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Theorem fidomndrnglem 17829
Description: Lemma for fidomndrng 17830. (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 3473 . . 3  |-  ( ph  ->  A  e.  B )
3 eldifsni 4141 . . . . . . . . . . . 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 6288 . . . . . . . . . . . . . . . . 17  |-  ( x  =  y  ->  (
x  .x.  A )  =  ( y  .x.  A ) )
7 fidomndrng.f . . . . . . . . . . . . . . . . 17  |-  F  =  ( x  e.  B  |->  ( x  .x.  A
) )
8 ovex 6309 . . . . . . . . . . . . . . . . 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 2445 . . . . . . . . . . . . . 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 17820 . . . . . . . . . . . . . . 15  |-  ( ( R  e. Domn  /\  y  e.  B  /\  A  e.  B )  ->  (
( y  .x.  A
)  =  .0.  <->  ( y  =  .0.  \/  A  =  .0.  ) ) )
2013, 14, 15, 19syl3anc 1229 . . . . . . . . . . . . . 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 2659 . . . . . . . . . 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 2857 . . . . . . 7  |-  ( ph  ->  A. y  e.  B  ( ( F `  y )  =  .0. 
->  y  =  .0.  ) )
28 domnring 17819 . . . . . . . . . . 11  |-  ( R  e. Domn  ->  R  e.  Ring )
2912, 28syl 16 . . . . . . . . . 10  |-  ( ph  ->  R  e.  Ring )
3016, 17ringrghm 17125 . . . . . . . . . 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 2535 . . . . . . . 8  |-  ( ph  ->  F  e.  ( R 
GrpHom  R ) )
3316, 16, 18, 18ghmf1 16169 . . . . . . . 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 7549 . . . . . . . 8  |-  ( B  e.  Fin  ->  B  ~~  B )
3836, 37syl 16 . . . . . . 7  |-  ( ph  ->  B  ~~  B )
39 f1finf1o 7748 . . . . . . 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 5818 . . . . 5  |-  ( F : B -1-1-onto-> B  ->  `' F : B -1-1-onto-> B )
43 f1of 5806 . . . . 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, 45ringidcl 17093 . . . . 5  |-  ( R  e.  Ring  ->  .1.  e.  B )
4729, 46syl 16 . . . 4  |-  ( ph  ->  .1.  e.  B )
4844, 47ffvelrnd 6017 . . 3  |-  ( ph  ->  ( `' F `  .1.  )  e.  B
)
49 fidomndrng.d . . . 4  |-  .||  =  (
||r `  R )
5016, 49, 17dvdsrmul 17171 . . 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 6288 . . . . 5  |-  ( y  =  ( `' F `  .1.  )  ->  (
y  .x.  A )  =  ( ( `' F `  .1.  )  .x.  A ) )
536cbvmptv 4528 . . . . . 6  |-  ( x  e.  B  |->  ( x 
.x.  A ) )  =  ( y  e.  B  |->  ( y  .x.  A ) )
547, 53eqtri 2472 . . . . 5  |-  F  =  ( y  e.  B  |->  ( y  .x.  A
) )
55 ovex 6309 . . . . 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 6168 . . . 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 2486 . 2  |-  ( ph  ->  ( ( `' F `  .1.  )  .x.  A
)  =  .1.  )
6151, 60breqtrd 4461 1  |-  ( ph  ->  A  .||  .1.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1383    e. wcel 1804    =/= wne 2638   A.wral 2793    \ cdif 3458   {csn 4014   class class class wbr 4437    |-> cmpt 4495   `'ccnv 4988   -->wf 5574   -1-1->wf1 5575   -1-1-onto->wf1o 5577   ` cfv 5578  (class class class)co 6281    ~~ cen 7515   Fincfn 7518   Basecbs 14509   .rcmulr 14575   0gc0g 14714    GrpHom cghm 16138   1rcur 17027   Ringcrg 17072   ||rcdsr 17161  Domncdomn 17802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-er 7313  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10543  df-2 10600  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-plusg 14587  df-0g 14716  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15931  df-minusg 15932  df-sbg 15933  df-ghm 16139  df-mgp 17016  df-ur 17028  df-ring 17074  df-dvdsr 17164  df-nzr 17780  df-domn 17806
This theorem is referenced by:  fidomndrng  17830
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