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Theorem imasmnd 15579
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 1009 . . . 4  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  R  e.  Mnd )
8 simp2 989 . . . . 5  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  x  e.  V )
923ad2ant1 1009 . . . . 5  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  V  =  ( Base `  R )
)
108, 9eleqtrd 2544 . . . 4  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  x  e.  ( Base `  R )
)
11 simp3 990 . . . . 5  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  y  e.  V )
1211, 9eleqtrd 2544 . . . 4  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  y  e.  ( Base `  R )
)
13 eqid 2454 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
1413, 3mndcl 15540 . . . 4  |-  ( ( R  e.  Mnd  /\  x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
) )  ->  (
x  .+  y )  e.  ( Base `  R
) )
157, 10, 12, 14syl3anc 1219 . . 3  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  ( x  .+  y )  e.  (
Base `  R )
)
1615, 9eleqtrrd 2545 . 2  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  ( x  .+  y )  e.  V
)
176adantr 465 . . . 4  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  ->  R  e.  Mnd )
18103adant3r3 1199 . . . 4  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  ->  x  e.  ( Base `  R ) )
19123adant3r3 1199 . . . 4  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
y  e.  ( Base `  R ) )
20 simpr3 996 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
z  e.  V )
212adantr 465 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  ->  V  =  ( Base `  R ) )
2220, 21eleqtrd 2544 . . . 4  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
z  e.  ( Base `  R ) )
2313, 3mndass 15541 . . . 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 1221 . . 3  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
( ( x  .+  y )  .+  z
)  =  ( x 
.+  ( y  .+  z ) ) )
2524fveq2d 5804 . 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 15559 . . . 4  |-  ( R  e.  Mnd  ->  .0.  e.  ( Base `  R
) )
286, 27syl 16 . . 3  |-  ( ph  ->  .0.  e.  ( Base `  R ) )
2928, 2eleqtrrd 2545 . 2  |-  ( ph  ->  .0.  e.  V )
306adantr 465 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  R  e.  Mnd )
312eleq2d 2524 . . . . 5  |-  ( ph  ->  ( x  e.  V  <->  x  e.  ( Base `  R
) ) )
3231biimpa 484 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  x  e.  ( Base `  R
) )
3313, 3, 26mndlid 15561 . . . 4  |-  ( ( R  e.  Mnd  /\  x  e.  ( Base `  R ) )  -> 
(  .0.  .+  x
)  =  x )
3430, 32, 33syl2anc 661 . . 3  |-  ( (
ph  /\  x  e.  V )  ->  (  .0.  .+  x )  =  x )
3534fveq2d 5804 . 2  |-  ( (
ph  /\  x  e.  V )  ->  ( F `  (  .0.  .+  x ) )  =  ( F `  x
) )
3613, 3, 26mndrid 15562 . . . 4  |-  ( ( R  e.  Mnd  /\  x  e.  ( Base `  R ) )  -> 
( x  .+  .0.  )  =  x )
3730, 32, 36syl2anc 661 . . 3  |-  ( (
ph  /\  x  e.  V )  ->  (
x  .+  .0.  )  =  x )
3837fveq2d 5804 . 2  |-  ( (
ph  /\  x  e.  V )  ->  ( F `  ( x  .+  .0.  ) )  =  ( F `  x
) )
391, 2, 3, 4, 5, 6, 16, 25, 29, 35, 38imasmnd2 15578 1  |-  ( ph  ->  ( U  e.  Mnd  /\  ( F `  .0.  )  =  ( 0g `  U ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   -onto->wfo 5525   ` cfv 5527  (class class class)co 6201   Basecbs 14293   +g cplusg 14358   0gc0g 14498    "s cimas 14562   Mndcmnd 15529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-1o 7031  df-oadd 7035  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-sup 7803  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-2 10492  df-3 10493  df-4 10494  df-5 10495  df-6 10496  df-7 10497  df-8 10498  df-9 10499  df-10 10500  df-n0 10692  df-z 10759  df-dec 10868  df-uz 10974  df-fz 11556  df-struct 14295  df-ndx 14296  df-slot 14297  df-base 14298  df-plusg 14371  df-mulr 14372  df-sca 14374  df-vsca 14375  df-ip 14376  df-tset 14377  df-ple 14378  df-ds 14380  df-0g 14500  df-imas 14566  df-mnd 15535
This theorem is referenced by:  imasmndf1  15580
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