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Theorem omndmul 28037
Description: In a commutative ordered monoid, the ordering is compatible with group power. (Contributed by Thierry Arnoux, 30-Jan-2018.)
Hypotheses
Ref Expression
omndmul.0  |-  B  =  ( Base `  M
)
omndmul.1  |-  .<_  =  ( le `  M )
omndmul.2  |-  .x.  =  (.g
`  M )
omndmul.o  |-  ( ph  ->  M  e. oMnd )
omndmul.c  |-  ( ph  ->  M  e. CMnd )
omndmul.x  |-  ( ph  ->  X  e.  B )
omndmul.y  |-  ( ph  ->  Y  e.  B )
omndmul.n  |-  ( ph  ->  N  e.  NN0 )
omndmul.l  |-  ( ph  ->  X  .<_  Y )
Assertion
Ref Expression
omndmul  |-  ( ph  ->  ( N  .x.  X
)  .<_  ( N  .x.  Y ) )

Proof of Theorem omndmul
Dummy variables  m  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 omndmul.n . 2  |-  ( ph  ->  N  e.  NN0 )
2 oveq1 6239 . . . 4  |-  ( m  =  0  ->  (
m  .x.  X )  =  ( 0  .x. 
X ) )
3 oveq1 6239 . . . 4  |-  ( m  =  0  ->  (
m  .x.  Y )  =  ( 0  .x. 
Y ) )
42, 3breq12d 4405 . . 3  |-  ( m  =  0  ->  (
( m  .x.  X
)  .<_  ( m  .x.  Y )  <->  ( 0 
.x.  X )  .<_  ( 0  .x.  Y
) ) )
5 oveq1 6239 . . . 4  |-  ( m  =  n  ->  (
m  .x.  X )  =  ( n  .x.  X ) )
6 oveq1 6239 . . . 4  |-  ( m  =  n  ->  (
m  .x.  Y )  =  ( n  .x.  Y ) )
75, 6breq12d 4405 . . 3  |-  ( m  =  n  ->  (
( m  .x.  X
)  .<_  ( m  .x.  Y )  <->  ( n  .x.  X )  .<_  ( n 
.x.  Y ) ) )
8 oveq1 6239 . . . 4  |-  ( m  =  ( n  + 
1 )  ->  (
m  .x.  X )  =  ( ( n  +  1 )  .x.  X ) )
9 oveq1 6239 . . . 4  |-  ( m  =  ( n  + 
1 )  ->  (
m  .x.  Y )  =  ( ( n  +  1 )  .x.  Y ) )
108, 9breq12d 4405 . . 3  |-  ( m  =  ( n  + 
1 )  ->  (
( m  .x.  X
)  .<_  ( m  .x.  Y )  <->  ( (
n  +  1 ) 
.x.  X )  .<_  ( ( n  + 
1 )  .x.  Y
) ) )
11 oveq1 6239 . . . 4  |-  ( m  =  N  ->  (
m  .x.  X )  =  ( N  .x.  X ) )
12 oveq1 6239 . . . 4  |-  ( m  =  N  ->  (
m  .x.  Y )  =  ( N  .x.  Y ) )
1311, 12breq12d 4405 . . 3  |-  ( m  =  N  ->  (
( m  .x.  X
)  .<_  ( m  .x.  Y )  <->  ( N  .x.  X )  .<_  ( N 
.x.  Y ) ) )
14 omndmul.o . . . . . 6  |-  ( ph  ->  M  e. oMnd )
15 omndtos 28028 . . . . . 6  |-  ( M  e. oMnd  ->  M  e. Toset )
16 tospos 27979 . . . . . 6  |-  ( M  e. Toset  ->  M  e.  Poset )
1714, 15, 163syl 20 . . . . 5  |-  ( ph  ->  M  e.  Poset )
18 omndmul.y . . . . . . 7  |-  ( ph  ->  Y  e.  B )
19 omndmul.0 . . . . . . . 8  |-  B  =  ( Base `  M
)
20 eqid 2400 . . . . . . . 8  |-  ( 0g
`  M )  =  ( 0g `  M
)
21 omndmul.2 . . . . . . . 8  |-  .x.  =  (.g
`  M )
2219, 20, 21mulg0 16361 . . . . . . 7  |-  ( Y  e.  B  ->  (
0  .x.  Y )  =  ( 0g `  M ) )
2318, 22syl 17 . . . . . 6  |-  ( ph  ->  ( 0  .x.  Y
)  =  ( 0g
`  M ) )
24 omndmnd 28027 . . . . . . 7  |-  ( M  e. oMnd  ->  M  e.  Mnd )
2519, 20mndidcl 16152 . . . . . . 7  |-  ( M  e.  Mnd  ->  ( 0g `  M )  e.  B )
2614, 24, 253syl 20 . . . . . 6  |-  ( ph  ->  ( 0g `  M
)  e.  B )
2723, 26eqeltrd 2488 . . . . 5  |-  ( ph  ->  ( 0  .x.  Y
)  e.  B )
28 omndmul.1 . . . . . 6  |-  .<_  =  ( le `  M )
2919, 28posref 15794 . . . . 5  |-  ( ( M  e.  Poset  /\  (
0  .x.  Y )  e.  B )  ->  (
0  .x.  Y )  .<_  ( 0  .x.  Y
) )
3017, 27, 29syl2anc 659 . . . 4  |-  ( ph  ->  ( 0  .x.  Y
)  .<_  ( 0  .x. 
Y ) )
31 omndmul.x . . . . 5  |-  ( ph  ->  X  e.  B )
3219, 20, 21mulg0 16361 . . . . . . . 8  |-  ( X  e.  B  ->  (
0  .x.  X )  =  ( 0g `  M ) )
3332adantr 463 . . . . . . 7  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( 0  .x.  X
)  =  ( 0g
`  M ) )
3422adantl 464 . . . . . . 7  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( 0  .x.  Y
)  =  ( 0g
`  M ) )
3533, 34eqtr4d 2444 . . . . . 6  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( 0  .x.  X
)  =  ( 0 
.x.  Y ) )
3635breq1d 4402 . . . . 5  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( ( 0  .x. 
X )  .<_  ( 0 
.x.  Y )  <->  ( 0 
.x.  Y )  .<_  ( 0  .x.  Y
) ) )
3731, 18, 36syl2anc 659 . . . 4  |-  ( ph  ->  ( ( 0  .x. 
X )  .<_  ( 0 
.x.  Y )  <->  ( 0 
.x.  Y )  .<_  ( 0  .x.  Y
) ) )
3830, 37mpbird 232 . . 3  |-  ( ph  ->  ( 0  .x.  X
)  .<_  ( 0  .x. 
Y ) )
39 eqid 2400 . . . . 5  |-  ( +g  `  M )  =  ( +g  `  M )
4014ad2antrr 724 . . . . 5  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  (
n  .x.  X )  .<_  ( n  .x.  Y
) )  ->  M  e. oMnd )
4118ad2antrr 724 . . . . 5  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  (
n  .x.  X )  .<_  ( n  .x.  Y
) )  ->  Y  e.  B )
4240, 24syl 17 . . . . . 6  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  (
n  .x.  X )  .<_  ( n  .x.  Y
) )  ->  M  e.  Mnd )
43 simplr 754 . . . . . 6  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  (
n  .x.  X )  .<_  ( n  .x.  Y
) )  ->  n  e.  NN0 )
4431ad2antrr 724 . . . . . 6  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  (
n  .x.  X )  .<_  ( n  .x.  Y
) )  ->  X  e.  B )
4519, 21mulgnn0cl 16372 . . . . . 6  |-  ( ( M  e.  Mnd  /\  n  e.  NN0  /\  X  e.  B )  ->  (
n  .x.  X )  e.  B )
4642, 43, 44, 45syl3anc 1228 . . . . 5  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  (
n  .x.  X )  .<_  ( n  .x.  Y
) )  ->  (
n  .x.  X )  e.  B )
4719, 21mulgnn0cl 16372 . . . . . 6  |-  ( ( M  e.  Mnd  /\  n  e.  NN0  /\  Y  e.  B )  ->  (
n  .x.  Y )  e.  B )
4842, 43, 41, 47syl3anc 1228 . . . . 5  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  (
n  .x.  X )  .<_  ( n  .x.  Y
) )  ->  (
n  .x.  Y )  e.  B )
49 simpr 459 . . . . 5  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  (
n  .x.  X )  .<_  ( n  .x.  Y
) )  ->  (
n  .x.  X )  .<_  ( n  .x.  Y
) )
50 omndmul.l . . . . . 6  |-  ( ph  ->  X  .<_  Y )
5150ad2antrr 724 . . . . 5  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  (
n  .x.  X )  .<_  ( n  .x.  Y
) )  ->  X  .<_  Y )
52 omndmul.c . . . . . 6  |-  ( ph  ->  M  e. CMnd )
5352ad2antrr 724 . . . . 5  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  (
n  .x.  X )  .<_  ( n  .x.  Y
) )  ->  M  e. CMnd )
5419, 28, 39, 40, 41, 46, 44, 48, 49, 51, 53omndadd2d 28031 . . . 4  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  (
n  .x.  X )  .<_  ( n  .x.  Y
) )  ->  (
( n  .x.  X
) ( +g  `  M
) X )  .<_  ( ( n  .x.  Y ) ( +g  `  M ) Y ) )
5519, 21, 39mulgnn0p1 16367 . . . . 5  |-  ( ( M  e.  Mnd  /\  n  e.  NN0  /\  X  e.  B )  ->  (
( n  +  1 )  .x.  X )  =  ( ( n 
.x.  X ) ( +g  `  M ) X ) )
5642, 43, 44, 55syl3anc 1228 . . . 4  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  (
n  .x.  X )  .<_  ( n  .x.  Y
) )  ->  (
( n  +  1 )  .x.  X )  =  ( ( n 
.x.  X ) ( +g  `  M ) X ) )
5719, 21, 39mulgnn0p1 16367 . . . . 5  |-  ( ( M  e.  Mnd  /\  n  e.  NN0  /\  Y  e.  B )  ->  (
( n  +  1 )  .x.  Y )  =  ( ( n 
.x.  Y ) ( +g  `  M ) Y ) )
5842, 43, 41, 57syl3anc 1228 . . . 4  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  (
n  .x.  X )  .<_  ( n  .x.  Y
) )  ->  (
( n  +  1 )  .x.  Y )  =  ( ( n 
.x.  Y ) ( +g  `  M ) Y ) )
5954, 56, 583brtr4d 4422 . . 3  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  (
n  .x.  X )  .<_  ( n  .x.  Y
) )  ->  (
( n  +  1 )  .x.  X ) 
.<_  ( ( n  + 
1 )  .x.  Y
) )
604, 7, 10, 13, 38, 59nn0indd 10918 . 2  |-  ( (
ph  /\  N  e.  NN0 )  ->  ( N  .x.  X )  .<_  ( N 
.x.  Y ) )
611, 60mpdan 666 1  |-  ( ph  ->  ( N  .x.  X
)  .<_  ( N  .x.  Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1403    e. wcel 1840   class class class wbr 4392   ` cfv 5523  (class class class)co 6232   0cc0 9440   1c1 9441    + caddc 9443   NN0cn0 10754   Basecbs 14731   +g cplusg 14799   lecple 14806   0gc0g 14944   Posetcpo 15783  Tosetctos 15877   Mndcmnd 16133  .gcmg 16270  CMndccmn 17012  oMndcomnd 28020
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-inf2 8009  ax-cnex 9496  ax-resscn 9497  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-addrcl 9501  ax-mulcl 9502  ax-mulrcl 9503  ax-mulcom 9504  ax-addass 9505  ax-mulass 9506  ax-distr 9507  ax-i2m1 9508  ax-1ne0 9509  ax-1rid 9510  ax-rnegex 9511  ax-rrecex 9512  ax-cnre 9513  ax-pre-lttri 9514  ax-pre-lttrn 9515  ax-pre-ltadd 9516  ax-pre-mulgt0 9517
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-om 6637  df-1st 6736  df-2nd 6737  df-recs 6997  df-rdg 7031  df-er 7266  df-en 7473  df-dom 7474  df-sdom 7475  df-pnf 9578  df-mnf 9579  df-xr 9580  df-ltxr 9581  df-le 9582  df-sub 9761  df-neg 9762  df-nn 10495  df-n0 10755  df-z 10824  df-uz 11044  df-fz 11642  df-seq 12060  df-0g 14946  df-preset 15771  df-poset 15789  df-toset 15878  df-mgm 16086  df-sgrp 16125  df-mnd 16135  df-mulg 16274  df-cmn 17014  df-omnd 28022
This theorem is referenced by: (None)
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