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Theorem imasmnd 16172
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 1016 . . . 4  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  R  e.  Mnd )
8 simp2 996 . . . . 5  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  x  e.  V )
923ad2ant1 1016 . . . . 5  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  V  =  ( Base `  R )
)
108, 9eleqtrd 2490 . . . 4  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  x  e.  ( Base `  R )
)
11 simp3 997 . . . . 5  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  y  e.  V )
1211, 9eleqtrd 2490 . . . 4  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  y  e.  ( Base `  R )
)
13 eqid 2400 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
1413, 3mndcl 16143 . . . 4  |-  ( ( R  e.  Mnd  /\  x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
) )  ->  (
x  .+  y )  e.  ( Base `  R
) )
157, 10, 12, 14syl3anc 1228 . . 3  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  ( x  .+  y )  e.  (
Base `  R )
)
1615, 9eleqtrrd 2491 . 2  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  ( x  .+  y )  e.  V
)
176adantr 463 . . . 4  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  ->  R  e.  Mnd )
18103adant3r3 1206 . . . 4  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  ->  x  e.  ( Base `  R ) )
19123adant3r3 1206 . . . 4  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
y  e.  ( Base `  R ) )
20 simpr3 1003 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
z  e.  V )
212adantr 463 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  ->  V  =  ( Base `  R ) )
2220, 21eleqtrd 2490 . . . 4  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
z  e.  ( Base `  R ) )
2313, 3mndass 16144 . . . 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 1230 . . 3  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
( ( x  .+  y )  .+  z
)  =  ( x 
.+  ( y  .+  z ) ) )
2524fveq2d 5807 . 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 16152 . . . 4  |-  ( R  e.  Mnd  ->  .0.  e.  ( Base `  R
) )
286, 27syl 17 . . 3  |-  ( ph  ->  .0.  e.  ( Base `  R ) )
2928, 2eleqtrrd 2491 . 2  |-  ( ph  ->  .0.  e.  V )
306adantr 463 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  R  e.  Mnd )
312eleq2d 2470 . . . . 5  |-  ( ph  ->  ( x  e.  V  <->  x  e.  ( Base `  R
) ) )
3231biimpa 482 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  x  e.  ( Base `  R
) )
3313, 3, 26mndlid 16155 . . . 4  |-  ( ( R  e.  Mnd  /\  x  e.  ( Base `  R ) )  -> 
(  .0.  .+  x
)  =  x )
3430, 32, 33syl2anc 659 . . 3  |-  ( (
ph  /\  x  e.  V )  ->  (  .0.  .+  x )  =  x )
3534fveq2d 5807 . 2  |-  ( (
ph  /\  x  e.  V )  ->  ( F `  (  .0.  .+  x ) )  =  ( F `  x
) )
3613, 3, 26mndrid 16156 . . . 4  |-  ( ( R  e.  Mnd  /\  x  e.  ( Base `  R ) )  -> 
( x  .+  .0.  )  =  x )
3730, 32, 36syl2anc 659 . . 3  |-  ( (
ph  /\  x  e.  V )  ->  (
x  .+  .0.  )  =  x )
3837fveq2d 5807 . 2  |-  ( (
ph  /\  x  e.  V )  ->  ( F `  ( x  .+  .0.  ) )  =  ( F `  x
) )
391, 2, 3, 4, 5, 6, 16, 25, 29, 35, 38imasmnd2 16171 1  |-  ( ph  ->  ( U  e.  Mnd  /\  ( F `  .0.  )  =  ( 0g `  U ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 972    = wceq 1403    e. wcel 1840   -onto->wfo 5521   ` cfv 5523  (class class class)co 6232   Basecbs 14731   +g cplusg 14799   0gc0g 14944    "s cimas 15008   Mndcmnd 16133
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-cnex 9496  ax-resscn 9497  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-addrcl 9501  ax-mulcl 9502  ax-mulrcl 9503  ax-mulcom 9504  ax-addass 9505  ax-mulass 9506  ax-distr 9507  ax-i2m1 9508  ax-1ne0 9509  ax-1rid 9510  ax-rnegex 9511  ax-rrecex 9512  ax-cnre 9513  ax-pre-lttri 9514  ax-pre-lttrn 9515  ax-pre-ltadd 9516  ax-pre-mulgt0 9517
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-int 4225  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-om 6637  df-1st 6736  df-2nd 6737  df-recs 6997  df-rdg 7031  df-1o 7085  df-oadd 7089  df-er 7266  df-en 7473  df-dom 7474  df-sdom 7475  df-fin 7476  df-sup 7853  df-pnf 9578  df-mnf 9579  df-xr 9580  df-ltxr 9581  df-le 9582  df-sub 9761  df-neg 9762  df-nn 10495  df-2 10553  df-3 10554  df-4 10555  df-5 10556  df-6 10557  df-7 10558  df-8 10559  df-9 10560  df-10 10561  df-n0 10755  df-z 10824  df-dec 10938  df-uz 11044  df-fz 11642  df-struct 14733  df-ndx 14734  df-slot 14735  df-base 14736  df-plusg 14812  df-mulr 14813  df-sca 14815  df-vsca 14816  df-ip 14817  df-tset 14818  df-ple 14819  df-ds 14821  df-0g 14946  df-imas 15012  df-mgm 16086  df-sgrp 16125  df-mnd 16135
This theorem is referenced by:  imasmndf1  16173
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