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Theorem fidomndrnglem 17356
Description: Lemma for fidomndrng 17357. (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 3337 . . 3  |-  ( ph  ->  A  e.  B )
3 eldifsni 3998 . . . . . . . . . . . 12  |-  ( A  e.  ( B  \  {  .0.  } )  ->  A  =/=  .0.  )
41, 3syl 16 . . . . . . . . . . 11  |-  ( ph  ->  A  =/=  .0.  )
54ad2antrr 720 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  B )  /\  ( F `  y )  =  .0.  )  ->  A  =/=  .0.  )
6 oveq1 6097 . . . . . . . . . . . . . . . . 17  |-  ( x  =  y  ->  (
x  .x.  A )  =  ( y  .x.  A ) )
7 fidomndrng.f . . . . . . . . . . . . . . . . 17  |-  F  =  ( x  e.  B  |->  ( x  .x.  A
) )
8 ovex 6115 . . . . . . . . . . . . . . . . 17  |-  ( y 
.x.  A )  e. 
_V
96, 7, 8fvmpt 5771 . . . . . . . . . . . . . . . 16  |-  ( y  e.  B  ->  ( F `  y )  =  ( y  .x.  A ) )
109adantl 463 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  y  e.  B )  ->  ( F `  y )  =  ( y  .x.  A ) )
1110eqeq1d 2449 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  y  e.  B )  ->  (
( F `  y
)  =  .0.  <->  ( y  .x.  A )  =  .0.  ) )
12 fidomndrng.r . . . . . . . . . . . . . . . 16  |-  ( ph  ->  R  e. Domn )
1312adantr 462 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  y  e.  B )  ->  R  e. Domn )
14 simpr 458 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  y  e.  B )  ->  y  e.  B )
152adantr 462 . . . . . . . . . . . . . . 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 17347 . . . . . . . . . . . . . . 15  |-  ( ( R  e. Domn  /\  y  e.  B  /\  A  e.  B )  ->  (
( y  .x.  A
)  =  .0.  <->  ( y  =  .0.  \/  A  =  .0.  ) ) )
2013, 14, 15, 19syl3anc 1213 . . . . . . . . . . . . . 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 481 . . . . . . . . . . . 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 2797 . . . . . . 7  |-  ( ph  ->  A. y  e.  B  ( ( F `  y )  =  .0. 
->  y  =  .0.  ) )
28 domnrng 17346 . . . . . . . . . . 11  |-  ( R  e. Domn  ->  R  e.  Ring )
2912, 28syl 16 . . . . . . . . . 10  |-  ( ph  ->  R  e.  Ring )
3016, 17rngrghm 16684 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  A  e.  B )  ->  (
x  e.  B  |->  ( x  .x.  A ) )  e.  ( R 
GrpHom  R ) )
3129, 2, 30syl2anc 656 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  B  |->  ( x  .x.  A
) )  e.  ( R  GrpHom  R ) )
327, 31syl5eqel 2525 . . . . . . . 8  |-  ( ph  ->  F  e.  ( R 
GrpHom  R ) )
3316, 16, 18, 18ghmf1 15768 . . . . . . . 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 7337 . . . . . . . 8  |-  ( B  e.  Fin  ->  B  ~~  B )
3836, 37syl 16 . . . . . . 7  |-  ( ph  ->  B  ~~  B )
39 f1finf1o 7535 . . . . . . 7  |-  ( ( B  ~~  B  /\  B  e.  Fin )  ->  ( F : B -1-1-> B  <-> 
F : B -1-1-onto-> B ) )
4038, 36, 39syl2anc 656 . . . . . 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 5650 . . . . 5  |-  ( F : B -1-1-onto-> B  ->  `' F : B -1-1-onto-> B )
43 f1of 5638 . . . . 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 16655 . . . . 5  |-  ( R  e.  Ring  ->  .1.  e.  B )
4729, 46syl 16 . . . 4  |-  ( ph  ->  .1.  e.  B )
4844, 47ffvelrnd 5841 . . 3  |-  ( ph  ->  ( `' F `  .1.  )  e.  B
)
49 fidomndrng.d . . . 4  |-  .||  =  (
||r `  R )
5016, 49, 17dvdsrmul 16730 . . 3  |-  ( ( A  e.  B  /\  ( `' F `  .1.  )  e.  B )  ->  A  .||  ( ( `' F `  .1.  )  .x.  A
) )
512, 48, 50syl2anc 656 . 2  |-  ( ph  ->  A  .||  ( ( `' F `  .1.  )  .x.  A ) )
52 oveq1 6097 . . . . 5  |-  ( y  =  ( `' F `  .1.  )  ->  (
y  .x.  A )  =  ( ( `' F `  .1.  )  .x.  A ) )
536cbvmptv 4380 . . . . . 6  |-  ( x  e.  B  |->  ( x 
.x.  A ) )  =  ( y  e.  B  |->  ( y  .x.  A ) )
547, 53eqtri 2461 . . . . 5  |-  F  =  ( y  e.  B  |->  ( y  .x.  A
) )
55 ovex 6115 . . . . 5  |-  ( ( `' F `  .1.  )  .x.  A )  e.  _V
5652, 54, 55fvmpt 5771 . . . 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 5981 . . . 4  |-  ( ( F : B -1-1-onto-> B  /\  .1.  e.  B )  -> 
( F `  ( `' F `  .1.  )
)  =  .1.  )
5941, 47, 58syl2anc 656 . . 3  |-  ( ph  ->  ( F `  ( `' F `  .1.  )
)  =  .1.  )
6057, 59eqtr3d 2475 . 2  |-  ( ph  ->  ( ( `' F `  .1.  )  .x.  A
)  =  .1.  )
6151, 60breqtrd 4313 1  |-  ( ph  ->  A  .||  .1.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1364    e. wcel 1761    =/= wne 2604   A.wral 2713    \ cdif 3322   {csn 3874   class class class wbr 4289    e. cmpt 4347   `'ccnv 4835   -->wf 5411   -1-1->wf1 5412   -1-1-onto->wf1o 5414   ` cfv 5415  (class class class)co 6090    ~~ cen 7303   Fincfn 7306   Basecbs 14170   .rcmulr 14235   0gc0g 14374    GrpHom cghm 15737   1rcur 16593   Ringcrg 16635   ||rcdsr 16720  Domncdomn 17329
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 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-plusg 14247  df-0g 14376  df-mnd 15411  df-grp 15538  df-minusg 15539  df-sbg 15540  df-ghm 15738  df-mgp 16582  df-ur 16594  df-rng 16637  df-dvdsr 16723  df-nzr 17318  df-domn 17333
This theorem is referenced by:  fidomndrng  17357
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