Theorem ring1eq0 14326

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 6090	. . . 4 |- ( ( ( R e. Ring /\ X e. B /\ Y e. B ) /\ .1. = .0. ) -> ( .1. ( .r ` R ) X ) = ( .0. ( .r ` R ) X ) )
3	1	oveq1d 6090	. . . . 5 |- ( ( ( R e. Ring /\ X e. B /\ Y e. B ) /\ .1. = .0. ) -> ( .1. ( .r ` R ) Y ) = ( .0. ( .r ` R ) Y ) )
4		simpl1 1031	. . . . . . 7 |- ( ( ( R e. Ring /\ X e. B /\ Y e. B ) /\ .1. = .0. ) -> R e. Ring )
5		simpl2 1032	. . . . . . 7 |- ( ( ( R e. Ring /\ X e. B /\ Y e. B ) /\ .1. = .0. ) -> X e. B )
6		ring1eq0.b	. . . . . . . 8 |- B = ( Base ` R )
7		eqid 2238	. . . . . . . 8 |- ( .r ` R ) = ( .r ` R )
8		ring1eq0.z	. . . . . . . 8 |- .0. = ( 0g ` R )
9	6, 7, 8	ringlz 14321	. . . . . . 7 |- ( ( R e. Ring /\ X e. B ) -> ( .0. ( .r ` R ) X ) = .0. )
10	4, 5, 9	syl2anc 415	. . . . . 6 |- ( ( ( R e. Ring /\ X e. B /\ Y e. B ) /\ .1. = .0. ) -> ( .0. ( .r ` R ) X ) = .0. )
11		simpl3 1033	. . . . . . 7 |- ( ( ( R e. Ring /\ X e. B /\ Y e. B ) /\ .1. = .0. ) -> Y e. B )
12	6, 7, 8	ringlz 14321	. . . . . . 7 |- ( ( R e. Ring /\ Y e. B ) -> ( .0. ( .r ` R ) Y ) = .0. )
13	4, 11, 12	syl2anc 415	. . . . . 6 |- ( ( ( R e. Ring /\ X e. B /\ Y e. B ) /\ .1. = .0. ) -> ( .0. ( .r ` R ) Y ) = .0. )
14	10, 13	eqtr4d 2274	. . . . 5 |- ( ( ( R e. Ring /\ X e. B /\ Y e. B ) /\ .1. = .0. ) -> ( .0. ( .r ` R ) X ) = ( .0. ( .r ` R ) Y ) )
15	3, 14	eqtr4d 2274	. . . 4 |- ( ( ( R e. Ring /\ X e. B /\ Y e. B ) /\ .1. = .0. ) -> ( .1. ( .r ` R ) Y ) = ( .0. ( .r ` R ) X ) )
16	2, 15	eqtr4d 2274	. . 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 14301	. . . 4 |- ( ( R e. Ring /\ X e. B ) -> ( .1. ( .r ` R ) X ) = X )
19	4, 5, 18	syl2anc 415	. . 3 |- ( ( ( R e. Ring /\ X e. B /\ Y e. B ) /\ .1. = .0. ) -> ( .1. ( .r ` R ) X ) = X )
20	6, 7, 17	ringlidm 14301	. . . 4 |- ( ( R e. Ring /\ Y e. B ) -> ( .1. ( .r ` R ) Y ) = Y )
21	4, 11, 20	syl2anc 415	. . 3 |- ( ( ( R e. Ring /\ X e. B /\ Y e. B ) /\ .1. = .0. ) -> ( .1. ( .r ` R ) Y ) = Y )
22	16, 19, 21	3eqtr3d 2279	. 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 1009   = wceq 1402   e. wcel 2209   ` cfv 5372  (class class class)co 6075   Basecbs 13330   .rcmulr 13409   0gc0g 13587   1rcur 14237   Ringcrg 14274

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-pre-ltirr 8281  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-ltxr 8355  df-inn 9284  df-2 9342  df-3 9343  df-ndx 13333  df-slot 13334  df-base 13336  df-sets 13337  df-plusg 13421  df-mulr 13422  df-0g 13589  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-grp 13785  df-minusg 13786  df-mgp 14195  df-ur 14238  df-ring 14276

This theorem is referenced by:  isnzr2  14464  ringelnzr  14467  01eq0ring  14469