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Theorem imasmnd 15765
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 1012 . . . 4  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  R  e.  Mnd )
8 simp2 992 . . . . 5  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  x  e.  V )
923ad2ant1 1012 . . . . 5  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  V  =  ( Base `  R )
)
108, 9eleqtrd 2550 . . . 4  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  x  e.  ( Base `  R )
)
11 simp3 993 . . . . 5  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  y  e.  V )
1211, 9eleqtrd 2550 . . . 4  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  y  e.  ( Base `  R )
)
13 eqid 2460 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
1413, 3mndcl 15726 . . . 4  |-  ( ( R  e.  Mnd  /\  x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
) )  ->  (
x  .+  y )  e.  ( Base `  R
) )
157, 10, 12, 14syl3anc 1223 . . 3  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  ( x  .+  y )  e.  (
Base `  R )
)
1615, 9eleqtrrd 2551 . 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 1202 . . . 4  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  ->  x  e.  ( Base `  R ) )
19123adant3r3 1202 . . . 4  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
y  e.  ( Base `  R ) )
20 simpr3 999 . . . . 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 2550 . . . 4  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
z  e.  ( Base `  R ) )
2313, 3mndass 15727 . . . 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 1225 . . 3  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
( ( x  .+  y )  .+  z
)  =  ( x 
.+  ( y  .+  z ) ) )
2524fveq2d 5861 . 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 15745 . . . 4  |-  ( R  e.  Mnd  ->  .0.  e.  ( Base `  R
) )
286, 27syl 16 . . 3  |-  ( ph  ->  .0.  e.  ( Base `  R ) )
2928, 2eleqtrrd 2551 . 2  |-  ( ph  ->  .0.  e.  V )
306adantr 465 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  R  e.  Mnd )
312eleq2d 2530 . . . . 5  |-  ( ph  ->  ( x  e.  V  <->  x  e.  ( Base `  R
) ) )
3231biimpa 484 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  x  e.  ( Base `  R
) )
3313, 3, 26mndlid 15747 . . . 4  |-  ( ( R  e.  Mnd  /\  x  e.  ( Base `  R ) )  -> 
(  .0.  .+  x
)  =  x )
3430, 32, 33syl2anc 661 . . 3  |-  ( (
ph  /\  x  e.  V )  ->  (  .0.  .+  x )  =  x )
3534fveq2d 5861 . 2  |-  ( (
ph  /\  x  e.  V )  ->  ( F `  (  .0.  .+  x ) )  =  ( F `  x
) )
3613, 3, 26mndrid 15748 . . . 4  |-  ( ( R  e.  Mnd  /\  x  e.  ( Base `  R ) )  -> 
( x  .+  .0.  )  =  x )
3730, 32, 36syl2anc 661 . . 3  |-  ( (
ph  /\  x  e.  V )  ->  (
x  .+  .0.  )  =  x )
3837fveq2d 5861 . 2  |-  ( (
ph  /\  x  e.  V )  ->  ( F `  ( x  .+  .0.  ) )  =  ( F `  x
) )
391, 2, 3, 4, 5, 6, 16, 25, 29, 35, 38imasmnd2 15764 1  |-  ( ph  ->  ( U  e.  Mnd  /\  ( F `  .0.  )  =  ( 0g `  U ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   -onto->wfo 5577   ` cfv 5579  (class class class)co 6275   Basecbs 14479   +g cplusg 14544   0gc0g 14684    "s cimas 14748   Mndcmnd 15715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-sup 7890  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-10 10591  df-n0 10785  df-z 10854  df-dec 10966  df-uz 11072  df-fz 11662  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-plusg 14557  df-mulr 14558  df-sca 14560  df-vsca 14561  df-ip 14562  df-tset 14563  df-ple 14564  df-ds 14566  df-0g 14686  df-imas 14752  df-mnd 15721
This theorem is referenced by:  imasmndf1  15766
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