Theorem ring1eq0 13225

Description: If one and zero are equal, then any two elements of a ring are equal. Alternately, every ring has one distinct from zero except the zero ring containing the single element { 0 }. (Contributed by Mario Carneiro, 10-Sep-2014.)

Hypotheses

Ref	Expression
ring1eq0.b	|- B = ( Base ` R )
ring1eq0.u	|- .1. = ( 1r ` R )
ring1eq0.z	|- .0. = ( 0g ` R )

Assertion

Ref	Expression
ring1eq0	|- ( ( R e. Ring /\ X e. B /\ Y e. B ) -> ( .1. = .0. -> X = Y ) )

Proof of Theorem ring1eq0

Step	Hyp	Ref	Expression
1		simpr 110	. . . . 5 |- ( ( ( R e. Ring /\ X e. B /\ Y e. B ) /\ .1. = .0. ) -> .1. = .0. )
2	1	oveq1d 5890	. . . 4 |- ( ( ( R e. Ring /\ X e. B /\ Y e. B ) /\ .1. = .0. ) -> ( .1. ( .r ` R ) X ) = ( .0. ( .r ` R ) X ) )
3	1	oveq1d 5890	. . . . 5 |- ( ( ( R e. Ring /\ X e. B /\ Y e. B ) /\ .1. = .0. ) -> ( .1. ( .r ` R ) Y ) = ( .0. ( .r ` R ) Y ) )
4		simpl1 1000	. . . . . . 7 |- ( ( ( R e. Ring /\ X e. B /\ Y e. B ) /\ .1. = .0. ) -> R e. Ring )
5		simpl2 1001	. . . . . . 7 |- ( ( ( R e. Ring /\ X e. B /\ Y e. B ) /\ .1. = .0. ) -> X e. B )
6		ring1eq0.b	. . . . . . . 8 |- B = ( Base ` R )
7		eqid 2177	. . . . . . . 8 |- ( .r ` R ) = ( .r ` R )
8		ring1eq0.z	. . . . . . . 8 |- .0. = ( 0g ` R )
9	6, 7, 8	ringlz 13222	. . . . . . 7 |- ( ( R e. Ring /\ X e. B ) -> ( .0. ( .r ` R ) X ) = .0. )
10	4, 5, 9	syl2anc 411	. . . . . 6 |- ( ( ( R e. Ring /\ X e. B /\ Y e. B ) /\ .1. = .0. ) -> ( .0. ( .r ` R ) X ) = .0. )
11		simpl3 1002	. . . . . . 7 |- ( ( ( R e. Ring /\ X e. B /\ Y e. B ) /\ .1. = .0. ) -> Y e. B )
12	6, 7, 8	ringlz 13222	. . . . . . 7 |- ( ( R e. Ring /\ Y e. B ) -> ( .0. ( .r ` R ) Y ) = .0. )
13	4, 11, 12	syl2anc 411	. . . . . 6 |- ( ( ( R e. Ring /\ X e. B /\ Y e. B ) /\ .1. = .0. ) -> ( .0. ( .r ` R ) Y ) = .0. )
14	10, 13	eqtr4d 2213	. . . . 5 |- ( ( ( R e. Ring /\ X e. B /\ Y e. B ) /\ .1. = .0. ) -> ( .0. ( .r ` R ) X ) = ( .0. ( .r ` R ) Y ) )
15	3, 14	eqtr4d 2213	. . . 4 |- ( ( ( R e. Ring /\ X e. B /\ Y e. B ) /\ .1. = .0. ) -> ( .1. ( .r ` R ) Y ) = ( .0. ( .r ` R ) X ) )
16	2, 15	eqtr4d 2213	. . 3 |- ( ( ( R e. Ring /\ X e. B /\ Y e. B ) /\ .1. = .0. ) -> ( .1. ( .r ` R ) X ) = ( .1. ( .r ` R ) Y ) )
17		ring1eq0.u	. . . . 5 |- .1. = ( 1r ` R )
18	6, 7, 17	ringlidm 13206	. . . 4 |- ( ( R e. Ring /\ X e. B ) -> ( .1. ( .r ` R ) X ) = X )
19	4, 5, 18	syl2anc 411	. . 3 |- ( ( ( R e. Ring /\ X e. B /\ Y e. B ) /\ .1. = .0. ) -> ( .1. ( .r ` R ) X ) = X )
20	6, 7, 17	ringlidm 13206	. . . 4 |- ( ( R e. Ring /\ Y e. B ) -> ( .1. ( .r ` R ) Y ) = Y )
21	4, 11, 20	syl2anc 411	. . 3 |- ( ( ( R e. Ring /\ X e. B /\ Y e. B ) /\ .1. = .0. ) -> ( .1. ( .r ` R ) Y ) = Y )
22	16, 19, 21	3eqtr3d 2218	. 2 |- ( ( ( R e. Ring /\ X e. B /\ Y e. B ) /\ .1. = .0. ) -> X = Y )
23	22	ex 115	1 |- ( ( R e. Ring /\ X e. B /\ Y e. B ) -> ( .1. = .0. -> X = Y ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 978   = wceq 1353   e. wcel 2148   ` cfv 5217  (class class class)co 5875   Basecbs 12462   .rcmulr 12537   0gc0g 12705   1rcur 13142   Ringcrg 13179

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 4119  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-addcom 7911  ax-addass 7913  ax-i2m1 7916  ax-0lt1 7917  ax-0id 7919  ax-rnegex 7920  ax-pre-ltirr 7923  ax-pre-ltadd 7927

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-rmo 2463  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-pnf 7994  df-mnf 7995  df-ltxr 7997  df-inn 8920  df-2 8978  df-3 8979  df-ndx 12465  df-slot 12466  df-base 12468  df-sets 12469  df-plusg 12549  df-mulr 12550  df-0g 12707  df-mgm 12775  df-sgrp 12808  df-mnd 12818  df-grp 12880  df-minusg 12881  df-mgp 13131  df-ur 13143  df-ring 13181

This theorem is referenced by:  ringelnzr  13328  01eq0ring  13330