Theorem ring1eq0 14060

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 6032	. . . 4 |- ( ( ( R e. Ring /\ X e. B /\ Y e. B ) /\ .1. = .0. ) -> ( .1. ( .r ` R ) X ) = ( .0. ( .r ` R ) X ) )
3	1	oveq1d 6032	. . . . 5 |- ( ( ( R e. Ring /\ X e. B /\ Y e. B ) /\ .1. = .0. ) -> ( .1. ( .r ` R ) Y ) = ( .0. ( .r ` R ) Y ) )
4		simpl1 1026	. . . . . . 7 |- ( ( ( R e. Ring /\ X e. B /\ Y e. B ) /\ .1. = .0. ) -> R e. Ring )
5		simpl2 1027	. . . . . . 7 |- ( ( ( R e. Ring /\ X e. B /\ Y e. B ) /\ .1. = .0. ) -> X e. B )
6		ring1eq0.b	. . . . . . . 8 |- B = ( Base ` R )
7		eqid 2231	. . . . . . . 8 |- ( .r ` R ) = ( .r ` R )
8		ring1eq0.z	. . . . . . . 8 |- .0. = ( 0g ` R )
9	6, 7, 8	ringlz 14055	. . . . . . 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 1028	. . . . . . 7 |- ( ( ( R e. Ring /\ X e. B /\ Y e. B ) /\ .1. = .0. ) -> Y e. B )
12	6, 7, 8	ringlz 14055	. . . . . . 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 2267	. . . . 5 |- ( ( ( R e. Ring /\ X e. B /\ Y e. B ) /\ .1. = .0. ) -> ( .0. ( .r ` R ) X ) = ( .0. ( .r ` R ) Y ) )
15	3, 14	eqtr4d 2267	. . . 4 |- ( ( ( R e. Ring /\ X e. B /\ Y e. B ) /\ .1. = .0. ) -> ( .1. ( .r ` R ) Y ) = ( .0. ( .r ` R ) X ) )
16	2, 15	eqtr4d 2267	. . 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 14035	. . . 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 14035	. . . 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 2272	. 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 1004   = wceq 1397   e. wcel 2202   ` cfv 5326  (class class class)co 6017   Basecbs 13081   .rcmulr 13160   0gc0g 13338   1rcur 13971   Ringcrg 14008

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-pre-ltirr 8143  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-ltxr 8218  df-inn 9143  df-2 9201  df-3 9202  df-ndx 13084  df-slot 13085  df-base 13087  df-sets 13088  df-plusg 13172  df-mulr 13173  df-0g 13340  df-mgm 13438  df-sgrp 13484  df-mnd 13499  df-grp 13585  df-minusg 13586  df-mgp 13933  df-ur 13972  df-ring 14010

This theorem is referenced by:  isnzr2  14197  ringelnzr  14200  01eq0ring  14202