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Theorem imasmnd 14660
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 978 . . . 4  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  R  e.  Mnd )
8 simp2 958 . . . . 5  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  x  e.  V )
923ad2ant1 978 . . . . 5  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  V  =  ( Base `  R )
)
108, 9eleqtrd 2463 . . . 4  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  x  e.  ( Base `  R )
)
11 simp3 959 . . . . 5  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  y  e.  V )
1211, 9eleqtrd 2463 . . . 4  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  y  e.  ( Base `  R )
)
13 eqid 2387 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
1413, 3mndcl 14622 . . . 4  |-  ( ( R  e.  Mnd  /\  x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
) )  ->  (
x  .+  y )  e.  ( Base `  R
) )
157, 10, 12, 14syl3anc 1184 . . 3  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  ( x  .+  y )  e.  (
Base `  R )
)
1615, 9eleqtrrd 2464 . 2  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  ( x  .+  y )  e.  V
)
176adantr 452 . . . 4  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  ->  R  e.  Mnd )
18103adant3r3 1164 . . . 4  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  ->  x  e.  ( Base `  R ) )
19123adant3r3 1164 . . . 4  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
y  e.  ( Base `  R ) )
20 simpr3 965 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
z  e.  V )
212adantr 452 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  ->  V  =  ( Base `  R ) )
2220, 21eleqtrd 2463 . . . 4  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
z  e.  ( Base `  R ) )
2313, 3mndass 14623 . . . 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 1186 . . 3  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
( ( x  .+  y )  .+  z
)  =  ( x 
.+  ( y  .+  z ) ) )
2524fveq2d 5672 . 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 14641 . . . 4  |-  ( R  e.  Mnd  ->  .0.  e.  ( Base `  R
) )
286, 27syl 16 . . 3  |-  ( ph  ->  .0.  e.  ( Base `  R ) )
2928, 2eleqtrrd 2464 . 2  |-  ( ph  ->  .0.  e.  V )
306adantr 452 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  R  e.  Mnd )
312eleq2d 2454 . . . . 5  |-  ( ph  ->  ( x  e.  V  <->  x  e.  ( Base `  R
) ) )
3231biimpa 471 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  x  e.  ( Base `  R
) )
3313, 3, 26mndlid 14643 . . . 4  |-  ( ( R  e.  Mnd  /\  x  e.  ( Base `  R ) )  -> 
(  .0.  .+  x
)  =  x )
3430, 32, 33syl2anc 643 . . 3  |-  ( (
ph  /\  x  e.  V )  ->  (  .0.  .+  x )  =  x )
3534fveq2d 5672 . 2  |-  ( (
ph  /\  x  e.  V )  ->  ( F `  (  .0.  .+  x ) )  =  ( F `  x
) )
3613, 3, 26mndrid 14644 . . . 4  |-  ( ( R  e.  Mnd  /\  x  e.  ( Base `  R ) )  -> 
( x  .+  .0.  )  =  x )
3730, 32, 36syl2anc 643 . . 3  |-  ( (
ph  /\  x  e.  V )  ->  (
x  .+  .0.  )  =  x )
3837fveq2d 5672 . 2  |-  ( (
ph  /\  x  e.  V )  ->  ( F `  ( x  .+  .0.  ) )  =  ( F `  x
) )
391, 2, 3, 4, 5, 6, 16, 25, 29, 35, 38imasmnd2 14659 1  |-  ( ph  ->  ( U  e.  Mnd  /\  ( F `  .0.  )  =  ( 0g `  U ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   -onto->wfo 5392   ` cfv 5394  (class class class)co 6020   Basecbs 13396   +g cplusg 13456   0gc0g 13650    "s cimas 13657   Mndcmnd 14611
This theorem is referenced by:  imasmndf1  14661
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-oadd 6664  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-sup 7381  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-2 9990  df-3 9991  df-4 9992  df-5 9993  df-6 9994  df-7 9995  df-8 9996  df-9 9997  df-10 9998  df-n0 10154  df-z 10215  df-dec 10315  df-uz 10421  df-fz 10976  df-struct 13398  df-ndx 13399  df-slot 13400  df-base 13401  df-plusg 13469  df-mulr 13470  df-sca 13472  df-vsca 13473  df-tset 13475  df-ple 13476  df-ds 13478  df-0g 13654  df-imas 13661  df-mnd 14617
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