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Theorem srg1zr 12983
Description: The only semiring with a base set consisting of one element is the zero ring (at least if its operations are internal binary operations). (Contributed by FL, 13-Feb-2010.) (Revised by AV, 25-Jan-2020.)
Hypotheses
Ref Expression
srg1zr.b  |-  B  =  ( Base `  R
)
srg1zr.p  |-  .+  =  ( +g  `  R )
srg1zr.t  |-  .*  =  ( .r `  R )
Assertion
Ref Expression
srg1zr  |-  ( ( ( R  e. SRing  /\  .+  Fn  ( B  X.  B
)  /\  .*  Fn  ( B  X.  B
) )  /\  Z  e.  B )  ->  ( B  =  { Z } 
<->  (  .+  =  { <. <. Z ,  Z >. ,  Z >. }  /\  .*  =  { <. <. Z ,  Z >. ,  Z >. } ) ) )

Proof of Theorem srg1zr
StepHypRef Expression
1 pm4.24 395 . 2  |-  ( B  =  { Z }  <->  ( B  =  { Z }  /\  B  =  { Z } ) )
2 srgmnd 12963 . . . . . . 7  |-  ( R  e. SRing  ->  R  e.  Mnd )
323ad2ant1 1018 . . . . . 6  |-  ( ( R  e. SRing  /\  .+  Fn  ( B  X.  B
)  /\  .*  Fn  ( B  X.  B
) )  ->  R  e.  Mnd )
43adantr 276 . . . . 5  |-  ( ( ( R  e. SRing  /\  .+  Fn  ( B  X.  B
)  /\  .*  Fn  ( B  X.  B
) )  /\  Z  e.  B )  ->  R  e.  Mnd )
5 mndmgm 12702 . . . . 5  |-  ( R  e.  Mnd  ->  R  e. Mgm )
64, 5syl 14 . . . 4  |-  ( ( ( R  e. SRing  /\  .+  Fn  ( B  X.  B
)  /\  .*  Fn  ( B  X.  B
) )  /\  Z  e.  B )  ->  R  e. Mgm )
7 simpr 110 . . . 4  |-  ( ( ( R  e. SRing  /\  .+  Fn  ( B  X.  B
)  /\  .*  Fn  ( B  X.  B
) )  /\  Z  e.  B )  ->  Z  e.  B )
8 simpl2 1001 . . . 4  |-  ( ( ( R  e. SRing  /\  .+  Fn  ( B  X.  B
)  /\  .*  Fn  ( B  X.  B
) )  /\  Z  e.  B )  ->  .+  Fn  ( B  X.  B
) )
9 srg1zr.b . . . . 5  |-  B  =  ( Base `  R
)
10 srg1zr.p . . . . 5  |-  .+  =  ( +g  `  R )
119, 10mgmb1mgm1 12666 . . . 4  |-  ( ( R  e. Mgm  /\  Z  e.  B  /\  .+  Fn  ( B  X.  B
) )  ->  ( B  =  { Z } 
<-> 
.+  =  { <. <. Z ,  Z >. ,  Z >. } ) )
126, 7, 8, 11syl3anc 1238 . . 3  |-  ( ( ( R  e. SRing  /\  .+  Fn  ( B  X.  B
)  /\  .*  Fn  ( B  X.  B
) )  /\  Z  e.  B )  ->  ( B  =  { Z } 
<-> 
.+  =  { <. <. Z ,  Z >. ,  Z >. } ) )
13 eqid 2177 . . . . . . . 8  |-  (mulGrp `  R )  =  (mulGrp `  R )
1413, 9mgpbasg 12950 . . . . . . 7  |-  ( R  e. SRing  ->  B  =  (
Base `  (mulGrp `  R
) ) )
15143ad2ant1 1018 . . . . . 6  |-  ( ( R  e. SRing  /\  .+  Fn  ( B  X.  B
)  /\  .*  Fn  ( B  X.  B
) )  ->  B  =  ( Base `  (mulGrp `  R ) ) )
1615adantr 276 . . . . 5  |-  ( ( ( R  e. SRing  /\  .+  Fn  ( B  X.  B
)  /\  .*  Fn  ( B  X.  B
) )  /\  Z  e.  B )  ->  B  =  ( Base `  (mulGrp `  R ) ) )
1716eqeq1d 2186 . . . 4  |-  ( ( ( R  e. SRing  /\  .+  Fn  ( B  X.  B
)  /\  .*  Fn  ( B  X.  B
) )  /\  Z  e.  B )  ->  ( B  =  { Z } 
<->  ( Base `  (mulGrp `  R ) )  =  { Z } ) )
18 simpl1 1000 . . . . . 6  |-  ( ( ( R  e. SRing  /\  .+  Fn  ( B  X.  B
)  /\  .*  Fn  ( B  X.  B
) )  /\  Z  e.  B )  ->  R  e. SRing )
1913srgmgp 12964 . . . . . 6  |-  ( R  e. SRing  ->  (mulGrp `  R )  e.  Mnd )
20 mndmgm 12702 . . . . . 6  |-  ( (mulGrp `  R )  e.  Mnd  ->  (mulGrp `  R )  e. Mgm )
2118, 19, 203syl 17 . . . . 5  |-  ( ( ( R  e. SRing  /\  .+  Fn  ( B  X.  B
)  /\  .*  Fn  ( B  X.  B
) )  /\  Z  e.  B )  ->  (mulGrp `  R )  e. Mgm )
227, 16eleqtrd 2256 . . . . 5  |-  ( ( ( R  e. SRing  /\  .+  Fn  ( B  X.  B
)  /\  .*  Fn  ( B  X.  B
) )  /\  Z  e.  B )  ->  Z  e.  ( Base `  (mulGrp `  R ) ) )
23 srg1zr.t . . . . . . . . . . 11  |-  .*  =  ( .r `  R )
2413, 23mgpplusgg 12948 . . . . . . . . . 10  |-  ( R  e. SRing  ->  .*  =  ( +g  `  (mulGrp `  R
) ) )
2524fneq1d 5301 . . . . . . . . 9  |-  ( R  e. SRing  ->  (  .*  Fn  ( B  X.  B
)  <->  ( +g  `  (mulGrp `  R ) )  Fn  ( B  X.  B
) ) )
2625biimpa 296 . . . . . . . 8  |-  ( ( R  e. SRing  /\  .*  Fn  ( B  X.  B
) )  ->  ( +g  `  (mulGrp `  R
) )  Fn  ( B  X.  B ) )
27263adant2 1016 . . . . . . 7  |-  ( ( R  e. SRing  /\  .+  Fn  ( B  X.  B
)  /\  .*  Fn  ( B  X.  B
) )  ->  ( +g  `  (mulGrp `  R
) )  Fn  ( B  X.  B ) )
2827adantr 276 . . . . . 6  |-  ( ( ( R  e. SRing  /\  .+  Fn  ( B  X.  B
)  /\  .*  Fn  ( B  X.  B
) )  /\  Z  e.  B )  ->  ( +g  `  (mulGrp `  R
) )  Fn  ( B  X.  B ) )
2916sqxpeqd 4648 . . . . . . 7  |-  ( ( ( R  e. SRing  /\  .+  Fn  ( B  X.  B
)  /\  .*  Fn  ( B  X.  B
) )  /\  Z  e.  B )  ->  ( B  X.  B )  =  ( ( Base `  (mulGrp `  R ) )  X.  ( Base `  (mulGrp `  R ) ) ) )
3029fneq2d 5302 . . . . . 6  |-  ( ( ( R  e. SRing  /\  .+  Fn  ( B  X.  B
)  /\  .*  Fn  ( B  X.  B
) )  /\  Z  e.  B )  ->  (
( +g  `  (mulGrp `  R ) )  Fn  ( B  X.  B
)  <->  ( +g  `  (mulGrp `  R ) )  Fn  ( ( Base `  (mulGrp `  R ) )  X.  ( Base `  (mulGrp `  R ) ) ) ) )
3128, 30mpbid 147 . . . . 5  |-  ( ( ( R  e. SRing  /\  .+  Fn  ( B  X.  B
)  /\  .*  Fn  ( B  X.  B
) )  /\  Z  e.  B )  ->  ( +g  `  (mulGrp `  R
) )  Fn  (
( Base `  (mulGrp `  R
) )  X.  ( Base `  (mulGrp `  R
) ) ) )
32 eqid 2177 . . . . . 6  |-  ( Base `  (mulGrp `  R )
)  =  ( Base `  (mulGrp `  R )
)
33 eqid 2177 . . . . . 6  |-  ( +g  `  (mulGrp `  R )
)  =  ( +g  `  (mulGrp `  R )
)
3432, 33mgmb1mgm1 12666 . . . . 5  |-  ( ( (mulGrp `  R )  e. Mgm  /\  Z  e.  (
Base `  (mulGrp `  R
) )  /\  ( +g  `  (mulGrp `  R
) )  Fn  (
( Base `  (mulGrp `  R
) )  X.  ( Base `  (mulGrp `  R
) ) ) )  ->  ( ( Base `  (mulGrp `  R )
)  =  { Z } 
<->  ( +g  `  (mulGrp `  R ) )  =  { <. <. Z ,  Z >. ,  Z >. } ) )
3521, 22, 31, 34syl3anc 1238 . . . 4  |-  ( ( ( R  e. SRing  /\  .+  Fn  ( B  X.  B
)  /\  .*  Fn  ( B  X.  B
) )  /\  Z  e.  B )  ->  (
( Base `  (mulGrp `  R
) )  =  { Z }  <->  ( +g  `  (mulGrp `  R ) )  =  { <. <. Z ,  Z >. ,  Z >. } ) )
3624eqcomd 2183 . . . . . 6  |-  ( R  e. SRing  ->  ( +g  `  (mulGrp `  R ) )  =  .*  )
3718, 36syl 14 . . . . 5  |-  ( ( ( R  e. SRing  /\  .+  Fn  ( B  X.  B
)  /\  .*  Fn  ( B  X.  B
) )  /\  Z  e.  B )  ->  ( +g  `  (mulGrp `  R
) )  =  .*  )
3837eqeq1d 2186 . . . 4  |-  ( ( ( R  e. SRing  /\  .+  Fn  ( B  X.  B
)  /\  .*  Fn  ( B  X.  B
) )  /\  Z  e.  B )  ->  (
( +g  `  (mulGrp `  R ) )  =  { <. <. Z ,  Z >. ,  Z >. }  <->  .*  =  { <. <. Z ,  Z >. ,  Z >. } ) )
3917, 35, 383bitrd 214 . . 3  |-  ( ( ( R  e. SRing  /\  .+  Fn  ( B  X.  B
)  /\  .*  Fn  ( B  X.  B
) )  /\  Z  e.  B )  ->  ( B  =  { Z } 
<->  .*  =  { <. <. Z ,  Z >. ,  Z >. } ) )
4012, 39anbi12d 473 . 2  |-  ( ( ( R  e. SRing  /\  .+  Fn  ( B  X.  B
)  /\  .*  Fn  ( B  X.  B
) )  /\  Z  e.  B )  ->  (
( B  =  { Z }  /\  B  =  { Z } )  <-> 
(  .+  =  { <. <. Z ,  Z >. ,  Z >. }  /\  .*  =  { <. <. Z ,  Z >. ,  Z >. } ) ) )
411, 40bitrid 192 1  |-  ( ( ( R  e. SRing  /\  .+  Fn  ( B  X.  B
)  /\  .*  Fn  ( B  X.  B
) )  /\  Z  e.  B )  ->  ( B  =  { Z } 
<->  (  .+  =  { <. <. Z ,  Z >. ,  Z >. }  /\  .*  =  { <. <. Z ,  Z >. ,  Z >. } ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 978    = wceq 1353    e. wcel 2148   {csn 3591   <.cop 3594    X. cxp 4620    Fn wfn 5206   ` cfv 5211   Basecbs 12432   +g cplusg 12505   .rcmulr 12506  Mgmcmgm 12652   Mndcmnd 12696  mulGrpcmgp 12944  SRingcsrg 12959
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-in1 614  ax-in2 615  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-coll 4115  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-cnex 7880  ax-resscn 7881  ax-1cn 7882  ax-1re 7883  ax-icn 7884  ax-addcl 7885  ax-addrcl 7886  ax-mulcl 7887  ax-addcom 7889  ax-addass 7891  ax-i2m1 7894  ax-0lt1 7895  ax-0id 7897  ax-rnegex 7898  ax-pre-ltirr 7901  ax-pre-ltadd 7905
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  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-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-riota 5824  df-ov 5871  df-oprab 5872  df-mpo 5873  df-1st 6134  df-2nd 6135  df-pnf 7971  df-mnf 7972  df-ltxr 7974  df-inn 8896  df-2 8954  df-3 8955  df-ndx 12435  df-slot 12436  df-base 12438  df-sets 12439  df-plusg 12518  df-mulr 12519  df-0g 12642  df-plusf 12653  df-mgm 12654  df-sgrp 12687  df-mnd 12697  df-cmn 12904  df-mgp 12945  df-srg 12960
This theorem is referenced by:  srgen1zr  12984
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