ( ph  ->  ( Z  .x.  X
)  e.  B )
651, 2ringcl 17083 . . . 4  |-  ( ( R  e.  Ring  /\  Z  e.  B  /\  Y  e.  B )  ->  ( Z  .x.  Y )  e.  B )
6615, 7, 6, 65syl3anc 1227 . . 3  |-  ( ph  ->  ( Z  .x.  Y
)  e.  B )
678, 10pltval 15461 . . 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 1227 . 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 920 1  |-  ( ph  ->  ( Z  .x.  X
)  .<  ( Z  .x.  Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 972    = wceq 1381    e. wcel 1802    =/= wne 2636   class class class wbr 4434   ` cfv 5575  (class class class)co 6278   Basecbs 14506   .rcmulr 14572   lecple 14578   0gc0g 14711   ltcplt 15441   Grpcgrp 15924   1rcur 17024   Ringcrg 17069  Unitcui 17159   invrcinvr 17191   DivRingcdr 17267  oRingcorng 27655
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4545  ax-sep 4555  ax-nul 4563  ax-pow 4612  ax-pr 4673  ax-un 6574  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3419  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-pss 3475  df-nul 3769  df-if 3924  df-pw 3996  df-sn 4012  df-pr 4014  df-tp 4016  df-op 4018  df-uni 4232  df-iun 4314  df-br 4435  df-opab 4493  df-mpt 4494  df-tr 4528  df-eprel 4778  df-id 4782  df-po 4787  df-so 4788  df-fr 4825  df-we 4827  df-ord 4868  df-on 4869  df-lim 4870  df-suc 4871  df-xp 4992  df-rel 4993  df-cnv 4994  df-co 4995  df-dm 4996  df-rn 4997  df-res 4998  df-ima 4999  df-iota 5538  df-fun 5577  df-fn 5578  df-f 5579  df-f1 5580  df-fo 5581  df-f1o 5582  df-fv 5583  df-riota 6239  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6683  df-1st 6782  df-2nd 6783  df-tpos 6954  df-recs 7041  df-rdg 7075  df-er 7310  df-en 7516  df-dom 7517  df-sdom 7518  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9809  df-neg 9810  df-nn 10540  df-2 10597  df-3 10598  df-ndx 14509  df-slot 14510  df-base 14511  df-sets 14512  df-ress 14513  df-plusg 14584  df-mulr 14585  df-0g 14713  df-plt 15459  df-mgm 15743  df-sgrp 15782  df-mnd 15792  df-grp 15928  df-minusg 15929  df-sbg 15930  df-mgp 17013  df-ur 17025  df-ring 17071  df-oppr 17143  df-dvdsr 17161  df-unit 17162  df-invr 17192  df-drng 17269  df-omnd 27559  df-ogrp 27560  df-orng 27657
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator