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Theorem orngmullt 28034
Description: In an ordered ring, the strict ordering is compatible with the ring multiplication operation. (Contributed by Thierry Arnoux, 9-Sep-2018.)
Hypotheses
Ref Expression
orngmullt.b  |-  B  =  ( Base `  R
)
orngmullt.t  |-  .x.  =  ( .r `  R )
orngmullt.0  |-  .0.  =  ( 0g `  R )
orngmullt.l  |-  .<  =  ( lt `  R )
orngmullt.1  |-  ( ph  ->  R  e. oRing )
orngmullt.4  |-  ( ph  ->  R  e.  DivRing )
orngmullt.2  |-  ( ph  ->  X  e.  B )
orngmullt.3  |-  ( ph  ->  Y  e.  B )
orngmullt.x  |-  ( ph  ->  .0.  .<  X )
orngmullt.y  |-  ( ph  ->  .0.  .<  Y )
Assertion
Ref Expression
orngmullt  |-  ( ph  ->  .0.  .<  ( X  .x.  Y ) )

Proof of Theorem orngmullt
StepHypRef Expression
1 orngmullt.1 . . 3  |-  ( ph  ->  R  e. oRing )
2 orngmullt.2 . . 3  |-  ( ph  ->  X  e.  B )
3 orngmullt.x . . . . 5  |-  ( ph  ->  .0.  .<  X )
4 orngring 28025 . . . . . . 7  |-  ( R  e. oRing  ->  R  e.  Ring )
5 ringgrp 17398 . . . . . . 7  |-  ( R  e.  Ring  ->  R  e. 
Grp )
6 orngmullt.b . . . . . . . 8  |-  B  =  ( Base `  R
)
7 orngmullt.0 . . . . . . . 8  |-  .0.  =  ( 0g `  R )
86, 7grpidcl 16277 . . . . . . 7  |-  ( R  e.  Grp  ->  .0.  e.  B )
91, 4, 5, 84syl 21 . . . . . 6  |-  ( ph  ->  .0.  e.  B )
10 eqid 2454 . . . . . . 7  |-  ( le
`  R )  =  ( le `  R
)
11 orngmullt.l . . . . . . 7  |-  .<  =  ( lt `  R )
1210, 11pltval 15789 . . . . . 6  |-  ( ( R  e. oRing  /\  .0.  e.  B  /\  X  e.  B
)  ->  (  .0.  .<  X 
<->  (  .0.  ( le
`  R ) X  /\  .0.  =/=  X
) ) )
131, 9, 2, 12syl3anc 1226 . . . . 5  |-  ( ph  ->  (  .0.  .<  X  <->  (  .0.  ( le `  R ) X  /\  .0.  =/=  X ) ) )
143, 13mpbid 210 . . . 4  |-  ( ph  ->  (  .0.  ( le
`  R ) X  /\  .0.  =/=  X
) )
1514simpld 457 . . 3  |-  ( ph  ->  .0.  ( le `  R ) X )
16 orngmullt.3 . . 3  |-  ( ph  ->  Y  e.  B )
17 orngmullt.y . . . . 5  |-  ( ph  ->  .0.  .<  Y )
1810, 11pltval 15789 . . . . . 6  |-  ( ( R  e. oRing  /\  .0.  e.  B  /\  Y  e.  B
)  ->  (  .0.  .<  Y 
<->  (  .0.  ( le
`  R ) Y  /\  .0.  =/=  Y
) ) )
191, 9, 16, 18syl3anc 1226 . . . . 5  |-  ( ph  ->  (  .0.  .<  Y  <->  (  .0.  ( le `  R ) Y  /\  .0.  =/=  Y ) ) )
2017, 19mpbid 210 . . . 4  |-  ( ph  ->  (  .0.  ( le
`  R ) Y  /\  .0.  =/=  Y
) )
2120simpld 457 . . 3  |-  ( ph  ->  .0.  ( le `  R ) Y )
22 orngmullt.t . . . 4  |-  .x.  =  ( .r `  R )
236, 10, 7, 22orngmul 28028 . . 3  |-  ( ( R  e. oRing  /\  ( X  e.  B  /\  .0.  ( le `  R
) X )  /\  ( Y  e.  B  /\  .0.  ( le `  R ) Y ) )  ->  .0.  ( le `  R ) ( X  .x.  Y ) )
241, 2, 15, 16, 21, 23syl122anc 1235 . 2  |-  ( ph  ->  .0.  ( le `  R ) ( X 
.x.  Y ) )
2514simprd 461 . . . . 5  |-  ( ph  ->  .0.  =/=  X )
2625necomd 2725 . . . 4  |-  ( ph  ->  X  =/=  .0.  )
2720simprd 461 . . . . 5  |-  ( ph  ->  .0.  =/=  Y )
2827necomd 2725 . . . 4  |-  ( ph  ->  Y  =/=  .0.  )
29 orngmullt.4 . . . . 5  |-  ( ph  ->  R  e.  DivRing )
306, 7, 22, 29, 2, 16drngmulne0 17613 . . . 4  |-  ( ph  ->  ( ( X  .x.  Y )  =/=  .0.  <->  ( X  =/=  .0.  /\  Y  =/=  .0.  ) ) )
3126, 28, 30mpbir2and 920 . . 3  |-  ( ph  ->  ( X  .x.  Y
)  =/=  .0.  )
3231necomd 2725 . 2  |-  ( ph  ->  .0.  =/=  ( X 
.x.  Y ) )
331, 4syl 16 . . . 4  |-  ( ph  ->  R  e.  Ring )
346, 22ringcl 17407 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .x.  Y )  e.  B )
3533, 2, 16, 34syl3anc 1226 . . 3  |-  ( ph  ->  ( X  .x.  Y
)  e.  B )
3610, 11pltval 15789 . . 3  |-  ( ( R  e. oRing  /\  .0.  e.  B  /\  ( X  .x.  Y )  e.  B
)  ->  (  .0.  .< 
( X  .x.  Y
)  <->  (  .0.  ( le `  R ) ( X  .x.  Y )  /\  .0.  =/=  ( X  .x.  Y ) ) ) )
371, 9, 35, 36syl3anc 1226 . 2  |-  ( ph  ->  (  .0.  .<  ( X  .x.  Y )  <->  (  .0.  ( le `  R ) ( X  .x.  Y
)  /\  .0.  =/=  ( X  .x.  Y ) ) ) )
3824, 32, 37mpbir2and 920 1  |-  ( ph  ->  .0.  .<  ( X  .x.  Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   Basecbs 14716   .rcmulr 14785   lecple 14791   0gc0g 14929   ltcplt 15769   Grpcgrp 16252   Ringcrg 17393   DivRingcdr 17591  oRingcorng 28020
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  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 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-tpos 6947  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-mulr 14798  df-0g 14931  df-plt 15787  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-grp 16256  df-minusg 16257  df-mgp 17337  df-ur 17349  df-ring 17395  df-oppr 17467  df-dvdsr 17485  df-unit 17486  df-invr 17516  df-drng 17593  df-orng 28022
This theorem is referenced by: (None)
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