ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  rinvmod Unicode version

Theorem rinvmod 12926
Description: Uniqueness of a right inverse element in a commutative monoid, if it exists. Corresponds to caovimo 6061. (Contributed by AV, 31-Dec-2023.)
Hypotheses
Ref Expression
rinvmod.b  |-  B  =  ( Base `  G
)
rinvmod.0  |-  .0.  =  ( 0g `  G )
rinvmod.p  |-  .+  =  ( +g  `  G )
rinvmod.m  |-  ( ph  ->  G  e. CMnd )
rinvmod.a  |-  ( ph  ->  A  e.  B )
Assertion
Ref Expression
rinvmod  |-  ( ph  ->  E* w  e.  B  ( A  .+  w )  =  .0.  )
Distinct variable groups:    w, A    w, B    w,  .0.    w,  .+    ph, w
Allowed substitution hint:    G( w)

Proof of Theorem rinvmod
StepHypRef Expression
1 rinvmod.m . . . . . . . . 9  |-  ( ph  ->  G  e. CMnd )
21adantr 276 . . . . . . . 8  |-  ( (
ph  /\  w  e.  B )  ->  G  e. CMnd )
3 simpr 110 . . . . . . . 8  |-  ( (
ph  /\  w  e.  B )  ->  w  e.  B )
4 rinvmod.a . . . . . . . . 9  |-  ( ph  ->  A  e.  B )
54adantr 276 . . . . . . . 8  |-  ( (
ph  /\  w  e.  B )  ->  A  e.  B )
6 rinvmod.b . . . . . . . . 9  |-  B  =  ( Base `  G
)
7 rinvmod.p . . . . . . . . 9  |-  .+  =  ( +g  `  G )
86, 7cmncom 12919 . . . . . . . 8  |-  ( ( G  e. CMnd  /\  w  e.  B  /\  A  e.  B )  ->  (
w  .+  A )  =  ( A  .+  w ) )
92, 3, 5, 8syl3anc 1238 . . . . . . 7  |-  ( (
ph  /\  w  e.  B )  ->  (
w  .+  A )  =  ( A  .+  w ) )
109adantr 276 . . . . . 6  |-  ( ( ( ph  /\  w  e.  B )  /\  ( A  .+  w )  =  .0.  )  ->  (
w  .+  A )  =  ( A  .+  w ) )
11 simpr 110 . . . . . 6  |-  ( ( ( ph  /\  w  e.  B )  /\  ( A  .+  w )  =  .0.  )  ->  ( A  .+  w )  =  .0.  )
1210, 11eqtrd 2210 . . . . 5  |-  ( ( ( ph  /\  w  e.  B )  /\  ( A  .+  w )  =  .0.  )  ->  (
w  .+  A )  =  .0.  )
1312, 11jca 306 . . . 4  |-  ( ( ( ph  /\  w  e.  B )  /\  ( A  .+  w )  =  .0.  )  ->  (
( w  .+  A
)  =  .0.  /\  ( A  .+  w )  =  .0.  ) )
1413ex 115 . . 3  |-  ( (
ph  /\  w  e.  B )  ->  (
( A  .+  w
)  =  .0.  ->  ( ( w  .+  A
)  =  .0.  /\  ( A  .+  w )  =  .0.  ) ) )
1514ralrimiva 2550 . 2  |-  ( ph  ->  A. w  e.  B  ( ( A  .+  w )  =  .0. 
->  ( ( w  .+  A )  =  .0. 
/\  ( A  .+  w )  =  .0.  ) ) )
16 rinvmod.0 . . 3  |-  .0.  =  ( 0g `  G )
17 cmnmnd 12918 . . . 4  |-  ( G  e. CMnd  ->  G  e.  Mnd )
181, 17syl 14 . . 3  |-  ( ph  ->  G  e.  Mnd )
196, 16, 7, 18, 4mndinvmod 12723 . 2  |-  ( ph  ->  E* w  e.  B  ( ( w  .+  A )  =  .0. 
/\  ( A  .+  w )  =  .0.  ) )
20 rmoim 2938 . 2  |-  ( A. w  e.  B  (
( A  .+  w
)  =  .0.  ->  ( ( w  .+  A
)  =  .0.  /\  ( A  .+  w )  =  .0.  ) )  ->  ( E* w  e.  B  ( (
w  .+  A )  =  .0.  /\  ( A 
.+  w )  =  .0.  )  ->  E* w  e.  B  ( A  .+  w )  =  .0.  ) )
2115, 19, 20sylc 62 1  |-  ( ph  ->  E* w  e.  B  ( A  .+  w )  =  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   A.wral 2455   E*wrmo 2458   ` cfv 5211  (class class class)co 5868   Basecbs 12432   +g cplusg 12505   0gc0g 12640   Mndcmnd 12696  CMndccmn 12902
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-cnex 7880  ax-resscn 7881  ax-1re 7883  ax-addrcl 7886
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-iota 5173  df-fun 5213  df-fn 5214  df-fv 5219  df-riota 5824  df-ov 5871  df-inn 8896  df-2 8954  df-ndx 12435  df-slot 12436  df-base 12438  df-plusg 12518  df-0g 12642  df-mgm 12654  df-sgrp 12687  df-mnd 12697  df-cmn 12904
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator