MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fidomndrnglem Unicode version

Theorem fidomndrnglem 16047
Description: Lemma for fidomndrng 16048. (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.  }
) )
2 eldifi 3298 . . . 4  |-  ( A  e.  ( B  \  {  .0.  } )  ->  A  e.  B )
31, 2syl 15 . . 3  |-  ( ph  ->  A  e.  B )
4 eldifsni 3750 . . . . . . . . . . . 12  |-  ( A  e.  ( B  \  {  .0.  } )  ->  A  =/=  .0.  )
51, 4syl 15 . . . . . . . . . . 11  |-  ( ph  ->  A  =/=  .0.  )
65ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  B )  /\  ( F `  y )  =  .0.  )  ->  A  =/=  .0.  )
7 oveq1 5865 . . . . . . . . . . . . . . . . 17  |-  ( x  =  y  ->  (
x  .x.  A )  =  ( y  .x.  A ) )
8 fidomndrng.f . . . . . . . . . . . . . . . . 17  |-  F  =  ( x  e.  B  |->  ( x  .x.  A
) )
9 ovex 5883 . . . . . . . . . . . . . . . . 17  |-  ( y 
.x.  A )  e. 
_V
107, 8, 9fvmpt 5602 . . . . . . . . . . . . . . . 16  |-  ( y  e.  B  ->  ( F `  y )  =  ( y  .x.  A ) )
1110adantl 452 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  y  e.  B )  ->  ( F `  y )  =  ( y  .x.  A ) )
1211eqeq1d 2291 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  y  e.  B )  ->  (
( F `  y
)  =  .0.  <->  ( y  .x.  A )  =  .0.  ) )
13 fidomndrng.r . . . . . . . . . . . . . . . 16  |-  ( ph  ->  R  e. Domn )
1413adantr 451 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  y  e.  B )  ->  R  e. Domn )
15 simpr 447 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  y  e.  B )  ->  y  e.  B )
163adantr 451 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  y  e.  B )  ->  A  e.  B )
17 fidomndrng.b . . . . . . . . . . . . . . . 16  |-  B  =  ( Base `  R
)
18 fidomndrng.t . . . . . . . . . . . . . . . 16  |-  .x.  =  ( .r `  R )
19 fidomndrng.z . . . . . . . . . . . . . . . 16  |-  .0.  =  ( 0g `  R )
2017, 18, 19domneq0 16038 . . . . . . . . . . . . . . 15  |-  ( ( R  e. Domn  /\  y  e.  B  /\  A  e.  B )  ->  (
( y  .x.  A
)  =  .0.  <->  ( y  =  .0.  \/  A  =  .0.  ) ) )
2114, 15, 16, 20syl3anc 1182 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  y  e.  B )  ->  (
( y  .x.  A
)  =  .0.  <->  ( y  =  .0.  \/  A  =  .0.  ) ) )
2212, 21bitrd 244 . . . . . . . . . . . . 13  |-  ( (
ph  /\  y  e.  B )  ->  (
( F `  y
)  =  .0.  <->  ( y  =  .0.  \/  A  =  .0.  ) ) )
2322biimpa 470 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  y  e.  B )  /\  ( F `  y )  =  .0.  )  ->  (
y  =  .0.  \/  A  =  .0.  )
)
2423ord 366 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  B )  /\  ( F `  y )  =  .0.  )  ->  ( -.  y  =  .0.  ->  A  =  .0.  )
)
2524necon1ad 2513 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  B )  /\  ( F `  y )  =  .0.  )  ->  ( A  =/=  .0.  ->  y  =  .0.  ) )
266, 25mpd 14 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  B )  /\  ( F `  y )  =  .0.  )  ->  y  =  .0.  )
2726ex 423 . . . . . . . 8  |-  ( (
ph  /\  y  e.  B )  ->  (
( F `  y
)  =  .0.  ->  y  =  .0.  ) )
2827ralrimiva 2626 . . . . . . 7  |-  ( ph  ->  A. y  e.  B  ( ( F `  y )  =  .0. 
->  y  =  .0.  ) )
29 domnrng 16037 . . . . . . . . . . 11  |-  ( R  e. Domn  ->  R  e.  Ring )
3013, 29syl 15 . . . . . . . . . 10  |-  ( ph  ->  R  e.  Ring )
3117, 18rngrghm 15389 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  A  e.  B )  ->  (
x  e.  B  |->  ( x  .x.  A ) )  e.  ( R 
GrpHom  R ) )
3230, 3, 31syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  B  |->  ( x  .x.  A
) )  e.  ( R  GrpHom  R ) )
338, 32syl5eqel 2367 . . . . . . . 8  |-  ( ph  ->  F  e.  ( R 
GrpHom  R ) )
3417, 17, 19, 19ghmf1 14711 . . . . . . . 8  |-  ( F  e.  ( R  GrpHom  R )  ->  ( F : B -1-1-> B  <->  A. y  e.  B  ( ( F `  y )  =  .0. 
->  y  =  .0.  ) ) )
3533, 34syl 15 . . . . . . 7  |-  ( ph  ->  ( F : B -1-1-> B  <->  A. y  e.  B  ( ( F `  y )  =  .0. 
->  y  =  .0.  ) ) )
3628, 35mpbird 223 . . . . . 6  |-  ( ph  ->  F : B -1-1-> B
)
37 fidomndrng.x . . . . . . . 8  |-  ( ph  ->  B  e.  Fin )
38 enrefg 6893 . . . . . . . 8  |-  ( B  e.  Fin  ->  B  ~~  B )
3937, 38syl 15 . . . . . . 7  |-  ( ph  ->  B  ~~  B )
40 f1finf1o 7086 . . . . . . 7  |-  ( ( B  ~~  B  /\  B  e.  Fin )  ->  ( F : B -1-1-> B  <-> 
F : B -1-1-onto-> B ) )
4139, 37, 40syl2anc 642 . . . . . 6  |-  ( ph  ->  ( F : B -1-1-> B  <-> 
F : B -1-1-onto-> B ) )
4236, 41mpbid 201 . . . . 5  |-  ( ph  ->  F : B -1-1-onto-> B )
43 f1ocnv 5485 . . . . 5  |-  ( F : B -1-1-onto-> B  ->  `' F : B -1-1-onto-> B )
44 f1of 5472 . . . . 5  |-  ( `' F : B -1-1-onto-> B  ->  `' F : B --> B )
4542, 43, 443syl 18 . . . 4  |-  ( ph  ->  `' F : B --> B )
46 fidomndrng.o . . . . . 6  |-  .1.  =  ( 1r `  R )
4717, 46rngidcl 15361 . . . . 5  |-  ( R  e.  Ring  ->  .1.  e.  B )
4830, 47syl 15 . . . 4  |-  ( ph  ->  .1.  e.  B )
49 ffvelrn 5663 . . . 4  |-  ( ( `' F : B --> B  /\  .1.  e.  B )  -> 
( `' F `  .1.  )  e.  B
)
5045, 48, 49syl2anc 642 . . 3  |-  ( ph  ->  ( `' F `  .1.  )  e.  B
)
51 fidomndrng.d . . . 4  |-  .||  =  (
||r `  R )
5217, 51, 18dvdsrmul 15430 . . 3  |-  ( ( A  e.  B  /\  ( `' F `  .1.  )  e.  B )  ->  A  .||  ( ( `' F `  .1.  )  .x.  A
) )
533, 50, 52syl2anc 642 . 2  |-  ( ph  ->  A  .||  ( ( `' F `  .1.  )  .x.  A ) )
54 oveq1 5865 . . . . 5  |-  ( y  =  ( `' F `  .1.  )  ->  (
y  .x.  A )  =  ( ( `' F `  .1.  )  .x.  A ) )
557cbvmptv 4111 . . . . . 6  |-  ( x  e.  B  |->  ( x 
.x.  A ) )  =  ( y  e.  B  |->  ( y  .x.  A ) )
568, 55eqtri 2303 . . . . 5  |-  F  =  ( y  e.  B  |->  ( y  .x.  A
) )
57 ovex 5883 . . . . 5  |-  ( ( `' F `  .1.  )  .x.  A )  e.  _V
5854, 56, 57fvmpt 5602 . . . 4  |-  ( ( `' F `  .1.  )  e.  B  ->  ( F `
 ( `' F `  .1.  ) )  =  ( ( `' F `  .1.  )  .x.  A
) )
5950, 58syl 15 . . 3  |-  ( ph  ->  ( F `  ( `' F `  .1.  )
)  =  ( ( `' F `  .1.  )  .x.  A ) )
60 f1ocnvfv2 5793 . . . 4  |-  ( ( F : B -1-1-onto-> B  /\  .1.  e.  B )  -> 
( F `  ( `' F `  .1.  )
)  =  .1.  )
6142, 48, 60syl2anc 642 . . 3  |-  ( ph  ->  ( F `  ( `' F `  .1.  )
)  =  .1.  )
6259, 61eqtr3d 2317 . 2  |-  ( ph  ->  ( ( `' F `  .1.  )  .x.  A
)  =  .1.  )
6353, 62breqtrd 4047 1  |-  ( ph  ->  A  .||  .1.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543    \ cdif 3149   {csn 3640   class class class wbr 4023    e. cmpt 4077   `'ccnv 4688   -->wf 5251   -1-1->wf1 5252   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858    ~~ cen 6860   Fincfn 6863   Basecbs 13148   .rcmulr 13209   0gc0g 13400    GrpHom cghm 14680   Ringcrg 15337   1rcur 15339   ||rcdsr 15420  Domncdomn 16021
This theorem is referenced by:  fidomndrng  16048
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-plusg 13221  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491  df-ghm 14681  df-mgp 15326  df-rng 15340  df-ur 15342  df-dvdsr 15423  df-nzr 16010  df-domn 16025
  Copyright terms: Public domain W3C validator