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Theorem zrrnghm 40425
Description: The constant mapping to zero is a nonunital ring homomorphism from the zero ring to any nonunital ring. (Contributed by AV, 17-Apr-2020.)
Hypotheses
Ref Expression
zrrhm.b  |-  B  =  ( Base `  T
)
zrrhm.0  |-  .0.  =  ( 0g `  S )
zrrhm.h  |-  H  =  ( x  e.  B  |->  .0.  )
Assertion
Ref Expression
zrrnghm  |-  ( ( S  e. Rng  /\  T  e.  ( Ring  \ NzRing ) )  ->  H  e.  ( T RngHomo  S ) )
Distinct variable groups:    x, B    x, S    x, T    x,  .0.
Allowed substitution hint:    H( x)

Proof of Theorem zrrnghm
Dummy variables  a 
c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eldifi 3544 . . . . 5  |-  ( T  e.  ( Ring  \ NzRing )  ->  T  e.  Ring )
2 ringrng 40387 . . . . 5  |-  ( T  e.  Ring  ->  T  e. Rng )
31, 2syl 17 . . . 4  |-  ( T  e.  ( Ring  \ NzRing )  ->  T  e. Rng )
43anim1i 578 . . 3  |-  ( ( T  e.  ( Ring  \ NzRing )  /\  S  e. Rng )  ->  ( T  e. Rng  /\  S  e. Rng )
)
54ancoms 460 . 2  |-  ( ( S  e. Rng  /\  T  e.  ( Ring  \ NzRing ) )  ->  ( T  e. Rng  /\  S  e. Rng )
)
6 rngabl 40385 . . . . . 6  |-  ( S  e. Rng  ->  S  e.  Abel )
7 ablgrp 17513 . . . . . 6  |-  ( S  e.  Abel  ->  S  e. 
Grp )
86, 7syl 17 . . . . 5  |-  ( S  e. Rng  ->  S  e.  Grp )
98adantr 472 . . . 4  |-  ( ( S  e. Rng  /\  T  e.  ( Ring  \ NzRing ) )  ->  S  e.  Grp )
10 ringgrp 17863 . . . . . 6  |-  ( T  e.  Ring  ->  T  e. 
Grp )
111, 10syl 17 . . . . 5  |-  ( T  e.  ( Ring  \ NzRing )  ->  T  e.  Grp )
1211adantl 473 . . . 4  |-  ( ( S  e. Rng  /\  T  e.  ( Ring  \ NzRing ) )  ->  T  e.  Grp )
13 zrrhm.b . . . . . 6  |-  B  =  ( Base `  T
)
14 eqid 2471 . . . . . 6  |-  ( 0g
`  T )  =  ( 0g `  T
)
1513, 140ringbas 40379 . . . . 5  |-  ( T  e.  ( Ring  \ NzRing )  ->  B  =  { ( 0g `  T ) } )
1615adantl 473 . . . 4  |-  ( ( S  e. Rng  /\  T  e.  ( Ring  \ NzRing ) )  ->  B  =  {
( 0g `  T
) } )
17 zrrhm.0 . . . . 5  |-  .0.  =  ( 0g `  S )
18 zrrhm.h . . . . 5  |-  H  =  ( x  e.  B  |->  .0.  )
1913, 17, 18, 14c0snghm 40424 . . . 4  |-  ( ( S  e.  Grp  /\  T  e.  Grp  /\  B  =  { ( 0g `  T ) } )  ->  H  e.  ( T  GrpHom  S ) )
209, 12, 16, 19syl3anc 1292 . . 3  |-  ( ( S  e. Rng  /\  T  e.  ( Ring  \ NzRing ) )  ->  H  e.  ( T  GrpHom  S ) )
2118a1i 11 . . . . . . . 8  |-  ( ( ( S  e. Rng  /\  T  e.  ( Ring  \ NzRing
) )  /\  B  =  { ( 0g `  T ) } )  ->  H  =  ( x  e.  B  |->  .0.  ) )
22 eqidd 2472 . . . . . . . 8  |-  ( ( ( ( S  e. Rng  /\  T  e.  ( Ring  \ NzRing ) )  /\  B  =  { ( 0g `  T ) } )  /\  x  =  ( 0g `  T
) )  ->  .0.  =  .0.  )
2313, 14ring0cl 17880 . . . . . . . . . 10  |-  ( T  e.  Ring  ->  ( 0g
`  T )  e.  B )
241, 23syl 17 . . . . . . . . 9  |-  ( T  e.  ( Ring  \ NzRing )  -> 
( 0g `  T
)  e.  B )
2524ad2antlr 741 . . . . . . . 8  |-  ( ( ( S  e. Rng  /\  T  e.  ( Ring  \ NzRing
) )  /\  B  =  { ( 0g `  T ) } )  ->  ( 0g `  T )  e.  B
)
26 fvex 5889 . . . . . . . . . 10  |-  ( 0g
`  S )  e. 
_V
2717, 26eqeltri 2545 . . . . . . . . 9  |-  .0.  e.  _V
2827a1i 11 . . . . . . . 8  |-  ( ( ( S  e. Rng  /\  T  e.  ( Ring  \ NzRing
) )  /\  B  =  { ( 0g `  T ) } )  ->  .0.  e.  _V )
2921, 22, 25, 28fvmptd 5969 . . . . . . 7  |-  ( ( ( S  e. Rng  /\  T  e.  ( Ring  \ NzRing
) )  /\  B  =  { ( 0g `  T ) } )  ->  ( H `  ( 0g `  T ) )  =  .0.  )
30 eqid 2471 . . . . . . . . . . . . . 14  |-  ( Base `  S )  =  (
Base `  S )
3130, 17grpidcl 16772 . . . . . . . . . . . . 13  |-  ( S  e.  Grp  ->  .0.  e.  ( Base `  S
) )
328, 31syl 17 . . . . . . . . . . . 12  |-  ( S  e. Rng  ->  .0.  e.  ( Base `  S ) )
33 eqid 2471 . . . . . . . . . . . . 13  |-  ( .r
`  S )  =  ( .r `  S
)
3430, 33, 17rnglz 40392 . . . . . . . . . . . 12  |-  ( ( S  e. Rng  /\  .0.  e.  ( Base `  S
) )  ->  (  .0.  ( .r `  S
)  .0.  )  =  .0.  )
3532, 34mpdan 681 . . . . . . . . . . 11  |-  ( S  e. Rng  ->  (  .0.  ( .r `  S )  .0.  )  =  .0.  )
3635adantr 472 . . . . . . . . . 10  |-  ( ( S  e. Rng  /\  T  e.  ( Ring  \ NzRing ) )  ->  (  .0.  ( .r `  S )  .0.  )  =  .0.  )
3736adantr 472 . . . . . . . . 9  |-  ( ( ( S  e. Rng  /\  T  e.  ( Ring  \ NzRing
) )  /\  B  =  { ( 0g `  T ) } )  ->  (  .0.  ( .r `  S )  .0.  )  =  .0.  )
3837adantr 472 . . . . . . . 8  |-  ( ( ( ( S  e. Rng  /\  T  e.  ( Ring  \ NzRing ) )  /\  B  =  { ( 0g `  T ) } )  /\  ( H `
 ( 0g `  T ) )  =  .0.  )  ->  (  .0.  ( .r `  S
)  .0.  )  =  .0.  )
39 simpr 468 . . . . . . . . 9  |-  ( ( ( ( S  e. Rng  /\  T  e.  ( Ring  \ NzRing ) )  /\  B  =  { ( 0g `  T ) } )  /\  ( H `
 ( 0g `  T ) )  =  .0.  )  ->  ( H `  ( 0g `  T ) )  =  .0.  )
4039, 39oveq12d 6326 . . . . . . . 8  |-  ( ( ( ( S  e. Rng  /\  T  e.  ( Ring  \ NzRing ) )  /\  B  =  { ( 0g `  T ) } )  /\  ( H `
 ( 0g `  T ) )  =  .0.  )  ->  (
( H `  ( 0g `  T ) ) ( .r `  S
) ( H `  ( 0g `  T ) ) )  =  (  .0.  ( .r `  S )  .0.  )
)
41 eqid 2471 . . . . . . . . . . . . . . 15  |-  ( .r
`  T )  =  ( .r `  T
)
4213, 41, 14ringlz 17895 . . . . . . . . . . . . . 14  |-  ( ( T  e.  Ring  /\  ( 0g `  T )  e.  B )  ->  (
( 0g `  T
) ( .r `  T ) ( 0g
`  T ) )  =  ( 0g `  T ) )
4323, 42mpdan 681 . . . . . . . . . . . . 13  |-  ( T  e.  Ring  ->  ( ( 0g `  T ) ( .r `  T
) ( 0g `  T ) )  =  ( 0g `  T
) )
441, 43syl 17 . . . . . . . . . . . 12  |-  ( T  e.  ( Ring  \ NzRing )  -> 
( ( 0g `  T ) ( .r
`  T ) ( 0g `  T ) )  =  ( 0g
`  T ) )
4544ad2antlr 741 . . . . . . . . . . 11  |-  ( ( ( S  e. Rng  /\  T  e.  ( Ring  \ NzRing
) )  /\  B  =  { ( 0g `  T ) } )  ->  ( ( 0g
`  T ) ( .r `  T ) ( 0g `  T
) )  =  ( 0g `  T ) )
4645adantr 472 . . . . . . . . . 10  |-  ( ( ( ( S  e. Rng  /\  T  e.  ( Ring  \ NzRing ) )  /\  B  =  { ( 0g `  T ) } )  /\  ( H `
 ( 0g `  T ) )  =  .0.  )  ->  (
( 0g `  T
) ( .r `  T ) ( 0g
`  T ) )  =  ( 0g `  T ) )
4746fveq2d 5883 . . . . . . . . 9  |-  ( ( ( ( S  e. Rng  /\  T  e.  ( Ring  \ NzRing ) )  /\  B  =  { ( 0g `  T ) } )  /\  ( H `
 ( 0g `  T ) )  =  .0.  )  ->  ( H `  ( ( 0g `  T ) ( .r `  T ) ( 0g `  T
) ) )  =  ( H `  ( 0g `  T ) ) )
4847, 39eqtrd 2505 . . . . . . . 8  |-  ( ( ( ( S  e. Rng  /\  T  e.  ( Ring  \ NzRing ) )  /\  B  =  { ( 0g `  T ) } )  /\  ( H `
 ( 0g `  T ) )  =  .0.  )  ->  ( H `  ( ( 0g `  T ) ( .r `  T ) ( 0g `  T
) ) )  =  .0.  )
4938, 40, 483eqtr4rd 2516 . . . . . . 7  |-  ( ( ( ( S  e. Rng  /\  T  e.  ( Ring  \ NzRing ) )  /\  B  =  { ( 0g `  T ) } )  /\  ( H `
 ( 0g `  T ) )  =  .0.  )  ->  ( H `  ( ( 0g `  T ) ( .r `  T ) ( 0g `  T
) ) )  =  ( ( H `  ( 0g `  T ) ) ( .r `  S ) ( H `
 ( 0g `  T ) ) ) )
5029, 49mpdan 681 . . . . . 6  |-  ( ( ( S  e. Rng  /\  T  e.  ( Ring  \ NzRing
) )  /\  B  =  { ( 0g `  T ) } )  ->  ( H `  ( ( 0g `  T ) ( .r
`  T ) ( 0g `  T ) ) )  =  ( ( H `  ( 0g `  T ) ) ( .r `  S
) ( H `  ( 0g `  T ) ) ) )
5123, 23jca 541 . . . . . . . . 9  |-  ( T  e.  Ring  ->  ( ( 0g `  T )  e.  B  /\  ( 0g `  T )  e.  B ) )
521, 51syl 17 . . . . . . . 8  |-  ( T  e.  ( Ring  \ NzRing )  -> 
( ( 0g `  T )  e.  B  /\  ( 0g `  T
)  e.  B ) )
5352ad2antlr 741 . . . . . . 7  |-  ( ( ( S  e. Rng  /\  T  e.  ( Ring  \ NzRing
) )  /\  B  =  { ( 0g `  T ) } )  ->  ( ( 0g
`  T )  e.  B  /\  ( 0g
`  T )  e.  B ) )
54 oveq1 6315 . . . . . . . . . 10  |-  ( a  =  ( 0g `  T )  ->  (
a ( .r `  T ) c )  =  ( ( 0g
`  T ) ( .r `  T ) c ) )
5554fveq2d 5883 . . . . . . . . 9  |-  ( a  =  ( 0g `  T )  ->  ( H `  ( a
( .r `  T
) c ) )  =  ( H `  ( ( 0g `  T ) ( .r
`  T ) c ) ) )
56 fveq2 5879 . . . . . . . . . 10  |-  ( a  =  ( 0g `  T )  ->  ( H `  a )  =  ( H `  ( 0g `  T ) ) )
5756oveq1d 6323 . . . . . . . . 9  |-  ( a  =  ( 0g `  T )  ->  (
( H `  a
) ( .r `  S ) ( H `
 c ) )  =  ( ( H `
 ( 0g `  T ) ) ( .r `  S ) ( H `  c
) ) )
5855, 57eqeq12d 2486 . . . . . . . 8  |-  ( a  =  ( 0g `  T )  ->  (
( H `  (
a ( .r `  T ) c ) )  =  ( ( H `  a ) ( .r `  S
) ( H `  c ) )  <->  ( H `  ( ( 0g `  T ) ( .r
`  T ) c ) )  =  ( ( H `  ( 0g `  T ) ) ( .r `  S
) ( H `  c ) ) ) )
59 oveq2 6316 . . . . . . . . . 10  |-  ( c  =  ( 0g `  T )  ->  (
( 0g `  T
) ( .r `  T ) c )  =  ( ( 0g
`  T ) ( .r `  T ) ( 0g `  T
) ) )
6059fveq2d 5883 . . . . . . . . 9  |-  ( c  =  ( 0g `  T )  ->  ( H `  ( ( 0g `  T ) ( .r `  T ) c ) )  =  ( H `  (
( 0g `  T
) ( .r `  T ) ( 0g
`  T ) ) ) )
61 fveq2 5879 . . . . . . . . . 10  |-  ( c  =  ( 0g `  T )  ->  ( H `  c )  =  ( H `  ( 0g `  T ) ) )
6261oveq2d 6324 . . . . . . . . 9  |-  ( c  =  ( 0g `  T )  ->  (
( H `  ( 0g `  T ) ) ( .r `  S
) ( H `  c ) )  =  ( ( H `  ( 0g `  T ) ) ( .r `  S ) ( H `
 ( 0g `  T ) ) ) )
6360, 62eqeq12d 2486 . . . . . . . 8  |-  ( c  =  ( 0g `  T )  ->  (
( H `  (
( 0g `  T
) ( .r `  T ) c ) )  =  ( ( H `  ( 0g
`  T ) ) ( .r `  S
) ( H `  c ) )  <->  ( H `  ( ( 0g `  T ) ( .r
`  T ) ( 0g `  T ) ) )  =  ( ( H `  ( 0g `  T ) ) ( .r `  S
) ( H `  ( 0g `  T ) ) ) ) )
6458, 632ralsng 3999 . . . . . . 7  |-  ( ( ( 0g `  T
)  e.  B  /\  ( 0g `  T )  e.  B )  -> 
( A. a  e. 
{ ( 0g `  T ) } A. c  e.  { ( 0g `  T ) }  ( H `  (
a ( .r `  T ) c ) )  =  ( ( H `  a ) ( .r `  S
) ( H `  c ) )  <->  ( H `  ( ( 0g `  T ) ( .r
`  T ) ( 0g `  T ) ) )  =  ( ( H `  ( 0g `  T ) ) ( .r `  S
) ( H `  ( 0g `  T ) ) ) ) )
6553, 64syl 17 . . . . . 6  |-  ( ( ( S  e. Rng  /\  T  e.  ( Ring  \ NzRing
) )  /\  B  =  { ( 0g `  T ) } )  ->  ( A. a  e.  { ( 0g `  T ) } A. c  e.  { ( 0g `  T ) }  ( H `  (
a ( .r `  T ) c ) )  =  ( ( H `  a ) ( .r `  S
) ( H `  c ) )  <->  ( H `  ( ( 0g `  T ) ( .r
`  T ) ( 0g `  T ) ) )  =  ( ( H `  ( 0g `  T ) ) ( .r `  S
) ( H `  ( 0g `  T ) ) ) ) )
6650, 65mpbird 240 . . . . 5  |-  ( ( ( S  e. Rng  /\  T  e.  ( Ring  \ NzRing
) )  /\  B  =  { ( 0g `  T ) } )  ->  A. a  e.  {
( 0g `  T
) } A. c  e.  { ( 0g `  T ) }  ( H `  ( a
( .r `  T
) c ) )  =  ( ( H `
 a ) ( .r `  S ) ( H `  c
) ) )
67 raleq 2973 . . . . . . 7  |-  ( B  =  { ( 0g
`  T ) }  ->  ( A. c  e.  B  ( H `  ( a ( .r
`  T ) c ) )  =  ( ( H `  a
) ( .r `  S ) ( H `
 c ) )  <->  A. c  e.  { ( 0g `  T ) }  ( H `  ( a ( .r
`  T ) c ) )  =  ( ( H `  a
) ( .r `  S ) ( H `
 c ) ) ) )
6867raleqbi1dv 2981 . . . . . 6  |-  ( B  =  { ( 0g
`  T ) }  ->  ( A. a  e.  B  A. c  e.  B  ( H `  ( a ( .r
`  T ) c ) )  =  ( ( H `  a
) ( .r `  S ) ( H `
 c ) )  <->  A. a  e.  { ( 0g `  T ) } A. c  e. 
{ ( 0g `  T ) }  ( H `  ( a
( .r `  T
) c ) )  =  ( ( H `
 a ) ( .r `  S ) ( H `  c
) ) ) )
6968adantl 473 . . . . 5  |-  ( ( ( S  e. Rng  /\  T  e.  ( Ring  \ NzRing
) )  /\  B  =  { ( 0g `  T ) } )  ->  ( A. a  e.  B  A. c  e.  B  ( H `  ( a ( .r
`  T ) c ) )  =  ( ( H `  a
) ( .r `  S ) ( H `
 c ) )  <->  A. a  e.  { ( 0g `  T ) } A. c  e. 
{ ( 0g `  T ) }  ( H `  ( a
( .r `  T
) c ) )  =  ( ( H `
 a ) ( .r `  S ) ( H `  c
) ) ) )
7066, 69mpbird 240 . . . 4  |-  ( ( ( S  e. Rng  /\  T  e.  ( Ring  \ NzRing
) )  /\  B  =  { ( 0g `  T ) } )  ->  A. a  e.  B  A. c  e.  B  ( H `  ( a ( .r `  T
) c ) )  =  ( ( H `
 a ) ( .r `  S ) ( H `  c
) ) )
7116, 70mpdan 681 . . 3  |-  ( ( S  e. Rng  /\  T  e.  ( Ring  \ NzRing ) )  ->  A. a  e.  B  A. c  e.  B  ( H `  ( a ( .r `  T
) c ) )  =  ( ( H `
 a ) ( .r `  S ) ( H `  c
) ) )
7220, 71jca 541 . 2  |-  ( ( S  e. Rng  /\  T  e.  ( Ring  \ NzRing ) )  ->  ( H  e.  ( T  GrpHom  S )  /\  A. a  e.  B  A. c  e.  B  ( H `  ( a ( .r
`  T ) c ) )  =  ( ( H `  a
) ( .r `  S ) ( H `
 c ) ) ) )
7313, 41, 33isrnghm 40400 . 2  |-  ( H  e.  ( T RngHomo  S
)  <->  ( ( T  e. Rng  /\  S  e. Rng )  /\  ( H  e.  ( T  GrpHom  S )  /\  A. a  e.  B  A. c  e.  B  ( H `  ( a ( .r
`  T ) c ) )  =  ( ( H `  a
) ( .r `  S ) ( H `
 c ) ) ) ) )
745, 72, 73sylanbrc 677 1  |-  ( ( S  e. Rng  /\  T  e.  ( Ring  \ NzRing ) )  ->  H  e.  ( T RngHomo  S ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904   A.wral 2756   _Vcvv 3031    \ cdif 3387   {csn 3959    |-> cmpt 4454   ` cfv 5589  (class class class)co 6308   Basecbs 15199   .rcmulr 15269   0gc0g 15416   Grpcgrp 16747    GrpHom cghm 16958   Abelcabl 17509   Ringcrg 17858  NzRingcnzr 18558  Rngcrng 40382   RngHomo crngh 40393
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-hash 12554  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-plusg 15281  df-0g 15418  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-mhm 16660  df-grp 16751  df-minusg 16752  df-ghm 16959  df-cmn 17510  df-abl 17511  df-mgp 17802  df-ur 17814  df-ring 17860  df-nzr 18559  df-mgmhm 40287  df-rng0 40383  df-rnghomo 40395
This theorem is referenced by:  zrinitorngc  40510
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