Theorem ring1eq0 13355

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 5903	. . . 4 |- ( ( ( R e. Ring /\ X e. B /\ Y e. B ) /\ .1. = .0. ) -> ( .1. ( .r ` R ) X ) = ( .0. ( .r ` R ) X ) )
3	1	oveq1d 5903	. . . . 5 |- ( ( ( R e. Ring /\ X e. B /\ Y e. B ) /\ .1. = .0. ) -> ( .1. ( .r ` R ) Y ) = ( .0. ( .r ` R ) Y ) )
4		simpl1 1001	. . . . . . 7 |- ( ( ( R e. Ring /\ X e. B /\ Y e. B ) /\ .1. = .0. ) -> R e. Ring )
5		simpl2 1002	. . . . . . 7 |- ( ( ( R e. Ring /\ X e. B /\ Y e. B ) /\ .1. = .0. ) -> X e. B )
6		ring1eq0.b	. . . . . . . 8 |- B = ( Base ` R )
7		eqid 2187	. . . . . . . 8 |- ( .r ` R ) = ( .r ` R )
8		ring1eq0.z	. . . . . . . 8 |- .0. = ( 0g ` R )
9	6, 7, 8	ringlz 13352	. . . . . . 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 1003	. . . . . . 7 |- ( ( ( R e. Ring /\ X e. B /\ Y e. B ) /\ .1. = .0. ) -> Y e. B )
12	6, 7, 8	ringlz 13352	. . . . . . 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 2223	. . . . 5 |- ( ( ( R e. Ring /\ X e. B /\ Y e. B ) /\ .1. = .0. ) -> ( .0. ( .r ` R ) X ) = ( .0. ( .r ` R ) Y ) )
15	3, 14	eqtr4d 2223	. . . 4 |- ( ( ( R e. Ring /\ X e. B /\ Y e. B ) /\ .1. = .0. ) -> ( .1. ( .r ` R ) Y ) = ( .0. ( .r ` R ) X ) )
16	2, 15	eqtr4d 2223	. . 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 13332	. . . 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 13332	. . . 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 2228	. 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 979   = wceq 1363   e. wcel 2158   ` cfv 5228  (class class class)co 5888   Basecbs 12476   .rcmulr 12552   0gc0g 12723   1rcur 13268   Ringcrg 13305

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7916  ax-resscn 7917  ax-1cn 7918  ax-1re 7919  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-addcom 7925  ax-addass 7927  ax-i2m1 7930  ax-0lt1 7931  ax-0id 7933  ax-rnegex 7934  ax-pre-ltirr 7937  ax-pre-ltadd 7941

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-pnf 8008  df-mnf 8009  df-ltxr 8011  df-inn 8934  df-2 8992  df-3 8993  df-ndx 12479  df-slot 12480  df-base 12482  df-sets 12483  df-plusg 12564  df-mulr 12565  df-0g 12725  df-mgm 12794  df-sgrp 12827  df-mnd 12840  df-grp 12909  df-minusg 12910  df-mgp 13230  df-ur 13269  df-ring 13307

This theorem is referenced by:  ringelnzr  13464  01eq0ring  13466