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Theorem submomnd 27573
Description: A submonoid of an ordered monoid is also ordered. (Contributed by Thierry Arnoux, 23-Mar-2018.)
Assertion
Ref Expression
submomnd  |-  ( ( M  e. oMnd  /\  ( Ms  A )  e.  Mnd )  ->  ( Ms  A )  e. oMnd )

Proof of Theorem submomnd
Dummy variables  a 
b  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 461 . 2  |-  ( ( M  e. oMnd  /\  ( Ms  A )  e.  Mnd )  ->  ( Ms  A )  e.  Mnd )
2 omndtos 27568 . . . 4  |-  ( M  e. oMnd  ->  M  e. Toset )
32adantr 465 . . 3  |-  ( ( M  e. oMnd  /\  ( Ms  A )  e.  Mnd )  ->  M  e. Toset )
4 reldmress 14560 . . . . . . . 8  |-  Rel  doms
54ovprc2 6313 . . . . . . 7  |-  ( -.  A  e.  _V  ->  ( Ms  A )  =  (/) )
65fveq2d 5860 . . . . . 6  |-  ( -.  A  e.  _V  ->  (
Base `  ( Ms  A
) )  =  (
Base `  (/) ) )
76adantl 466 . . . . 5  |-  ( ( ( M  e. oMnd  /\  ( Ms  A )  e.  Mnd )  /\  -.  A  e. 
_V )  ->  ( Base `  ( Ms  A ) )  =  ( Base `  (/) ) )
8 base0 14548 . . . . 5  |-  (/)  =  (
Base `  (/) )
97, 8syl6eqr 2502 . . . 4  |-  ( ( ( M  e. oMnd  /\  ( Ms  A )  e.  Mnd )  /\  -.  A  e. 
_V )  ->  ( Base `  ( Ms  A ) )  =  (/) )
10 eqid 2443 . . . . . . . 8  |-  ( Base `  ( Ms  A ) )  =  ( Base `  ( Ms  A ) )
11 eqid 2443 . . . . . . . 8  |-  ( 0g
`  ( Ms  A ) )  =  ( 0g
`  ( Ms  A ) )
1210, 11mndidcl 15812 . . . . . . 7  |-  ( ( Ms  A )  e.  Mnd  ->  ( 0g `  ( Ms  A ) )  e.  ( Base `  ( Ms  A ) ) )
13 ne0i 3776 . . . . . . 7  |-  ( ( 0g `  ( Ms  A ) )  e.  (
Base `  ( Ms  A
) )  ->  ( Base `  ( Ms  A ) )  =/=  (/) )
1412, 13syl 16 . . . . . 6  |-  ( ( Ms  A )  e.  Mnd  ->  ( Base `  ( Ms  A ) )  =/=  (/) )
1514ad2antlr 726 . . . . 5  |-  ( ( ( M  e. oMnd  /\  ( Ms  A )  e.  Mnd )  /\  -.  A  e. 
_V )  ->  ( Base `  ( Ms  A ) )  =/=  (/) )
1615neneqd 2645 . . . 4  |-  ( ( ( M  e. oMnd  /\  ( Ms  A )  e.  Mnd )  /\  -.  A  e. 
_V )  ->  -.  ( Base `  ( Ms  A
) )  =  (/) )
179, 16condan 794 . . 3  |-  ( ( M  e. oMnd  /\  ( Ms  A )  e.  Mnd )  ->  A  e.  _V )
18 resstos 27521 . . 3  |-  ( ( M  e. Toset  /\  A  e. 
_V )  ->  ( Ms  A )  e. Toset )
193, 17, 18syl2anc 661 . 2  |-  ( ( M  e. oMnd  /\  ( Ms  A )  e.  Mnd )  ->  ( Ms  A )  e. Toset )
20 simplll 759 . . . . . 6  |-  ( ( ( ( M  e. oMnd  /\  ( Ms  A )  e.  Mnd )  /\  ( a  e.  ( Base `  ( Ms  A ) )  /\  b  e.  ( Base `  ( Ms  A ) )  /\  c  e.  ( Base `  ( Ms  A ) ) ) )  /\  a ( le `  ( Ms  A ) ) b )  ->  M  e. oMnd )
21 eqid 2443 . . . . . . . . . . 11  |-  ( Ms  A )  =  ( Ms  A )
22 eqid 2443 . . . . . . . . . . 11  |-  ( Base `  M )  =  (
Base `  M )
2321, 22ressbas 14564 . . . . . . . . . 10  |-  ( A  e.  _V  ->  ( A  i^i  ( Base `  M
) )  =  (
Base `  ( Ms  A
) ) )
24 inss2 3704 . . . . . . . . . 10  |-  ( A  i^i  ( Base `  M
) )  C_  ( Base `  M )
2523, 24syl6eqssr 3540 . . . . . . . . 9  |-  ( A  e.  _V  ->  ( Base `  ( Ms  A ) )  C_  ( Base `  M ) )
2617, 25syl 16 . . . . . . . 8  |-  ( ( M  e. oMnd  /\  ( Ms  A )  e.  Mnd )  ->  ( Base `  ( Ms  A ) )  C_  ( Base `  M )
)
2726ad2antrr 725 . . . . . . 7  |-  ( ( ( ( M  e. oMnd  /\  ( Ms  A )  e.  Mnd )  /\  ( a  e.  ( Base `  ( Ms  A ) )  /\  b  e.  ( Base `  ( Ms  A ) )  /\  c  e.  ( Base `  ( Ms  A ) ) ) )  /\  a ( le `  ( Ms  A ) ) b )  ->  ( Base `  ( Ms  A ) )  C_  ( Base `  M )
)
28 simplr1 1039 . . . . . . 7  |-  ( ( ( ( M  e. oMnd  /\  ( Ms  A )  e.  Mnd )  /\  ( a  e.  ( Base `  ( Ms  A ) )  /\  b  e.  ( Base `  ( Ms  A ) )  /\  c  e.  ( Base `  ( Ms  A ) ) ) )  /\  a ( le `  ( Ms  A ) ) b )  ->  a  e.  (
Base `  ( Ms  A
) ) )
2927, 28sseldd 3490 . . . . . 6  |-  ( ( ( ( M  e. oMnd  /\  ( Ms  A )  e.  Mnd )  /\  ( a  e.  ( Base `  ( Ms  A ) )  /\  b  e.  ( Base `  ( Ms  A ) )  /\  c  e.  ( Base `  ( Ms  A ) ) ) )  /\  a ( le `  ( Ms  A ) ) b )  ->  a  e.  (
Base `  M )
)
30 simplr2 1040 . . . . . . 7  |-  ( ( ( ( M  e. oMnd  /\  ( Ms  A )  e.  Mnd )  /\  ( a  e.  ( Base `  ( Ms  A ) )  /\  b  e.  ( Base `  ( Ms  A ) )  /\  c  e.  ( Base `  ( Ms  A ) ) ) )  /\  a ( le `  ( Ms  A ) ) b )  ->  b  e.  (
Base `  ( Ms  A
) ) )
3127, 30sseldd 3490 . . . . . 6  |-  ( ( ( ( M  e. oMnd  /\  ( Ms  A )  e.  Mnd )  /\  ( a  e.  ( Base `  ( Ms  A ) )  /\  b  e.  ( Base `  ( Ms  A ) )  /\  c  e.  ( Base `  ( Ms  A ) ) ) )  /\  a ( le `  ( Ms  A ) ) b )  ->  b  e.  (
Base `  M )
)
32 simplr3 1041 . . . . . . 7  |-  ( ( ( ( M  e. oMnd  /\  ( Ms  A )  e.  Mnd )  /\  ( a  e.  ( Base `  ( Ms  A ) )  /\  b  e.  ( Base `  ( Ms  A ) )  /\  c  e.  ( Base `  ( Ms  A ) ) ) )  /\  a ( le `  ( Ms  A ) ) b )  ->  c  e.  (
Base `  ( Ms  A
) ) )
3327, 32sseldd 3490 . . . . . 6  |-  ( ( ( ( M  e. oMnd  /\  ( Ms  A )  e.  Mnd )  /\  ( a  e.  ( Base `  ( Ms  A ) )  /\  b  e.  ( Base `  ( Ms  A ) )  /\  c  e.  ( Base `  ( Ms  A ) ) ) )  /\  a ( le `  ( Ms  A ) ) b )  ->  c  e.  (
Base `  M )
)
34 eqid 2443 . . . . . . . . . . 11  |-  ( le
`  M )  =  ( le `  M
)
3521, 34ressle 14674 . . . . . . . . . 10  |-  ( A  e.  _V  ->  ( le `  M )  =  ( le `  ( Ms  A ) ) )
3617, 35syl 16 . . . . . . . . 9  |-  ( ( M  e. oMnd  /\  ( Ms  A )  e.  Mnd )  ->  ( le `  M )  =  ( le `  ( Ms  A ) ) )
3736adantr 465 . . . . . . . 8  |-  ( ( ( M  e. oMnd  /\  ( Ms  A )  e.  Mnd )  /\  ( a  e.  ( Base `  ( Ms  A ) )  /\  b  e.  ( Base `  ( Ms  A ) )  /\  c  e.  ( Base `  ( Ms  A ) ) ) )  ->  ( le `  M )  =  ( le `  ( Ms  A ) ) )
3837breqd 4448 . . . . . . 7  |-  ( ( ( M  e. oMnd  /\  ( Ms  A )  e.  Mnd )  /\  ( a  e.  ( Base `  ( Ms  A ) )  /\  b  e.  ( Base `  ( Ms  A ) )  /\  c  e.  ( Base `  ( Ms  A ) ) ) )  ->  ( a
( le `  M
) b  <->  a ( le `  ( Ms  A ) ) b ) )
3938biimpar 485 . . . . . 6  |-  ( ( ( ( M  e. oMnd  /\  ( Ms  A )  e.  Mnd )  /\  ( a  e.  ( Base `  ( Ms  A ) )  /\  b  e.  ( Base `  ( Ms  A ) )  /\  c  e.  ( Base `  ( Ms  A ) ) ) )  /\  a ( le `  ( Ms  A ) ) b )  ->  a ( le
`  M ) b )
40 eqid 2443 . . . . . . 7  |-  ( +g  `  M )  =  ( +g  `  M )
4122, 34, 40omndadd 27569 . . . . . 6  |-  ( ( M  e. oMnd  /\  (
a  e.  ( Base `  M )  /\  b  e.  ( Base `  M
)  /\  c  e.  ( Base `  M )
)  /\  a ( le `  M ) b )  ->  ( a
( +g  `  M ) c ) ( le
`  M ) ( b ( +g  `  M
) c ) )
4220, 29, 31, 33, 39, 41syl131anc 1242 . . . . 5  |-  ( ( ( ( M  e. oMnd  /\  ( Ms  A )  e.  Mnd )  /\  ( a  e.  ( Base `  ( Ms  A ) )  /\  b  e.  ( Base `  ( Ms  A ) )  /\  c  e.  ( Base `  ( Ms  A ) ) ) )  /\  a ( le `  ( Ms  A ) ) b )  ->  ( a ( +g  `  M ) c ) ( le
`  M ) ( b ( +g  `  M
) c ) )
4317adantr 465 . . . . . . . . 9  |-  ( ( ( M  e. oMnd  /\  ( Ms  A )  e.  Mnd )  /\  ( a  e.  ( Base `  ( Ms  A ) )  /\  b  e.  ( Base `  ( Ms  A ) )  /\  c  e.  ( Base `  ( Ms  A ) ) ) )  ->  A  e.  _V )
4421, 40ressplusg 14616 . . . . . . . . 9  |-  ( A  e.  _V  ->  ( +g  `  M )  =  ( +g  `  ( Ms  A ) ) )
4543, 44syl 16 . . . . . . . 8  |-  ( ( ( M  e. oMnd  /\  ( Ms  A )  e.  Mnd )  /\  ( a  e.  ( Base `  ( Ms  A ) )  /\  b  e.  ( Base `  ( Ms  A ) )  /\  c  e.  ( Base `  ( Ms  A ) ) ) )  ->  ( +g  `  M )  =  ( +g  `  ( Ms  A ) ) )
4645oveqd 6298 . . . . . . 7  |-  ( ( ( M  e. oMnd  /\  ( Ms  A )  e.  Mnd )  /\  ( a  e.  ( Base `  ( Ms  A ) )  /\  b  e.  ( Base `  ( Ms  A ) )  /\  c  e.  ( Base `  ( Ms  A ) ) ) )  ->  ( a
( +g  `  M ) c )  =  ( a ( +g  `  ( Ms  A ) ) c ) )
4743, 35syl 16 . . . . . . 7  |-  ( ( ( M  e. oMnd  /\  ( Ms  A )  e.  Mnd )  /\  ( a  e.  ( Base `  ( Ms  A ) )  /\  b  e.  ( Base `  ( Ms  A ) )  /\  c  e.  ( Base `  ( Ms  A ) ) ) )  ->  ( le `  M )  =  ( le `  ( Ms  A ) ) )
4845oveqd 6298 . . . . . . 7  |-  ( ( ( M  e. oMnd  /\  ( Ms  A )  e.  Mnd )  /\  ( a  e.  ( Base `  ( Ms  A ) )  /\  b  e.  ( Base `  ( Ms  A ) )  /\  c  e.  ( Base `  ( Ms  A ) ) ) )  ->  ( b
( +g  `  M ) c )  =  ( b ( +g  `  ( Ms  A ) ) c ) )
4946, 47, 48breq123d 4451 . . . . . 6  |-  ( ( ( M  e. oMnd  /\  ( Ms  A )  e.  Mnd )  /\  ( a  e.  ( Base `  ( Ms  A ) )  /\  b  e.  ( Base `  ( Ms  A ) )  /\  c  e.  ( Base `  ( Ms  A ) ) ) )  ->  ( (
a ( +g  `  M
) c ) ( le `  M ) ( b ( +g  `  M ) c )  <-> 
( a ( +g  `  ( Ms  A ) ) c ) ( le `  ( Ms  A ) ) ( b ( +g  `  ( Ms  A ) ) c ) ) )
5049adantr 465 . . . . 5  |-  ( ( ( ( M  e. oMnd  /\  ( Ms  A )  e.  Mnd )  /\  ( a  e.  ( Base `  ( Ms  A ) )  /\  b  e.  ( Base `  ( Ms  A ) )  /\  c  e.  ( Base `  ( Ms  A ) ) ) )  /\  a ( le `  ( Ms  A ) ) b )  ->  ( ( a ( +g  `  M
) c ) ( le `  M ) ( b ( +g  `  M ) c )  <-> 
( a ( +g  `  ( Ms  A ) ) c ) ( le `  ( Ms  A ) ) ( b ( +g  `  ( Ms  A ) ) c ) ) )
5142, 50mpbid 210 . . . 4  |-  ( ( ( ( M  e. oMnd  /\  ( Ms  A )  e.  Mnd )  /\  ( a  e.  ( Base `  ( Ms  A ) )  /\  b  e.  ( Base `  ( Ms  A ) )  /\  c  e.  ( Base `  ( Ms  A ) ) ) )  /\  a ( le `  ( Ms  A ) ) b )  ->  ( a ( +g  `  ( Ms  A ) ) c ) ( le `  ( Ms  A ) ) ( b ( +g  `  ( Ms  A ) ) c ) )
5251ex 434 . . 3  |-  ( ( ( M  e. oMnd  /\  ( Ms  A )  e.  Mnd )  /\  ( a  e.  ( Base `  ( Ms  A ) )  /\  b  e.  ( Base `  ( Ms  A ) )  /\  c  e.  ( Base `  ( Ms  A ) ) ) )  ->  ( a
( le `  ( Ms  A ) ) b  ->  ( a ( +g  `  ( Ms  A ) ) c ) ( le `  ( Ms  A ) ) ( b ( +g  `  ( Ms  A ) ) c ) ) )
5352ralrimivvva 2865 . 2  |-  ( ( M  e. oMnd  /\  ( Ms  A )  e.  Mnd )  ->  A. a  e.  (
Base `  ( Ms  A
) ) A. b  e.  ( Base `  ( Ms  A ) ) A. c  e.  ( Base `  ( Ms  A ) ) ( a ( le `  ( Ms  A ) ) b  ->  ( a ( +g  `  ( Ms  A ) ) c ) ( le `  ( Ms  A ) ) ( b ( +g  `  ( Ms  A ) ) c ) ) )
54 eqid 2443 . . 3  |-  ( +g  `  ( Ms  A ) )  =  ( +g  `  ( Ms  A ) )
55 eqid 2443 . . 3  |-  ( le
`  ( Ms  A ) )  =  ( le
`  ( Ms  A ) )
5610, 54, 55isomnd 27564 . 2  |-  ( ( Ms  A )  e. oMnd  <->  ( ( Ms  A )  e.  Mnd  /\  ( Ms  A )  e. Toset  /\  A. a  e.  ( Base `  ( Ms  A ) ) A. b  e.  ( Base `  ( Ms  A ) ) A. c  e.  ( Base `  ( Ms  A ) ) ( a ( le `  ( Ms  A ) ) b  ->  ( a ( +g  `  ( Ms  A ) ) c ) ( le `  ( Ms  A ) ) ( b ( +g  `  ( Ms  A ) ) c ) ) ) )
571, 19, 53, 56syl3anbrc 1181 1  |-  ( ( M  e. oMnd  /\  ( Ms  A )  e.  Mnd )  ->  ( Ms  A )  e. oMnd )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804    =/= wne 2638   A.wral 2793   _Vcvv 3095    i^i cin 3460    C_ wss 3461   (/)c0 3770   class class class wbr 4437   ` cfv 5578  (class class class)co 6281   Basecbs 14509   ↾s cress 14510   +g cplusg 14574   lecple 14581   0gc0g 14714  Tosetctos 15537   Mndcmnd 15793  oMndcomnd 27560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-recs 7044  df-rdg 7078  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10543  df-2 10600  df-3 10601  df-4 10602  df-5 10603  df-6 10604  df-7 10605  df-8 10606  df-9 10607  df-10 10608  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-ress 14516  df-plusg 14587  df-ple 14594  df-0g 14716  df-poset 15449  df-toset 15538  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-omnd 27562
This theorem is referenced by:  suborng  27678  nn0omnd  27704
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