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Theorem orngmullt 26415
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 . . . 4  |-  ( ph  ->  R  e. oRing )
2 orngmullt.2 . . . 4  |-  ( ph  ->  X  e.  B )
3 orngmullt.x . . . . . 6  |-  ( ph  ->  .0.  .<  X )
4 orngrng 26406 . . . . . . . 8  |-  ( R  e. oRing  ->  R  e.  Ring )
5 rnggrp 16765 . . . . . . . 8  |-  ( R  e.  Ring  ->  R  e. 
Grp )
6 orngmullt.b . . . . . . . . 9  |-  B  =  ( Base `  R
)
7 orngmullt.0 . . . . . . . . 9  |-  .0.  =  ( 0g `  R )
86, 7grpidcl 15677 . . . . . . . 8  |-  ( R  e.  Grp  ->  .0.  e.  B )
91, 4, 5, 84syl 21 . . . . . . 7  |-  ( ph  ->  .0.  e.  B )
10 eqid 2451 . . . . . . . 8  |-  ( le
`  R )  =  ( le `  R
)
11 orngmullt.l . . . . . . . 8  |-  .<  =  ( lt `  R )
1210, 11pltval 15241 . . . . . . 7  |-  ( ( R  e. oRing  /\  .0.  e.  B  /\  X  e.  B
)  ->  (  .0.  .<  X 
<->  (  .0.  ( le
`  R ) X  /\  .0.  =/=  X
) ) )
131, 9, 2, 12syl3anc 1219 . . . . . 6  |-  ( ph  ->  (  .0.  .<  X  <->  (  .0.  ( le `  R ) X  /\  .0.  =/=  X ) ) )
143, 13mpbid 210 . . . . 5  |-  ( ph  ->  (  .0.  ( le
`  R ) X  /\  .0.  =/=  X
) )
1514simpld 459 . . . 4  |-  ( ph  ->  .0.  ( le `  R ) X )
16 orngmullt.3 . . . 4  |-  ( ph  ->  Y  e.  B )
17 orngmullt.y . . . . . 6  |-  ( ph  ->  .0.  .<  Y )
1810, 11pltval 15241 . . . . . . 7  |-  ( ( R  e. oRing  /\  .0.  e.  B  /\  Y  e.  B
)  ->  (  .0.  .<  Y 
<->  (  .0.  ( le
`  R ) Y  /\  .0.  =/=  Y
) ) )
191, 9, 16, 18syl3anc 1219 . . . . . 6  |-  ( ph  ->  (  .0.  .<  Y  <->  (  .0.  ( le `  R ) Y  /\  .0.  =/=  Y ) ) )
2017, 19mpbid 210 . . . . 5  |-  ( ph  ->  (  .0.  ( le
`  R ) Y  /\  .0.  =/=  Y
) )
2120simpld 459 . . . 4  |-  ( ph  ->  .0.  ( le `  R ) Y )
22 orngmullt.t . . . . 5  |-  .x.  =  ( .r `  R )
236, 10, 7, 22orngmul 26409 . . . 4  |-  ( ( 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 1228 . . 3  |-  ( ph  ->  .0.  ( le `  R ) ( X 
.x.  Y ) )
2514simprd 463 . . . . . . 7  |-  ( ph  ->  .0.  =/=  X )
2625necomd 2719 . . . . . 6  |-  ( ph  ->  X  =/=  .0.  )
2720simprd 463 . . . . . . 7  |-  ( ph  ->  .0.  =/=  Y )
2827necomd 2719 . . . . . 6  |-  ( ph  ->  Y  =/=  .0.  )
2926, 28jca 532 . . . . 5  |-  ( ph  ->  ( X  =/=  .0.  /\  Y  =/=  .0.  )
)
30 orngmullt.4 . . . . . 6  |-  ( ph  ->  R  e.  DivRing )
316, 7, 22, 30, 2, 16drngmulne0 16969 . . . . 5  |-  ( ph  ->  ( ( X  .x.  Y )  =/=  .0.  <->  ( X  =/=  .0.  /\  Y  =/=  .0.  ) ) )
3229, 31mpbird 232 . . . 4  |-  ( ph  ->  ( X  .x.  Y
)  =/=  .0.  )
3332necomd 2719 . . 3  |-  ( ph  ->  .0.  =/=  ( X 
.x.  Y ) )
3424, 33jca 532 . 2  |-  ( ph  ->  (  .0.  ( le
`  R ) ( X  .x.  Y )  /\  .0.  =/=  ( X  .x.  Y ) ) )
351, 4syl 16 . . . 4  |-  ( ph  ->  R  e.  Ring )
366, 22rngcl 16773 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .x.  Y )  e.  B )
3735, 2, 16, 36syl3anc 1219 . . 3  |-  ( ph  ->  ( X  .x.  Y
)  e.  B )
3810, 11pltval 15241 . . 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 ) ) ) )
391, 9, 37, 38syl3anc 1219 . 2  |-  ( ph  ->  (  .0.  .<  ( X  .x.  Y )  <->  (  .0.  ( le `  R ) ( X  .x.  Y
)  /\  .0.  =/=  ( X  .x.  Y ) ) ) )
4034, 39mpbird 232 1  |-  ( ph  ->  .0.  .<  ( X  .x.  Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2644   class class class wbr 4393   ` cfv 5519  (class class class)co 6193   Basecbs 14285   .rcmulr 14350   lecple 14356   0gc0g 14489   ltcplt 15222   Grpcgrp 15521   Ringcrg 16760   DivRingcdr 16947  oRingcorng 26401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-tpos 6848  df-recs 6935  df-rdg 6969  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-nn 10427  df-2 10484  df-3 10485  df-ndx 14288  df-slot 14289  df-base 14290  df-sets 14291  df-ress 14292  df-plusg 14362  df-mulr 14363  df-0g 14491  df-plt 15239  df-mnd 15526  df-grp 15656  df-minusg 15657  df-mgp 16706  df-ur 16718  df-rng 16762  df-oppr 16830  df-dvdsr 16848  df-unit 16849  df-invr 16879  df-drng 16949  df-orng 26403
This theorem is referenced by: (None)
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