Theorem ringadd2 13215

Description: A ring element plus itself is two times the element. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 22-Dec-2013.) (Revised by AV, 24-Aug-2021.)

Hypotheses

Ref	Expression
ringadd2.b	|- B = ( Base ` R )
ringadd2.p	|- .+ = ( +g ` R )
ringadd2.t	|- .x. = ( .r ` R )

Assertion

Ref	Expression
ringadd2	|- ( ( R e. Ring /\ X e. B ) -> E. x e. B ( X .+ X ) = ( ( x .+ x ) .x. X ) )

Distinct variable groups:   x, B   x, R   x, X   x, .x.

Allowed substitution hint:   .+ ( x)

Proof of Theorem ringadd2

Step	Hyp	Ref	Expression
1		ringadd2.b	. . 3 |- B = ( Base ` R )
2		ringadd2.t	. . 3 |- .x. = ( .r ` R )
3	1, 2	ringid 13214	. 2 |- ( ( R e. Ring /\ X e. B ) -> E. x e. B ( ( x .x. X ) = X /\ ( X .x. x ) = X ) )
4		oveq12 5886	. . . . . . 7 |- ( ( ( x .x. X ) = X /\ ( x .x. X ) = X ) -> ( ( x .x. X ) .+ ( x .x. X ) ) = ( X .+ X ) )
5	4	anidms 397	. . . . . 6 |- ( ( x .x. X ) = X -> ( ( x .x. X ) .+ ( x .x. X ) ) = ( X .+ X ) )
6	5	eqcomd 2183	. . . . 5 |- ( ( x .x. X ) = X -> ( X .+ X ) = ( ( x .x. X ) .+ ( x .x. X ) ) )
7		simpll 527	. . . . . . 7 |- ( ( ( R e. Ring /\ X e. B ) /\ x e. B ) -> R e. Ring )
8		simpr 110	. . . . . . 7 |- ( ( ( R e. Ring /\ X e. B ) /\ x e. B ) -> x e. B )
9		simplr 528	. . . . . . 7 |- ( ( ( R e. Ring /\ X e. B ) /\ x e. B ) -> X e. B )
10		ringadd2.p	. . . . . . . 8 |- .+ = ( +g ` R )
11	1, 10, 2	ringdir 13207	. . . . . . 7 |- ( ( R e. Ring /\ ( x e. B /\ x e. B /\ X e. B ) ) -> ( ( x .+ x ) .x. X ) = ( ( x .x. X ) .+ ( x .x. X ) ) )
12	7, 8, 8, 9, 11	syl13anc 1240	. . . . . 6 |- ( ( ( R e. Ring /\ X e. B ) /\ x e. B ) -> ( ( x .+ x ) .x. X ) = ( ( x .x. X ) .+ ( x .x. X ) ) )
13	12	eqeq2d 2189	. . . . 5 |- ( ( ( R e. Ring /\ X e. B ) /\ x e. B ) -> ( ( X .+ X ) = ( ( x .+ x ) .x. X ) <-> ( X .+ X ) = ( ( x .x. X ) .+ ( x .x. X ) ) ) )
14	6, 13	imbitrrid 156	. . . 4 |- ( ( ( R e. Ring /\ X e. B ) /\ x e. B ) -> ( ( x .x. X ) = X -> ( X .+ X ) = ( ( x .+ x ) .x. X ) ) )
15	14	adantrd 279	. . 3 |- ( ( ( R e. Ring /\ X e. B ) /\ x e. B ) -> ( ( ( x .x. X ) = X /\ ( X .x. x ) = X ) -> ( X .+ X ) = ( ( x .+ x ) .x. X ) ) )
16	15	reximdva 2579	. 2 |- ( ( R e. Ring /\ X e. B ) -> ( E. x e. B ( ( x .x. X ) = X /\ ( X .x. x ) = X ) -> E. x e. B ( X .+ X ) = ( ( x .+ x ) .x. X ) ) )
17	3, 16	mpd 13	1 |- ( ( R e. Ring /\ X e. B ) -> E. x e. B ( X .+ X ) = ( ( x .+ x ) .x. X ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   E.wrex 2456   ` cfv 5218  (class class class)co 5877   Basecbs 12464   +g cplusg 12538   .rcmulr 12539   Ringcrg 13184

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-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-i2m1 7918  ax-0lt1 7919  ax-0id 7921  ax-rnegex 7922  ax-pre-ltirr 7925  ax-pre-ltadd 7929

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 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-pnf 7996  df-mnf 7997  df-ltxr 7999  df-inn 8922  df-2 8980  df-3 8981  df-ndx 12467  df-slot 12468  df-base 12470  df-sets 12471  df-plusg 12551  df-mulr 12552  df-0g 12712  df-mgm 12780  df-sgrp 12813  df-mnd 12823  df-mgp 13136  df-ur 13148  df-ring 13186

This theorem is referenced by: (None)