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Theorem omndmul 26342
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 6210 . . . 4  |-  ( m  =  0  ->  (
m  .x.  X )  =  ( 0  .x. 
X ) )
3 oveq1 6210 . . . 4  |-  ( m  =  0  ->  (
m  .x.  Y )  =  ( 0  .x. 
Y ) )
42, 3breq12d 4416 . . 3  |-  ( m  =  0  ->  (
( m  .x.  X
)  .<_  ( m  .x.  Y )  <->  ( 0 
.x.  X )  .<_  ( 0  .x.  Y
) ) )
5 oveq1 6210 . . . 4  |-  ( m  =  n  ->  (
m  .x.  X )  =  ( n  .x.  X ) )
6 oveq1 6210 . . . 4  |-  ( m  =  n  ->  (
m  .x.  Y )  =  ( n  .x.  Y ) )
75, 6breq12d 4416 . . 3  |-  ( m  =  n  ->  (
( m  .x.  X
)  .<_  ( m  .x.  Y )  <->  ( n  .x.  X )  .<_  ( n 
.x.  Y ) ) )
8 oveq1 6210 . . . 4  |-  ( m  =  ( n  + 
1 )  ->  (
m  .x.  X )  =  ( ( n  +  1 )  .x.  X ) )
9 oveq1 6210 . . . 4  |-  ( m  =  ( n  + 
1 )  ->  (
m  .x.  Y )  =  ( ( n  +  1 )  .x.  Y ) )
108, 9breq12d 4416 . . 3  |-  ( m  =  ( n  + 
1 )  ->  (
( m  .x.  X
)  .<_  ( m  .x.  Y )  <->  ( (
n  +  1 ) 
.x.  X )  .<_  ( ( n  + 
1 )  .x.  Y
) ) )
11 oveq1 6210 . . . 4  |-  ( m  =  N  ->  (
m  .x.  X )  =  ( N  .x.  X ) )
12 oveq1 6210 . . . 4  |-  ( m  =  N  ->  (
m  .x.  Y )  =  ( N  .x.  Y ) )
1311, 12breq12d 4416 . . 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 26333 . . . . . 6  |-  ( M  e. oMnd  ->  M  e. Toset )
16 tospos 26284 . . . . . 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 2454 . . . . . . . 8  |-  ( 0g
`  M )  =  ( 0g `  M
)
21 omndmul.2 . . . . . . . 8  |-  .x.  =  (.g
`  M )
2219, 20, 21mulg0 15754 . . . . . . 7  |-  ( Y  e.  B  ->  (
0  .x.  Y )  =  ( 0g `  M ) )
2318, 22syl 16 . . . . . 6  |-  ( ph  ->  ( 0  .x.  Y
)  =  ( 0g
`  M ) )
24 omndmnd 26332 . . . . . . 7  |-  ( M  e. oMnd  ->  M  e.  Mnd )
2519, 20mndidcl 15561 . . . . . . 7  |-  ( M  e.  Mnd  ->  ( 0g `  M )  e.  B )
2614, 24, 253syl 20 . . . . . 6  |-  ( ph  ->  ( 0g `  M
)  e.  B )
2723, 26eqeltrd 2542 . . . . 5  |-  ( ph  ->  ( 0  .x.  Y
)  e.  B )
28 omndmul.1 . . . . . 6  |-  .<_  =  ( le `  M )
2919, 28posref 15243 . . . . 5  |-  ( ( M  e.  Poset  /\  (
0  .x.  Y )  e.  B )  ->  (
0  .x.  Y )  .<_  ( 0  .x.  Y
) )
3017, 27, 29syl2anc 661 . . . 4  |-  ( ph  ->  ( 0  .x.  Y
)  .<_  ( 0  .x. 
Y ) )
31 omndmul.x . . . . 5  |-  ( ph  ->  X  e.  B )
3219, 20, 21mulg0 15754 . . . . . . . 8  |-  ( X  e.  B  ->  (
0  .x.  X )  =  ( 0g `  M ) )
3332adantr 465 . . . . . . 7  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( 0  .x.  X
)  =  ( 0g
`  M ) )
3422adantl 466 . . . . . . 7  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( 0  .x.  Y
)  =  ( 0g
`  M ) )
3533, 34eqtr4d 2498 . . . . . 6  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( 0  .x.  X
)  =  ( 0 
.x.  Y ) )
3635breq1d 4413 . . . . 5  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( ( 0  .x. 
X )  .<_  ( 0 
.x.  Y )  <->  ( 0 
.x.  Y )  .<_  ( 0  .x.  Y
) ) )
3731, 18, 36syl2anc 661 . . . 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 2454 . . . . 5  |-  ( +g  `  M )  =  ( +g  `  M )
4014ad2antrr 725 . . . . 5  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  (
n  .x.  X )  .<_  ( n  .x.  Y
) )  ->  M  e. oMnd )
4118ad2antrr 725 . . . . 5  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  (
n  .x.  X )  .<_  ( n  .x.  Y
) )  ->  Y  e.  B )
4240, 24syl 16 . . . . . 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 725 . . . . . 6  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  (
n  .x.  X )  .<_  ( n  .x.  Y
) )  ->  X  e.  B )
4519, 21mulgnn0cl 15765 . . . . . 6  |-  ( ( M  e.  Mnd  /\  n  e.  NN0  /\  X  e.  B )  ->  (
n  .x.  X )  e.  B )
4642, 43, 44, 45syl3anc 1219 . . . . 5  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  (
n  .x.  X )  .<_  ( n  .x.  Y
) )  ->  (
n  .x.  X )  e.  B )
4719, 21mulgnn0cl 15765 . . . . . 6  |-  ( ( M  e.  Mnd  /\  n  e.  NN0  /\  Y  e.  B )  ->  (
n  .x.  Y )  e.  B )
4842, 43, 41, 47syl3anc 1219 . . . . 5  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  (
n  .x.  X )  .<_  ( n  .x.  Y
) )  ->  (
n  .x.  Y )  e.  B )
49 simpr 461 . . . . 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 725 . . . . 5  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  (
n  .x.  X )  .<_  ( n  .x.  Y
) )  ->  X  .<_  Y )
52 omndmul.c . . . . . 6  |-  ( ph  ->  M  e. CMnd )
5352ad2antrr 725 . . . . 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 26336 . . . 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 15760 . . . . 5  |-  ( ( M  e.  Mnd  /\  n  e.  NN0  /\  X  e.  B )  ->  (
( n  +  1 )  .x.  X )  =  ( ( n 
.x.  X ) ( +g  `  M ) X ) )
5642, 43, 44, 55syl3anc 1219 . . . 4  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  (
n  .x.  X )  .<_  ( n  .x.  Y
) )  ->  (
( n  +  1 )  .x.  X )  =  ( ( n 
.x.  X ) ( +g  `  M ) X ) )
5719, 21, 39mulgnn0p1 15760 . . . . 5  |-  ( ( M  e.  Mnd  /\  n  e.  NN0  /\  Y  e.  B )  ->  (
( n  +  1 )  .x.  Y )  =  ( ( n 
.x.  Y ) ( +g  `  M ) Y ) )
5842, 43, 41, 57syl3anc 1219 . . . 4  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  (
n  .x.  X )  .<_  ( n  .x.  Y
) )  ->  (
( n  +  1 )  .x.  Y )  =  ( ( n 
.x.  Y ) ( +g  `  M ) Y ) )
5954, 56, 583brtr4d 4433 . . 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 26253 . 2  |-  ( (
ph  /\  N  e.  NN0 )  ->  ( N  .x.  X )  .<_  ( N 
.x.  Y ) )
611, 60mpdan 668 1  |-  ( ph  ->  ( N  .x.  X
)  .<_  ( N  .x.  Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   class class class wbr 4403   ` cfv 5529  (class class class)co 6203   0cc0 9396   1c1 9397    + caddc 9399   NN0cn0 10693   Basecbs 14295   +g cplusg 14360   lecple 14367   0gc0g 14500   Posetcpo 15232  Tosetctos 15325   Mndcmnd 15531  .gcmg 15536  CMndccmn 16401  oMndcomnd 26325
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 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7961  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-nn 10437  df-n0 10694  df-z 10761  df-uz 10976  df-fz 11558  df-seq 11927  df-0g 14502  df-poset 15238  df-toset 15326  df-mnd 15537  df-mulg 15670  df-cmn 16403  df-omnd 26327
This theorem is referenced by: (None)
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