Theorem srg1zr 13175

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

Step	Hyp	Ref	Expression
1		pm4.24 395	. 2 |- ( B = { Z } <-> ( B = { Z } /\ B = { Z } ) )
2		srgmnd 13155	. . . . . . 7 |- ( R e. SRing -> R e. Mnd )
3	2	3ad2ant1 1018	. . . . . 6 |- ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) -> R e. Mnd )
4	3	adantr 276	. . . . 5 |- ( ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> R e. Mnd )
5		mndmgm 12828	. . . . 5 |- ( R e. Mnd -> R e. Mgm )
6	4, 5	syl 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 )
11	9, 10	mgmb1mgm1 12792	. . . 4 |- ( ( R e. Mgm /\ Z e. B /\ .+ Fn ( B X. B ) ) -> ( B = { Z } <-> .+ = { <. <. Z , Z >. , Z >. } ) )
12	6, 7, 8, 11	syl3anc 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 )
14	13, 9	mgpbasg 13141	. . . . . . 7 |- ( R e. SRing -> B = ( Base ` (mulGrp ` R ) ) )
15	14	3ad2ant1 1018	. . . . . 6 |- ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) -> B = ( Base ` (mulGrp ` R ) ) )
16	15	adantr 276	. . . . 5 |- ( ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> B = ( Base ` (mulGrp ` R ) ) )
17	16	eqeq1d 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 )
19	13	srgmgp 13156	. . . . . 6 |- ( R e. SRing -> (mulGrp ` R ) e. Mnd )
20		mndmgm 12828	. . . . . 6 |- ( (mulGrp ` R ) e. Mnd -> (mulGrp ` R ) e. Mgm )
21	18, 19, 20	3syl 17	. . . . 5 |- ( ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> (mulGrp ` R ) e. Mgm )
22	7, 16	eleqtrd 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 )
24	13, 23	mgpplusgg 13139	. . . . . . . . . 10 |- ( R e. SRing -> .* = ( +g ` (mulGrp ` R ) ) )
25	24	fneq1d 5308	. . . . . . . . 9 |- ( R e. SRing -> ( .* Fn ( B X. B ) <-> ( +g ` (mulGrp ` R ) ) Fn ( B X. B ) ) )
26	25	biimpa 296	. . . . . . . 8 |- ( ( R e. SRing /\ .* Fn ( B X. B ) ) -> ( +g ` (mulGrp ` R ) ) Fn ( B X. B ) )
27	26	3adant2 1016	. . . . . . 7 |- ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) -> ( +g ` (mulGrp ` R ) ) Fn ( B X. B ) )
28	27	adantr 276	. . . . . 6 |- ( ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( +g ` (mulGrp ` R ) ) Fn ( B X. B ) )
29	16	sqxpeqd 4654	. . . . . . 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 ) ) ) )
30	29	fneq2d 5309	. . . . . 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 ) ) ) ) )
31	28, 30	mpbid 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 ) )
34	32, 33	mgmb1mgm1 12792	. . . . 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 >. } ) )
35	21, 22, 31, 34	syl3anc 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 >. } ) )
36	24	eqcomd 2183	. . . . . 6 |- ( R e. SRing -> ( +g ` (mulGrp ` R ) ) = .* )
37	18, 36	syl 14	. . . . 5 |- ( ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( +g ` (mulGrp ` R ) ) = .* )
38	37	eqeq1d 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 >. } ) )
39	17, 35, 38	3bitrd 214	. . 3 |- ( ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( B = { Z } <-> .* = { <. <. Z , Z >. , Z >. } ) )
40	12, 39	anbi12d 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 >. } ) ) )
41	1, 40	bitrid 192	1 |- ( ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( B = { Z } <-> ( .+ = { <. <. Z , Z >. , Z >. } /\ .* = { <. <. Z , Z >. , Z >. } ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 978   = wceq 1353   e. wcel 2148   {csn 3594   <.cop 3597   X. cxp 4626   Fn wfn 5213   ` cfv 5218   Basecbs 12464   +g cplusg 12538   .rcmulr 12539  Mgmcmgm 12778   Mndcmnd 12822  mulGrpcmgp 13135  SRingcsrg 13151

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 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-i2m1 7918  ax-0lt1 7919  ax-0id 7921  ax-rnegex 7922  ax-pre-ltirr 7925  ax-pre-ltadd 7929

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 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-pnf 7996  df-mnf 7997  df-ltxr 7999  df-inn 8922  df-2 8980  df-3 8981  df-ndx 12467  df-slot 12468  df-base 12470  df-sets 12471  df-plusg 12551  df-mulr 12552  df-0g 12712  df-plusf 12779  df-mgm 12780  df-sgrp 12813  df-mnd 12823  df-cmn 13095  df-mgp 13136  df-srg 13152

This theorem is referenced by:  srgen1zr  13176