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Theorem mndinvmod 12867
Description: Uniqueness of an inverse element in a monoid, if it exists. (Contributed by AV, 20-Jan-2024.)
Hypotheses
Ref Expression
mndinvmod.b  |-  B  =  ( Base `  G
)
mndinvmod.0  |-  .0.  =  ( 0g `  G )
mndinvmod.p  |-  .+  =  ( +g  `  G )
mndinvmod.m  |-  ( ph  ->  G  e.  Mnd )
mndinvmod.a  |-  ( ph  ->  A  e.  B )
Assertion
Ref Expression
mndinvmod  |-  ( ph  ->  E* w  e.  B  ( ( w  .+  A )  =  .0. 
/\  ( A  .+  w )  =  .0.  ) )
Distinct variable groups:    w, A    w, B    w,  .0.    w,  .+    ph, w
Allowed substitution hint:    G( w)

Proof of Theorem mndinvmod
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 mndinvmod.m . . . . . . . 8  |-  ( ph  ->  G  e.  Mnd )
2 simpl 109 . . . . . . . 8  |-  ( ( w  e.  B  /\  v  e.  B )  ->  w  e.  B )
3 mndinvmod.b . . . . . . . . 9  |-  B  =  ( Base `  G
)
4 mndinvmod.p . . . . . . . . 9  |-  .+  =  ( +g  `  G )
5 mndinvmod.0 . . . . . . . . 9  |-  .0.  =  ( 0g `  G )
63, 4, 5mndrid 12858 . . . . . . . 8  |-  ( ( G  e.  Mnd  /\  w  e.  B )  ->  ( w  .+  .0.  )  =  w )
71, 2, 6syl2an 289 . . . . . . 7  |-  ( (
ph  /\  ( w  e.  B  /\  v  e.  B ) )  -> 
( w  .+  .0.  )  =  w )
87eqcomd 2193 . . . . . 6  |-  ( (
ph  /\  ( w  e.  B  /\  v  e.  B ) )  ->  w  =  ( w  .+  .0.  ) )
98adantr 276 . . . . 5  |-  ( ( ( ph  /\  (
w  e.  B  /\  v  e.  B )
)  /\  ( (
( w  .+  A
)  =  .0.  /\  ( A  .+  w )  =  .0.  )  /\  ( ( v  .+  A )  =  .0. 
/\  ( A  .+  v )  =  .0.  ) ) )  ->  w  =  ( w  .+  .0.  ) )
10 oveq2 5896 . . . . . . . . 9  |-  (  .0.  =  ( A  .+  v )  ->  (
w  .+  .0.  )  =  ( w  .+  ( A  .+  v ) ) )
1110eqcoms 2190 . . . . . . . 8  |-  ( ( A  .+  v )  =  .0.  ->  (
w  .+  .0.  )  =  ( w  .+  ( A  .+  v ) ) )
1211adantl 277 . . . . . . 7  |-  ( ( ( v  .+  A
)  =  .0.  /\  ( A  .+  v )  =  .0.  )  -> 
( w  .+  .0.  )  =  ( w  .+  ( A  .+  v
) ) )
1312adantl 277 . . . . . 6  |-  ( ( ( ( w  .+  A )  =  .0. 
/\  ( A  .+  w )  =  .0.  )  /\  ( ( v  .+  A )  =  .0.  /\  ( A  .+  v )  =  .0.  ) )  -> 
( w  .+  .0.  )  =  ( w  .+  ( A  .+  v
) ) )
1413adantl 277 . . . . 5  |-  ( ( ( ph  /\  (
w  e.  B  /\  v  e.  B )
)  /\  ( (
( w  .+  A
)  =  .0.  /\  ( A  .+  w )  =  .0.  )  /\  ( ( v  .+  A )  =  .0. 
/\  ( A  .+  v )  =  .0.  ) ) )  -> 
( w  .+  .0.  )  =  ( w  .+  ( A  .+  v
) ) )
151adantr 276 . . . . . . . 8  |-  ( (
ph  /\  ( w  e.  B  /\  v  e.  B ) )  ->  G  e.  Mnd )
162adantl 277 . . . . . . . 8  |-  ( (
ph  /\  ( w  e.  B  /\  v  e.  B ) )  ->  w  e.  B )
17 mndinvmod.a . . . . . . . . 9  |-  ( ph  ->  A  e.  B )
1817adantr 276 . . . . . . . 8  |-  ( (
ph  /\  ( w  e.  B  /\  v  e.  B ) )  ->  A  e.  B )
19 simpr 110 . . . . . . . . 9  |-  ( ( w  e.  B  /\  v  e.  B )  ->  v  e.  B )
2019adantl 277 . . . . . . . 8  |-  ( (
ph  /\  ( w  e.  B  /\  v  e.  B ) )  -> 
v  e.  B )
213, 4mndass 12846 . . . . . . . . 9  |-  ( ( G  e.  Mnd  /\  ( w  e.  B  /\  A  e.  B  /\  v  e.  B
) )  ->  (
( w  .+  A
)  .+  v )  =  ( w  .+  ( A  .+  v ) ) )
2221eqcomd 2193 . . . . . . . 8  |-  ( ( G  e.  Mnd  /\  ( w  e.  B  /\  A  e.  B  /\  v  e.  B
) )  ->  (
w  .+  ( A  .+  v ) )  =  ( ( w  .+  A )  .+  v
) )
2315, 16, 18, 20, 22syl13anc 1250 . . . . . . 7  |-  ( (
ph  /\  ( w  e.  B  /\  v  e.  B ) )  -> 
( w  .+  ( A  .+  v ) )  =  ( ( w 
.+  A )  .+  v ) )
2423adantr 276 . . . . . 6  |-  ( ( ( ph  /\  (
w  e.  B  /\  v  e.  B )
)  /\  ( (
( w  .+  A
)  =  .0.  /\  ( A  .+  w )  =  .0.  )  /\  ( ( v  .+  A )  =  .0. 
/\  ( A  .+  v )  =  .0.  ) ) )  -> 
( w  .+  ( A  .+  v ) )  =  ( ( w 
.+  A )  .+  v ) )
25 oveq1 5895 . . . . . . . . 9  |-  ( ( w  .+  A )  =  .0.  ->  (
( w  .+  A
)  .+  v )  =  (  .0.  .+  v
) )
2625adantr 276 . . . . . . . 8  |-  ( ( ( w  .+  A
)  =  .0.  /\  ( A  .+  w )  =  .0.  )  -> 
( ( w  .+  A )  .+  v
)  =  (  .0.  .+  v ) )
2726adantr 276 . . . . . . 7  |-  ( ( ( ( w  .+  A )  =  .0. 
/\  ( A  .+  w )  =  .0.  )  /\  ( ( v  .+  A )  =  .0.  /\  ( A  .+  v )  =  .0.  ) )  -> 
( ( w  .+  A )  .+  v
)  =  (  .0.  .+  v ) )
2827adantl 277 . . . . . 6  |-  ( ( ( ph  /\  (
w  e.  B  /\  v  e.  B )
)  /\  ( (
( w  .+  A
)  =  .0.  /\  ( A  .+  w )  =  .0.  )  /\  ( ( v  .+  A )  =  .0. 
/\  ( A  .+  v )  =  .0.  ) ) )  -> 
( ( w  .+  A )  .+  v
)  =  (  .0.  .+  v ) )
293, 4, 5mndlid 12857 . . . . . . . 8  |-  ( ( G  e.  Mnd  /\  v  e.  B )  ->  (  .0.  .+  v
)  =  v )
301, 19, 29syl2an 289 . . . . . . 7  |-  ( (
ph  /\  ( w  e.  B  /\  v  e.  B ) )  -> 
(  .0.  .+  v
)  =  v )
3130adantr 276 . . . . . 6  |-  ( ( ( ph  /\  (
w  e.  B  /\  v  e.  B )
)  /\  ( (
( w  .+  A
)  =  .0.  /\  ( A  .+  w )  =  .0.  )  /\  ( ( v  .+  A )  =  .0. 
/\  ( A  .+  v )  =  .0.  ) ) )  -> 
(  .0.  .+  v
)  =  v )
3224, 28, 313eqtrd 2224 . . . . 5  |-  ( ( ( ph  /\  (
w  e.  B  /\  v  e.  B )
)  /\  ( (
( w  .+  A
)  =  .0.  /\  ( A  .+  w )  =  .0.  )  /\  ( ( v  .+  A )  =  .0. 
/\  ( A  .+  v )  =  .0.  ) ) )  -> 
( w  .+  ( A  .+  v ) )  =  v )
339, 14, 323eqtrd 2224 . . . 4  |-  ( ( ( ph  /\  (
w  e.  B  /\  v  e.  B )
)  /\  ( (
( w  .+  A
)  =  .0.  /\  ( A  .+  w )  =  .0.  )  /\  ( ( v  .+  A )  =  .0. 
/\  ( A  .+  v )  =  .0.  ) ) )  ->  w  =  v )
3433ex 115 . . 3  |-  ( (
ph  /\  ( w  e.  B  /\  v  e.  B ) )  -> 
( ( ( ( w  .+  A )  =  .0.  /\  ( A  .+  w )  =  .0.  )  /\  (
( v  .+  A
)  =  .0.  /\  ( A  .+  v )  =  .0.  ) )  ->  w  =  v ) )
3534ralrimivva 2569 . 2  |-  ( ph  ->  A. w  e.  B  A. v  e.  B  ( ( ( ( w  .+  A )  =  .0.  /\  ( A  .+  w )  =  .0.  )  /\  (
( v  .+  A
)  =  .0.  /\  ( A  .+  v )  =  .0.  ) )  ->  w  =  v ) )
36 oveq1 5895 . . . . 5  |-  ( w  =  v  ->  (
w  .+  A )  =  ( v  .+  A ) )
3736eqeq1d 2196 . . . 4  |-  ( w  =  v  ->  (
( w  .+  A
)  =  .0.  <->  ( v  .+  A )  =  .0.  ) )
38 oveq2 5896 . . . . 5  |-  ( w  =  v  ->  ( A  .+  w )  =  ( A  .+  v
) )
3938eqeq1d 2196 . . . 4  |-  ( w  =  v  ->  (
( A  .+  w
)  =  .0.  <->  ( A  .+  v )  =  .0.  ) )
4037, 39anbi12d 473 . . 3  |-  ( w  =  v  ->  (
( ( w  .+  A )  =  .0. 
/\  ( A  .+  w )  =  .0.  )  <->  ( ( v 
.+  A )  =  .0.  /\  ( A 
.+  v )  =  .0.  ) ) )
4140rmo4 2942 . 2  |-  ( E* w  e.  B  ( ( w  .+  A
)  =  .0.  /\  ( A  .+  w )  =  .0.  )  <->  A. w  e.  B  A. v  e.  B  ( (
( ( w  .+  A )  =  .0. 
/\  ( A  .+  w )  =  .0.  )  /\  ( ( v  .+  A )  =  .0.  /\  ( A  .+  v )  =  .0.  ) )  ->  w  =  v )
)
4235, 41sylibr 134 1  |-  ( ph  ->  E* w  e.  B  ( ( w  .+  A )  =  .0. 
/\  ( A  .+  w )  =  .0.  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 979    = wceq 1363    e. wcel 2158   A.wral 2465   E*wrmo 2468   ` cfv 5228  (class class class)co 5888   Basecbs 12475   +g cplusg 12550   0gc0g 12722   Mndcmnd 12838
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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-cnex 7915  ax-resscn 7916  ax-1re 7918  ax-addrcl 7921
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-iota 5190  df-fun 5230  df-fn 5231  df-fv 5236  df-riota 5844  df-ov 5891  df-inn 8933  df-2 8991  df-ndx 12478  df-slot 12479  df-base 12481  df-plusg 12563  df-0g 12724  df-mgm 12793  df-sgrp 12826  df-mnd 12839
This theorem is referenced by:  rinvmod  13145
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