Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  submomnd Structured version   Unicode version

Theorem submomnd 27934
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 459 . 2  |-  ( ( M  e. oMnd  /\  ( Ms  A )  e.  Mnd )  ->  ( Ms  A )  e.  Mnd )
2 omndtos 27929 . . . 4  |-  ( M  e. oMnd  ->  M  e. Toset )
32adantr 463 . . 3  |-  ( ( M  e. oMnd  /\  ( Ms  A )  e.  Mnd )  ->  M  e. Toset )
4 reldmress 14769 . . . . . . . 8  |-  Rel  doms
54ovprc2 6302 . . . . . . 7  |-  ( -.  A  e.  _V  ->  ( Ms  A )  =  (/) )
65fveq2d 5852 . . . . . 6  |-  ( -.  A  e.  _V  ->  (
Base `  ( Ms  A
) )  =  (
Base `  (/) ) )
76adantl 464 . . . . 5  |-  ( ( ( M  e. oMnd  /\  ( Ms  A )  e.  Mnd )  /\  -.  A  e. 
_V )  ->  ( Base `  ( Ms  A ) )  =  ( Base `  (/) ) )
8 base0 14757 . . . . 5  |-  (/)  =  (
Base `  (/) )
97, 8syl6eqr 2513 . . . 4  |-  ( ( ( M  e. oMnd  /\  ( Ms  A )  e.  Mnd )  /\  -.  A  e. 
_V )  ->  ( Base `  ( Ms  A ) )  =  (/) )
10 eqid 2454 . . . . . . . 8  |-  ( Base `  ( Ms  A ) )  =  ( Base `  ( Ms  A ) )
11 eqid 2454 . . . . . . . 8  |-  ( 0g
`  ( Ms  A ) )  =  ( 0g
`  ( Ms  A ) )
1210, 11mndidcl 16137 . . . . . . 7  |-  ( ( Ms  A )  e.  Mnd  ->  ( 0g `  ( Ms  A ) )  e.  ( Base `  ( Ms  A ) ) )
13 ne0i 3789 . . . . . . 7  |-  ( ( 0g `  ( Ms  A ) )  e.  (
Base `  ( Ms  A
) )  ->  ( Base `  ( Ms  A ) )  =/=  (/) )
1412, 13syl 16 . . . . . 6  |-  ( ( Ms  A )  e.  Mnd  ->  ( Base `  ( Ms  A ) )  =/=  (/) )
1514ad2antlr 724 . . . . 5  |-  ( ( ( M  e. oMnd  /\  ( Ms  A )  e.  Mnd )  /\  -.  A  e. 
_V )  ->  ( Base `  ( Ms  A ) )  =/=  (/) )
1615neneqd 2656 . . . 4  |-  ( ( ( M  e. oMnd  /\  ( Ms  A )  e.  Mnd )  /\  -.  A  e. 
_V )  ->  -.  ( Base `  ( Ms  A
) )  =  (/) )
179, 16condan 792 . . 3  |-  ( ( M  e. oMnd  /\  ( Ms  A )  e.  Mnd )  ->  A  e.  _V )
18 resstos 27882 . . 3  |-  ( ( M  e. Toset  /\  A  e. 
_V )  ->  ( Ms  A )  e. Toset )
193, 17, 18syl2anc 659 . 2  |-  ( ( M  e. oMnd  /\  ( Ms  A )  e.  Mnd )  ->  ( Ms  A )  e. Toset )
20 simplll 757 . . . . . 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 2454 . . . . . . . . . . 11  |-  ( Ms  A )  =  ( Ms  A )
22 eqid 2454 . . . . . . . . . . 11  |-  ( Base `  M )  =  (
Base `  M )
2321, 22ressbas 14773 . . . . . . . . . 10  |-  ( A  e.  _V  ->  ( A  i^i  ( Base `  M
) )  =  (
Base `  ( Ms  A
) ) )
24 inss2 3705 . . . . . . . . . 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 723 . . . . . . 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 1036 . . . . . . 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 1037 . . . . . . 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 1038 . . . . . . 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 2454 . . . . . . . . . . 11  |-  ( le
`  M )  =  ( le `  M
)
3521, 34ressle 14888 . . . . . . . . . 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 463 . . . . . . . 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 4450 . . . . . . 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 483 . . . . . 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 2454 . . . . . . 7  |-  ( +g  `  M )  =  ( +g  `  M )
4122, 34, 40omndadd 27930 . . . . . 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 1239 . . . . 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 463 . . . . . . . . 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 14830 . . . . . . . . 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 6287 . . . . . . 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 6287 . . . . . . 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 4453 . . . . . 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 463 . . . . 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 432 . . 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 2876 . 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 2454 . . 3  |-  ( +g  `  ( Ms  A ) )  =  ( +g  `  ( Ms  A ) )
55 eqid 2454 . . 3  |-  ( le
`  ( Ms  A ) )  =  ( le
`  ( Ms  A ) )
5610, 54, 55isomnd 27925 . 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 1178 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   _Vcvv 3106    i^i cin 3460    C_ wss 3461   (/)c0 3783   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   Basecbs 14716   ↾s cress 14717   +g cplusg 14784   lecple 14791   0gc0g 14929  Tosetctos 15862   Mndcmnd 16118  oMndcomnd 27921
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-ple 14804  df-0g 14931  df-poset 15774  df-toset 15863  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-omnd 27923
This theorem is referenced by:  suborng  28040  nn0omnd  28066
  Copyright terms: Public domain W3C validator