MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  imasmnd Unicode version

Theorem imasmnd 14410
Description: The image structure of a monoid is a monoid. (Contributed by Mario Carneiro, 24-Feb-2015.)
Hypotheses
Ref Expression
imasmnd.u  |-  ( ph  ->  U  =  ( F 
"s  R ) )
imasmnd.v  |-  ( ph  ->  V  =  ( Base `  R ) )
imasmnd.p  |-  .+  =  ( +g  `  R )
imasmnd.f  |-  ( ph  ->  F : V -onto-> B
)
imasmnd.e  |-  ( (
ph  /\  ( a  e.  V  /\  b  e.  V )  /\  (
p  e.  V  /\  q  e.  V )
)  ->  ( (
( F `  a
)  =  ( F `
 p )  /\  ( F `  b )  =  ( F `  q ) )  -> 
( F `  (
a  .+  b )
)  =  ( F `
 ( p  .+  q ) ) ) )
imasmnd.r  |-  ( ph  ->  R  e.  Mnd )
imasmnd.z  |-  .0.  =  ( 0g `  R )
Assertion
Ref Expression
imasmnd  |-  ( ph  ->  ( U  e.  Mnd  /\  ( F `  .0.  )  =  ( 0g `  U ) ) )
Distinct variable groups:    q, p,  .+    a, b, p, q, ph    U, a, b, p, q    .0. , p, q    B, p, q    F, a, b, p, q    R, p, q    V, a, b, p, q
Allowed substitution hints:    B( a, b)    .+ ( a, b)    R( a, b)    .0. ( a, b)

Proof of Theorem imasmnd
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 imasmnd.u . 2  |-  ( ph  ->  U  =  ( F 
"s  R ) )
2 imasmnd.v . 2  |-  ( ph  ->  V  =  ( Base `  R ) )
3 imasmnd.p . 2  |-  .+  =  ( +g  `  R )
4 imasmnd.f . 2  |-  ( ph  ->  F : V -onto-> B
)
5 imasmnd.e . 2  |-  ( (
ph  /\  ( a  e.  V  /\  b  e.  V )  /\  (
p  e.  V  /\  q  e.  V )
)  ->  ( (
( F `  a
)  =  ( F `
 p )  /\  ( F `  b )  =  ( F `  q ) )  -> 
( F `  (
a  .+  b )
)  =  ( F `
 ( p  .+  q ) ) ) )
6 imasmnd.r . 2  |-  ( ph  ->  R  e.  Mnd )
763ad2ant1 976 . . . 4  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  R  e.  Mnd )
8 simp2 956 . . . . 5  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  x  e.  V )
923ad2ant1 976 . . . . 5  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  V  =  ( Base `  R )
)
108, 9eleqtrd 2359 . . . 4  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  x  e.  ( Base `  R )
)
11 simp3 957 . . . . 5  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  y  e.  V )
1211, 9eleqtrd 2359 . . . 4  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  y  e.  ( Base `  R )
)
13 eqid 2283 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
1413, 3mndcl 14372 . . . 4  |-  ( ( R  e.  Mnd  /\  x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
) )  ->  (
x  .+  y )  e.  ( Base `  R
) )
157, 10, 12, 14syl3anc 1182 . . 3  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  ( x  .+  y )  e.  (
Base `  R )
)
1615, 9eleqtrrd 2360 . 2  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  ( x  .+  y )  e.  V
)
176adantr 451 . . . 4  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  ->  R  e.  Mnd )
18103adant3r3 1162 . . . 4  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  ->  x  e.  ( Base `  R ) )
19123adant3r3 1162 . . . 4  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
y  e.  ( Base `  R ) )
20 simpr3 963 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
z  e.  V )
212adantr 451 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  ->  V  =  ( Base `  R ) )
2220, 21eleqtrd 2359 . . . 4  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
z  e.  ( Base `  R ) )
2313, 3mndass 14373 . . . 4  |-  ( ( R  e.  Mnd  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( Base `  R )  /\  z  e.  ( Base `  R
) ) )  -> 
( ( x  .+  y )  .+  z
)  =  ( x 
.+  ( y  .+  z ) ) )
2417, 18, 19, 22, 23syl13anc 1184 . . 3  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
( ( x  .+  y )  .+  z
)  =  ( x 
.+  ( y  .+  z ) ) )
2524fveq2d 5529 . 2  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
( F `  (
( x  .+  y
)  .+  z )
)  =  ( F `
 ( x  .+  ( y  .+  z
) ) ) )
26 imasmnd.z . . . . 5  |-  .0.  =  ( 0g `  R )
2713, 26mndidcl 14391 . . . 4  |-  ( R  e.  Mnd  ->  .0.  e.  ( Base `  R
) )
286, 27syl 15 . . 3  |-  ( ph  ->  .0.  e.  ( Base `  R ) )
2928, 2eleqtrrd 2360 . 2  |-  ( ph  ->  .0.  e.  V )
306adantr 451 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  R  e.  Mnd )
312eleq2d 2350 . . . . 5  |-  ( ph  ->  ( x  e.  V  <->  x  e.  ( Base `  R
) ) )
3231biimpa 470 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  x  e.  ( Base `  R
) )
3313, 3, 26mndlid 14393 . . . 4  |-  ( ( R  e.  Mnd  /\  x  e.  ( Base `  R ) )  -> 
(  .0.  .+  x
)  =  x )
3430, 32, 33syl2anc 642 . . 3  |-  ( (
ph  /\  x  e.  V )  ->  (  .0.  .+  x )  =  x )
3534fveq2d 5529 . 2  |-  ( (
ph  /\  x  e.  V )  ->  ( F `  (  .0.  .+  x ) )  =  ( F `  x
) )
3613, 3, 26mndrid 14394 . . . 4  |-  ( ( R  e.  Mnd  /\  x  e.  ( Base `  R ) )  -> 
( x  .+  .0.  )  =  x )
3730, 32, 36syl2anc 642 . . 3  |-  ( (
ph  /\  x  e.  V )  ->  (
x  .+  .0.  )  =  x )
3837fveq2d 5529 . 2  |-  ( (
ph  /\  x  e.  V )  ->  ( F `  ( x  .+  .0.  ) )  =  ( F `  x
) )
391, 2, 3, 4, 5, 6, 16, 25, 29, 35, 38imasmnd2 14409 1  |-  ( ph  ->  ( U  e.  Mnd  /\  ( F `  .0.  )  =  ( 0g `  U ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   -onto->wfo 5253   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   0gc0g 13400    "s cimas 13407   Mndcmnd 14361
This theorem is referenced by:  imasmndf1  14411
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-0g 13404  df-imas 13411  df-mnd 14367
  Copyright terms: Public domain W3C validator