Theorem ring1eq0 13604

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 5937	. . . 4 |- ( ( ( R e. Ring /\ X e. B /\ Y e. B ) /\ .1. = .0. ) -> ( .1. ( .r ` R ) X ) = ( .0. ( .r ` R ) X ) )
3	1	oveq1d 5937	. . . . 5 |- ( ( ( R e. Ring /\ X e. B /\ Y e. B ) /\ .1. = .0. ) -> ( .1. ( .r ` R ) Y ) = ( .0. ( .r ` R ) Y ) )
4		simpl1 1002	. . . . . . 7 |- ( ( ( R e. Ring /\ X e. B /\ Y e. B ) /\ .1. = .0. ) -> R e. Ring )
5		simpl2 1003	. . . . . . 7 |- ( ( ( R e. Ring /\ X e. B /\ Y e. B ) /\ .1. = .0. ) -> X e. B )
6		ring1eq0.b	. . . . . . . 8 |- B = ( Base ` R )
7		eqid 2196	. . . . . . . 8 |- ( .r ` R ) = ( .r ` R )
8		ring1eq0.z	. . . . . . . 8 |- .0. = ( 0g ` R )
9	6, 7, 8	ringlz 13599	. . . . . . 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 1004	. . . . . . 7 |- ( ( ( R e. Ring /\ X e. B /\ Y e. B ) /\ .1. = .0. ) -> Y e. B )
12	6, 7, 8	ringlz 13599	. . . . . . 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 2232	. . . . 5 |- ( ( ( R e. Ring /\ X e. B /\ Y e. B ) /\ .1. = .0. ) -> ( .0. ( .r ` R ) X ) = ( .0. ( .r ` R ) Y ) )
15	3, 14	eqtr4d 2232	. . . 4 |- ( ( ( R e. Ring /\ X e. B /\ Y e. B ) /\ .1. = .0. ) -> ( .1. ( .r ` R ) Y ) = ( .0. ( .r ` R ) X ) )
16	2, 15	eqtr4d 2232	. . 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 13579	. . . 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 13579	. . . 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 2237	. 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 980   = wceq 1364   e. wcel 2167   ` cfv 5258  (class class class)co 5922   Basecbs 12678   .rcmulr 12756   0gc0g 12927   1rcur 13515   Ringcrg 13552

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-3 9050  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-plusg 12768  df-mulr 12769  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-grp 13135  df-minusg 13136  df-mgp 13477  df-ur 13516  df-ring 13554

This theorem is referenced by:  isnzr2  13740  ringelnzr  13743  01eq0ring  13745