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Theorem omndmul 28551
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 6315 . . . 4  |-  ( m  =  0  ->  (
m  .x.  X )  =  ( 0  .x. 
X ) )
3 oveq1 6315 . . . 4  |-  ( m  =  0  ->  (
m  .x.  Y )  =  ( 0  .x. 
Y ) )
42, 3breq12d 4408 . . 3  |-  ( m  =  0  ->  (
( m  .x.  X
)  .<_  ( m  .x.  Y )  <->  ( 0 
.x.  X )  .<_  ( 0  .x.  Y
) ) )
5 oveq1 6315 . . . 4  |-  ( m  =  n  ->  (
m  .x.  X )  =  ( n  .x.  X ) )
6 oveq1 6315 . . . 4  |-  ( m  =  n  ->  (
m  .x.  Y )  =  ( n  .x.  Y ) )
75, 6breq12d 4408 . . 3  |-  ( m  =  n  ->  (
( m  .x.  X
)  .<_  ( m  .x.  Y )  <->  ( n  .x.  X )  .<_  ( n 
.x.  Y ) ) )
8 oveq1 6315 . . . 4  |-  ( m  =  ( n  + 
1 )  ->  (
m  .x.  X )  =  ( ( n  +  1 )  .x.  X ) )
9 oveq1 6315 . . . 4  |-  ( m  =  ( n  + 
1 )  ->  (
m  .x.  Y )  =  ( ( n  +  1 )  .x.  Y ) )
108, 9breq12d 4408 . . 3  |-  ( m  =  ( n  + 
1 )  ->  (
( m  .x.  X
)  .<_  ( m  .x.  Y )  <->  ( (
n  +  1 ) 
.x.  X )  .<_  ( ( n  + 
1 )  .x.  Y
) ) )
11 oveq1 6315 . . . 4  |-  ( m  =  N  ->  (
m  .x.  X )  =  ( N  .x.  X ) )
12 oveq1 6315 . . . 4  |-  ( m  =  N  ->  (
m  .x.  Y )  =  ( N  .x.  Y ) )
1311, 12breq12d 4408 . . 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 28542 . . . . . 6  |-  ( M  e. oMnd  ->  M  e. Toset )
16 tospos 28494 . . . . . 6  |-  ( M  e. Toset  ->  M  e.  Poset )
1714, 15, 163syl 18 . . . . 5  |-  ( ph  ->  M  e.  Poset )
18 omndmul.y . . . . . . 7  |-  ( ph  ->  Y  e.  B )
19 omndmul.0 . . . . . . . 8  |-  B  =  ( Base `  M
)
20 eqid 2471 . . . . . . . 8  |-  ( 0g
`  M )  =  ( 0g `  M
)
21 omndmul.2 . . . . . . . 8  |-  .x.  =  (.g
`  M )
2219, 20, 21mulg0 16841 . . . . . . 7  |-  ( Y  e.  B  ->  (
0  .x.  Y )  =  ( 0g `  M ) )
2318, 22syl 17 . . . . . 6  |-  ( ph  ->  ( 0  .x.  Y
)  =  ( 0g
`  M ) )
24 omndmnd 28541 . . . . . . 7  |-  ( M  e. oMnd  ->  M  e.  Mnd )
2519, 20mndidcl 16632 . . . . . . 7  |-  ( M  e.  Mnd  ->  ( 0g `  M )  e.  B )
2614, 24, 253syl 18 . . . . . 6  |-  ( ph  ->  ( 0g `  M
)  e.  B )
2723, 26eqeltrd 2549 . . . . 5  |-  ( ph  ->  ( 0  .x.  Y
)  e.  B )
28 omndmul.1 . . . . . 6  |-  .<_  =  ( le `  M )
2919, 28posref 16274 . . . . 5  |-  ( ( M  e.  Poset  /\  (
0  .x.  Y )  e.  B )  ->  (
0  .x.  Y )  .<_  ( 0  .x.  Y
) )
3017, 27, 29syl2anc 673 . . . 4  |-  ( ph  ->  ( 0  .x.  Y
)  .<_  ( 0  .x. 
Y ) )
31 omndmul.x . . . . 5  |-  ( ph  ->  X  e.  B )
3219, 20, 21mulg0 16841 . . . . . . . 8  |-  ( X  e.  B  ->  (
0  .x.  X )  =  ( 0g `  M ) )
3332adantr 472 . . . . . . 7  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( 0  .x.  X
)  =  ( 0g
`  M ) )
3422adantl 473 . . . . . . 7  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( 0  .x.  Y
)  =  ( 0g
`  M ) )
3533, 34eqtr4d 2508 . . . . . 6  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( 0  .x.  X
)  =  ( 0 
.x.  Y ) )
3635breq1d 4405 . . . . 5  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( ( 0  .x. 
X )  .<_  ( 0 
.x.  Y )  <->  ( 0 
.x.  Y )  .<_  ( 0  .x.  Y
) ) )
3731, 18, 36syl2anc 673 . . . 4  |-  ( ph  ->  ( ( 0  .x. 
X )  .<_  ( 0 
.x.  Y )  <->  ( 0 
.x.  Y )  .<_  ( 0  .x.  Y
) ) )
3830, 37mpbird 240 . . 3  |-  ( ph  ->  ( 0  .x.  X
)  .<_  ( 0  .x. 
Y ) )
39 eqid 2471 . . . . 5  |-  ( +g  `  M )  =  ( +g  `  M )
4014ad2antrr 740 . . . . 5  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  (
n  .x.  X )  .<_  ( n  .x.  Y
) )  ->  M  e. oMnd )
4118ad2antrr 740 . . . . 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 770 . . . . . 6  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  (
n  .x.  X )  .<_  ( n  .x.  Y
) )  ->  n  e.  NN0 )
4431ad2antrr 740 . . . . . 6  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  (
n  .x.  X )  .<_  ( n  .x.  Y
) )  ->  X  e.  B )
4519, 21mulgnn0cl 16852 . . . . . 6  |-  ( ( M  e.  Mnd  /\  n  e.  NN0  /\  X  e.  B )  ->  (
n  .x.  X )  e.  B )
4642, 43, 44, 45syl3anc 1292 . . . . 5  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  (
n  .x.  X )  .<_  ( n  .x.  Y
) )  ->  (
n  .x.  X )  e.  B )
4719, 21mulgnn0cl 16852 . . . . . 6  |-  ( ( M  e.  Mnd  /\  n  e.  NN0  /\  Y  e.  B )  ->  (
n  .x.  Y )  e.  B )
4842, 43, 41, 47syl3anc 1292 . . . . 5  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  (
n  .x.  X )  .<_  ( n  .x.  Y
) )  ->  (
n  .x.  Y )  e.  B )
49 simpr 468 . . . . 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 740 . . . . 5  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  (
n  .x.  X )  .<_  ( n  .x.  Y
) )  ->  X  .<_  Y )
52 omndmul.c . . . . . 6  |-  ( ph  ->  M  e. CMnd )
5352ad2antrr 740 . . . . 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 28545 . . . 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 16847 . . . . 5  |-  ( ( M  e.  Mnd  /\  n  e.  NN0  /\  X  e.  B )  ->  (
( n  +  1 )  .x.  X )  =  ( ( n 
.x.  X ) ( +g  `  M ) X ) )
5642, 43, 44, 55syl3anc 1292 . . . 4  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  (
n  .x.  X )  .<_  ( n  .x.  Y
) )  ->  (
( n  +  1 )  .x.  X )  =  ( ( n 
.x.  X ) ( +g  `  M ) X ) )
5719, 21, 39mulgnn0p1 16847 . . . . 5  |-  ( ( M  e.  Mnd  /\  n  e.  NN0  /\  Y  e.  B )  ->  (
( n  +  1 )  .x.  Y )  =  ( ( n 
.x.  Y ) ( +g  `  M ) Y ) )
5842, 43, 41, 57syl3anc 1292 . . . 4  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  (
n  .x.  X )  .<_  ( n  .x.  Y
) )  ->  (
( n  +  1 )  .x.  Y )  =  ( ( n 
.x.  Y ) ( +g  `  M ) Y ) )
5954, 56, 583brtr4d 4426 . . 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 11055 . 2  |-  ( (
ph  /\  N  e.  NN0 )  ->  ( N  .x.  X )  .<_  ( N 
.x.  Y ) )
611, 60mpdan 681 1  |-  ( ph  ->  ( N  .x.  X
)  .<_  ( N  .x.  Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904   class class class wbr 4395   ` cfv 5589  (class class class)co 6308   0cc0 9557   1c1 9558    + caddc 9560   NN0cn0 10893   Basecbs 15199   +g cplusg 15268   lecple 15275   0gc0g 15416   Posetcpo 16263  Tosetctos 16357   Mndcmnd 16613  .gcmg 16750  CMndccmn 17508  oMndcomnd 28534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-seq 12252  df-0g 15418  df-preset 16251  df-poset 16269  df-toset 16358  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-mulg 16754  df-cmn 17510  df-omnd 28536
This theorem is referenced by: (None)
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