ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  srgisid Unicode version

Theorem srgisid 13100
Description: In a semiring, the only left-absorbing element is the additive identity. Remark in [Golan] p. 1. (Contributed by Thierry Arnoux, 1-May-2018.)
Hypotheses
Ref Expression
srgz.b  |-  B  =  ( Base `  R
)
srgz.t  |-  .x.  =  ( .r `  R )
srgz.z  |-  .0.  =  ( 0g `  R )
srgisid.1  |-  ( ph  ->  R  e. SRing )
srgisid.2  |-  ( ph  ->  Z  e.  B )
srgisid.3  |-  ( (
ph  /\  x  e.  B )  ->  ( Z  .x.  x )  =  Z )
Assertion
Ref Expression
srgisid  |-  ( ph  ->  Z  =  .0.  )
Distinct variable groups:    x, B    x, R    x,  .x.    x,  .0.    x, Z    ph, x

Proof of Theorem srgisid
StepHypRef Expression
1 srgisid.3 . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  ( Z  .x.  x )  =  Z )
21ralrimiva 2550 . . 3  |-  ( ph  ->  A. x  e.  B  ( Z  .x.  x )  =  Z )
3 srgisid.1 . . . 4  |-  ( ph  ->  R  e. SRing )
4 srgz.b . . . . 5  |-  B  =  ( Base `  R
)
5 srgz.z . . . . 5  |-  .0.  =  ( 0g `  R )
64, 5srg0cl 13091 . . . 4  |-  ( R  e. SRing  ->  .0.  e.  B
)
7 oveq2 5880 . . . . . 6  |-  ( x  =  .0.  ->  ( Z  .x.  x )  =  ( Z  .x.  .0.  ) )
87eqeq1d 2186 . . . . 5  |-  ( x  =  .0.  ->  (
( Z  .x.  x
)  =  Z  <->  ( Z  .x.  .0.  )  =  Z ) )
98rspcv 2837 . . . 4  |-  (  .0. 
e.  B  ->  ( A. x  e.  B  ( Z  .x.  x )  =  Z  ->  ( Z  .x.  .0.  )  =  Z ) )
103, 6, 93syl 17 . . 3  |-  ( ph  ->  ( A. x  e.  B  ( Z  .x.  x )  =  Z  ->  ( Z  .x.  .0.  )  =  Z
) )
112, 10mpd 13 . 2  |-  ( ph  ->  ( Z  .x.  .0.  )  =  Z )
12 srgisid.2 . . 3  |-  ( ph  ->  Z  e.  B )
13 srgz.t . . . 4  |-  .x.  =  ( .r `  R )
144, 13, 5srgrz 13098 . . 3  |-  ( ( R  e. SRing  /\  Z  e.  B )  ->  ( Z  .x.  .0.  )  =  .0.  )
153, 12, 14syl2anc 411 . 2  |-  ( ph  ->  ( Z  .x.  .0.  )  =  .0.  )
1611, 15eqtr3d 2212 1  |-  ( ph  ->  Z  =  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   A.wral 2455   ` cfv 5215  (class class class)co 5872   Basecbs 12454   .rcmulr 12529   0gc0g 12693  SRingcsrg 13077
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-cnex 7899  ax-resscn 7900  ax-1re 7902  ax-addrcl 7905
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-iota 5177  df-fun 5217  df-fn 5218  df-fv 5223  df-riota 5828  df-ov 5875  df-inn 8916  df-2 8974  df-3 8975  df-ndx 12457  df-slot 12458  df-base 12460  df-plusg 12541  df-mulr 12542  df-0g 12695  df-mgm 12707  df-sgrp 12740  df-mnd 12750  df-cmn 13021  df-srg 13078
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator