Theorem srg1zr 13749

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 13729	. . . . . . 7 |- ( R e. SRing -> R e. Mnd )
3	2	3ad2ant1 1021	. . . . . 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 13254	. . . . 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 1004	. . . 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 13200	. . . 4 |- ( ( R e. Mgm /\ Z e. B /\ .+ Fn ( B X. B ) ) -> ( B = { Z } <-> .+ = { <. <. Z , Z >. , Z >. } ) )
12	6, 7, 8, 11	syl3anc 1250	. . 3 |- ( ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( B = { Z } <-> .+ = { <. <. Z , Z >. , Z >. } ) )
13		eqid 2205	. . . . . . . 8 |- (mulGrp ` R ) = (mulGrp ` R )
14	13, 9	mgpbasg 13688	. . . . . . 7 |- ( R e. SRing -> B = ( Base ` (mulGrp ` R ) ) )
15	14	3ad2ant1 1021	. . . . . 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 2214	. . . 4 |- ( ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( B = { Z } <-> ( Base ` (mulGrp ` R ) ) = { Z } ) )
18		simpl1 1003	. . . . . 6 |- ( ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> R e. SRing )
19	13	srgmgp 13730	. . . . . 6 |- ( R e. SRing -> (mulGrp ` R ) e. Mnd )
20		mndmgm 13254	. . . . . 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 2284	. . . . 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 13686	. . . . . . . . . 10 |- ( R e. SRing -> .* = ( +g ` (mulGrp ` R ) ) )
25	24	fneq1d 5364	. . . . . . . . 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 1019	. . . . . . 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 4701	. . . . . . 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 5365	. . . . . 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 2205	. . . . . 6 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
33		eqid 2205	. . . . . 6 |- ( +g ` (mulGrp ` R ) ) = ( +g ` (mulGrp ` R ) )
34	32, 33	mgmb1mgm1 13200	. . . . 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 1250	. . . 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 2211	. . . . . 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 2214	. . . 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 981   = wceq 1373   e. wcel 2176   {csn 3633   <.cop 3636   X. cxp 4673   Fn wfn 5266   ` cfv 5271   Basecbs 12832   +g cplusg 12909   .rcmulr 12910  Mgmcmgm 13186   Mndcmnd 13248  mulGrpcmgp 13682  SRingcsrg 13725

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-addcom 8025  ax-addass 8027  ax-i2m1 8030  ax-0lt1 8031  ax-0id 8033  ax-rnegex 8034  ax-pre-ltirr 8037  ax-pre-ltadd 8041

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-pnf 8109  df-mnf 8110  df-ltxr 8112  df-inn 9037  df-2 9095  df-3 9096  df-ndx 12835  df-slot 12836  df-base 12838  df-sets 12839  df-plusg 12922  df-mulr 12923  df-0g 13090  df-plusf 13187  df-mgm 13188  df-sgrp 13234  df-mnd 13249  df-cmn 13622  df-mgp 13683  df-srg 13726

This theorem is referenced by:  srgen1zr  13750