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Theorem fidomndrnglem 17719
Description: Lemma for fidomndrng 17720. (Contributed by Mario Carneiro, 15-Jun-2015.)
Hypotheses
Ref Expression
fidomndrng.b  |-  B  =  ( Base `  R
)
fidomndrng.z  |-  .0.  =  ( 0g `  R )
fidomndrng.o  |-  .1.  =  ( 1r `  R )
fidomndrng.d  |-  .||  =  (
||r `  R )
fidomndrng.t  |-  .x.  =  ( .r `  R )
fidomndrng.r  |-  ( ph  ->  R  e. Domn )
fidomndrng.x  |-  ( ph  ->  B  e.  Fin )
fidomndrng.a  |-  ( ph  ->  A  e.  ( B 
\  {  .0.  }
) )
fidomndrng.f  |-  F  =  ( x  e.  B  |->  ( x  .x.  A
) )
Assertion
Ref Expression
fidomndrnglem  |-  ( ph  ->  A  .||  .1.  )
Distinct variable groups:    x, A    x, B    x, R    x,  .x.
Allowed substitution hints:    ph( x)    .|| ( x)    .1. ( x)    F( x)    .0. ( x)

Proof of Theorem fidomndrnglem
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 fidomndrng.a . . . 4  |-  ( ph  ->  A  e.  ( B 
\  {  .0.  }
) )
21eldifad 3481 . . 3  |-  ( ph  ->  A  e.  B )
3 eldifsni 4146 . . . . . . . . . . . 12  |-  ( A  e.  ( B  \  {  .0.  } )  ->  A  =/=  .0.  )
41, 3syl 16 . . . . . . . . . . 11  |-  ( ph  ->  A  =/=  .0.  )
54ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  B )  /\  ( F `  y )  =  .0.  )  ->  A  =/=  .0.  )
6 oveq1 6282 . . . . . . . . . . . . . . . . 17  |-  ( x  =  y  ->  (
x  .x.  A )  =  ( y  .x.  A ) )
7 fidomndrng.f . . . . . . . . . . . . . . . . 17  |-  F  =  ( x  e.  B  |->  ( x  .x.  A
) )
8 ovex 6300 . . . . . . . . . . . . . . . . 17  |-  ( y 
.x.  A )  e. 
_V
96, 7, 8fvmpt 5941 . . . . . . . . . . . . . . . 16  |-  ( y  e.  B  ->  ( F `  y )  =  ( y  .x.  A ) )
109adantl 466 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  y  e.  B )  ->  ( F `  y )  =  ( y  .x.  A ) )
1110eqeq1d 2462 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  y  e.  B )  ->  (
( F `  y
)  =  .0.  <->  ( y  .x.  A )  =  .0.  ) )
12 fidomndrng.r . . . . . . . . . . . . . . . 16  |-  ( ph  ->  R  e. Domn )
1312adantr 465 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  y  e.  B )  ->  R  e. Domn )
14 simpr 461 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  y  e.  B )  ->  y  e.  B )
152adantr 465 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  y  e.  B )  ->  A  e.  B )
16 fidomndrng.b . . . . . . . . . . . . . . . 16  |-  B  =  ( Base `  R
)
17 fidomndrng.t . . . . . . . . . . . . . . . 16  |-  .x.  =  ( .r `  R )
18 fidomndrng.z . . . . . . . . . . . . . . . 16  |-  .0.  =  ( 0g `  R )
1916, 17, 18domneq0 17710 . . . . . . . . . . . . . . 15  |-  ( ( R  e. Domn  /\  y  e.  B  /\  A  e.  B )  ->  (
( y  .x.  A
)  =  .0.  <->  ( y  =  .0.  \/  A  =  .0.  ) ) )
2013, 14, 15, 19syl3anc 1223 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  y  e.  B )  ->  (
( y  .x.  A
)  =  .0.  <->  ( y  =  .0.  \/  A  =  .0.  ) ) )
2111, 20bitrd 253 . . . . . . . . . . . . 13  |-  ( (
ph  /\  y  e.  B )  ->  (
( F `  y
)  =  .0.  <->  ( y  =  .0.  \/  A  =  .0.  ) ) )
2221biimpa 484 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  y  e.  B )  /\  ( F `  y )  =  .0.  )  ->  (
y  =  .0.  \/  A  =  .0.  )
)
2322ord 377 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  B )  /\  ( F `  y )  =  .0.  )  ->  ( -.  y  =  .0.  ->  A  =  .0.  )
)
2423necon1ad 2676 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  B )  /\  ( F `  y )  =  .0.  )  ->  ( A  =/=  .0.  ->  y  =  .0.  ) )
255, 24mpd 15 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  B )  /\  ( F `  y )  =  .0.  )  ->  y  =  .0.  )
2625ex 434 . . . . . . . 8  |-  ( (
ph  /\  y  e.  B )  ->  (
( F `  y
)  =  .0.  ->  y  =  .0.  ) )
2726ralrimiva 2871 . . . . . . 7  |-  ( ph  ->  A. y  e.  B  ( ( F `  y )  =  .0. 
->  y  =  .0.  ) )
28 domnrng 17709 . . . . . . . . . . 11  |-  ( R  e. Domn  ->  R  e.  Ring )
2912, 28syl 16 . . . . . . . . . 10  |-  ( ph  ->  R  e.  Ring )
3016, 17rngrghm 17028 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  A  e.  B )  ->  (
x  e.  B  |->  ( x  .x.  A ) )  e.  ( R 
GrpHom  R ) )
3129, 2, 30syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  B  |->  ( x  .x.  A
) )  e.  ( R  GrpHom  R ) )
327, 31syl5eqel 2552 . . . . . . . 8  |-  ( ph  ->  F  e.  ( R 
GrpHom  R ) )
3316, 16, 18, 18ghmf1 16083 . . . . . . . 8  |-  ( F  e.  ( R  GrpHom  R )  ->  ( F : B -1-1-> B  <->  A. y  e.  B  ( ( F `  y )  =  .0. 
->  y  =  .0.  ) ) )
3432, 33syl 16 . . . . . . 7  |-  ( ph  ->  ( F : B -1-1-> B  <->  A. y  e.  B  ( ( F `  y )  =  .0. 
->  y  =  .0.  ) ) )
3527, 34mpbird 232 . . . . . 6  |-  ( ph  ->  F : B -1-1-> B
)
36 fidomndrng.x . . . . . . . 8  |-  ( ph  ->  B  e.  Fin )
37 enrefg 7537 . . . . . . . 8  |-  ( B  e.  Fin  ->  B  ~~  B )
3836, 37syl 16 . . . . . . 7  |-  ( ph  ->  B  ~~  B )
39 f1finf1o 7736 . . . . . . 7  |-  ( ( B  ~~  B  /\  B  e.  Fin )  ->  ( F : B -1-1-> B  <-> 
F : B -1-1-onto-> B ) )
4038, 36, 39syl2anc 661 . . . . . 6  |-  ( ph  ->  ( F : B -1-1-> B  <-> 
F : B -1-1-onto-> B ) )
4135, 40mpbid 210 . . . . 5  |-  ( ph  ->  F : B -1-1-onto-> B )
42 f1ocnv 5819 . . . . 5  |-  ( F : B -1-1-onto-> B  ->  `' F : B -1-1-onto-> B )
43 f1of 5807 . . . . 5  |-  ( `' F : B -1-1-onto-> B  ->  `' F : B --> B )
4441, 42, 433syl 20 . . . 4  |-  ( ph  ->  `' F : B --> B )
45 fidomndrng.o . . . . . 6  |-  .1.  =  ( 1r `  R )
4616, 45rngidcl 16999 . . . . 5  |-  ( R  e.  Ring  ->  .1.  e.  B )
4729, 46syl 16 . . . 4  |-  ( ph  ->  .1.  e.  B )
4844, 47ffvelrnd 6013 . . 3  |-  ( ph  ->  ( `' F `  .1.  )  e.  B
)
49 fidomndrng.d . . . 4  |-  .||  =  (
||r `  R )
5016, 49, 17dvdsrmul 17074 . . 3  |-  ( ( A  e.  B  /\  ( `' F `  .1.  )  e.  B )  ->  A  .||  ( ( `' F `  .1.  )  .x.  A
) )
512, 48, 50syl2anc 661 . 2  |-  ( ph  ->  A  .||  ( ( `' F `  .1.  )  .x.  A ) )
52 oveq1 6282 . . . . 5  |-  ( y  =  ( `' F `  .1.  )  ->  (
y  .x.  A )  =  ( ( `' F `  .1.  )  .x.  A ) )
536cbvmptv 4531 . . . . . 6  |-  ( x  e.  B  |->  ( x 
.x.  A ) )  =  ( y  e.  B  |->  ( y  .x.  A ) )
547, 53eqtri 2489 . . . . 5  |-  F  =  ( y  e.  B  |->  ( y  .x.  A
) )
55 ovex 6300 . . . . 5  |-  ( ( `' F `  .1.  )  .x.  A )  e.  _V
5652, 54, 55fvmpt 5941 . . . 4  |-  ( ( `' F `  .1.  )  e.  B  ->  ( F `
 ( `' F `  .1.  ) )  =  ( ( `' F `  .1.  )  .x.  A
) )
5748, 56syl 16 . . 3  |-  ( ph  ->  ( F `  ( `' F `  .1.  )
)  =  ( ( `' F `  .1.  )  .x.  A ) )
58 f1ocnvfv2 6162 . . . 4  |-  ( ( F : B -1-1-onto-> B  /\  .1.  e.  B )  -> 
( F `  ( `' F `  .1.  )
)  =  .1.  )
5941, 47, 58syl2anc 661 . . 3  |-  ( ph  ->  ( F `  ( `' F `  .1.  )
)  =  .1.  )
6057, 59eqtr3d 2503 . 2  |-  ( ph  ->  ( ( `' F `  .1.  )  .x.  A
)  =  .1.  )
6151, 60breqtrd 4464 1  |-  ( ph  ->  A  .||  .1.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1374    e. wcel 1762    =/= wne 2655   A.wral 2807    \ cdif 3466   {csn 4020   class class class wbr 4440    |-> cmpt 4498   `'ccnv 4991   -->wf 5575   -1-1->wf1 5576   -1-1-onto->wf1o 5578   ` cfv 5579  (class class class)co 6275    ~~ cen 7503   Fincfn 7506   Basecbs 14479   .rcmulr 14545   0gc0g 14684    GrpHom cghm 16052   1rcur 16936   Ringcrg 16979   ||rcdsr 17064  Domncdomn 17692
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-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-er 7301  df-map 7412  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  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-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-plusg 14557  df-0g 14686  df-mnd 15721  df-grp 15851  df-minusg 15852  df-sbg 15853  df-ghm 16053  df-mgp 16925  df-ur 16937  df-rng 16981  df-dvdsr 17067  df-nzr 17681  df-domn 17696
This theorem is referenced by:  fidomndrng  17720
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