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Theorem fidomndrnglem 16321
Description: Lemma for fidomndrng 16322. (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 3292 . . 3  |-  ( ph  ->  A  e.  B )
3 eldifsni 3888 . . . . . . . . . . . 12  |-  ( A  e.  ( B  \  {  .0.  } )  ->  A  =/=  .0.  )
41, 3syl 16 . . . . . . . . . . 11  |-  ( ph  ->  A  =/=  .0.  )
54ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  B )  /\  ( F `  y )  =  .0.  )  ->  A  =/=  .0.  )
6 oveq1 6047 . . . . . . . . . . . . . . . . 17  |-  ( x  =  y  ->  (
x  .x.  A )  =  ( y  .x.  A ) )
7 fidomndrng.f . . . . . . . . . . . . . . . . 17  |-  F  =  ( x  e.  B  |->  ( x  .x.  A
) )
8 ovex 6065 . . . . . . . . . . . . . . . . 17  |-  ( y 
.x.  A )  e. 
_V
96, 7, 8fvmpt 5765 . . . . . . . . . . . . . . . 16  |-  ( y  e.  B  ->  ( F `  y )  =  ( y  .x.  A ) )
109adantl 453 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  y  e.  B )  ->  ( F `  y )  =  ( y  .x.  A ) )
1110eqeq1d 2412 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  y  e.  B )  ->  (
( F `  y
)  =  .0.  <->  ( y  .x.  A )  =  .0.  ) )
12 fidomndrng.r . . . . . . . . . . . . . . . 16  |-  ( ph  ->  R  e. Domn )
1312adantr 452 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  y  e.  B )  ->  R  e. Domn )
14 simpr 448 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  y  e.  B )  ->  y  e.  B )
152adantr 452 . . . . . . . . . . . . . . 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 16312 . . . . . . . . . . . . . . 15  |-  ( ( R  e. Domn  /\  y  e.  B  /\  A  e.  B )  ->  (
( y  .x.  A
)  =  .0.  <->  ( y  =  .0.  \/  A  =  .0.  ) ) )
2013, 14, 15, 19syl3anc 1184 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  y  e.  B )  ->  (
( y  .x.  A
)  =  .0.  <->  ( y  =  .0.  \/  A  =  .0.  ) ) )
2111, 20bitrd 245 . . . . . . . . . . . . 13  |-  ( (
ph  /\  y  e.  B )  ->  (
( F `  y
)  =  .0.  <->  ( y  =  .0.  \/  A  =  .0.  ) ) )
2221biimpa 471 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  y  e.  B )  /\  ( F `  y )  =  .0.  )  ->  (
y  =  .0.  \/  A  =  .0.  )
)
2322ord 367 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  B )  /\  ( F `  y )  =  .0.  )  ->  ( -.  y  =  .0.  ->  A  =  .0.  )
)
2423necon1ad 2634 . . . . . . . . . 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 424 . . . . . . . 8  |-  ( (
ph  /\  y  e.  B )  ->  (
( F `  y
)  =  .0.  ->  y  =  .0.  ) )
2726ralrimiva 2749 . . . . . . 7  |-  ( ph  ->  A. y  e.  B  ( ( F `  y )  =  .0. 
->  y  =  .0.  ) )
28 domnrng 16311 . . . . . . . . . . 11  |-  ( R  e. Domn  ->  R  e.  Ring )
2912, 28syl 16 . . . . . . . . . 10  |-  ( ph  ->  R  e.  Ring )
3016, 17rngrghm 15667 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  A  e.  B )  ->  (
x  e.  B  |->  ( x  .x.  A ) )  e.  ( R 
GrpHom  R ) )
3129, 2, 30syl2anc 643 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  B  |->  ( x  .x.  A
) )  e.  ( R  GrpHom  R ) )
327, 31syl5eqel 2488 . . . . . . . 8  |-  ( ph  ->  F  e.  ( R 
GrpHom  R ) )
3316, 16, 18, 18ghmf1 14989 . . . . . . . 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 224 . . . . . 6  |-  ( ph  ->  F : B -1-1-> B
)
36 fidomndrng.x . . . . . . . 8  |-  ( ph  ->  B  e.  Fin )
37 enrefg 7098 . . . . . . . 8  |-  ( B  e.  Fin  ->  B  ~~  B )
3836, 37syl 16 . . . . . . 7  |-  ( ph  ->  B  ~~  B )
39 f1finf1o 7294 . . . . . . 7  |-  ( ( B  ~~  B  /\  B  e.  Fin )  ->  ( F : B -1-1-> B  <-> 
F : B -1-1-onto-> B ) )
4038, 36, 39syl2anc 643 . . . . . 6  |-  ( ph  ->  ( F : B -1-1-> B  <-> 
F : B -1-1-onto-> B ) )
4135, 40mpbid 202 . . . . 5  |-  ( ph  ->  F : B -1-1-onto-> B )
42 f1ocnv 5646 . . . . 5  |-  ( F : B -1-1-onto-> B  ->  `' F : B -1-1-onto-> B )
43 f1of 5633 . . . . 5  |-  ( `' F : B -1-1-onto-> B  ->  `' F : B --> B )
4441, 42, 433syl 19 . . . 4  |-  ( ph  ->  `' F : B --> B )
45 fidomndrng.o . . . . . 6  |-  .1.  =  ( 1r `  R )
4616, 45rngidcl 15639 . . . . 5  |-  ( R  e.  Ring  ->  .1.  e.  B )
4729, 46syl 16 . . . 4  |-  ( ph  ->  .1.  e.  B )
4844, 47ffvelrnd 5830 . . 3  |-  ( ph  ->  ( `' F `  .1.  )  e.  B
)
49 fidomndrng.d . . . 4  |-  .||  =  (
||r `  R )
5016, 49, 17dvdsrmul 15708 . . 3  |-  ( ( A  e.  B  /\  ( `' F `  .1.  )  e.  B )  ->  A  .||  ( ( `' F `  .1.  )  .x.  A
) )
512, 48, 50syl2anc 643 . 2  |-  ( ph  ->  A  .||  ( ( `' F `  .1.  )  .x.  A ) )
52 oveq1 6047 . . . . 5  |-  ( y  =  ( `' F `  .1.  )  ->  (
y  .x.  A )  =  ( ( `' F `  .1.  )  .x.  A ) )
536cbvmptv 4260 . . . . . 6  |-  ( x  e.  B  |->  ( x 
.x.  A ) )  =  ( y  e.  B  |->  ( y  .x.  A ) )
547, 53eqtri 2424 . . . . 5  |-  F  =  ( y  e.  B  |->  ( y  .x.  A
) )
55 ovex 6065 . . . . 5  |-  ( ( `' F `  .1.  )  .x.  A )  e.  _V
5652, 54, 55fvmpt 5765 . . . 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 5974 . . . 4  |-  ( ( F : B -1-1-onto-> B  /\  .1.  e.  B )  -> 
( F `  ( `' F `  .1.  )
)  =  .1.  )
5941, 47, 58syl2anc 643 . . 3  |-  ( ph  ->  ( F `  ( `' F `  .1.  )
)  =  .1.  )
6057, 59eqtr3d 2438 . 2  |-  ( ph  ->  ( ( `' F `  .1.  )  .x.  A
)  =  .1.  )
6151, 60breqtrd 4196 1  |-  ( ph  ->  A  .||  .1.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666    \ cdif 3277   {csn 3774   class class class wbr 4172    e. cmpt 4226   `'ccnv 4836   -->wf 5409   -1-1->wf1 5410   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6040    ~~ cen 7065   Fincfn 7068   Basecbs 13424   .rcmulr 13485   0gc0g 13678    GrpHom cghm 14958   Ringcrg 15615   1rcur 15617   ||rcdsr 15698  Domncdomn 16295
This theorem is referenced by:  fidomndrng  16322
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-plusg 13497  df-0g 13682  df-mnd 14645  df-grp 14767  df-minusg 14768  df-sbg 14769  df-ghm 14959  df-mgp 15604  df-rng 15618  df-ur 15620  df-dvdsr 15701  df-nzr 16284  df-domn 16299
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