Theorem ringadd2 14027

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 14026	. 2 |- ( ( R e. Ring /\ X e. B ) -> E. x e. B ( ( x .x. X ) = X /\ ( X .x. x ) = X ) )
4		oveq12 6020	. . . . . . 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 2235	. . . . 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 14019	. . . . . . 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 1273	. . . . . 6 |- ( ( ( R e. Ring /\ X e. B ) /\ x e. B ) -> ( ( x .+ x ) .x. X ) = ( ( x .x. X ) .+ ( x .x. X ) ) )
13	12	eqeq2d 2241	. . . . 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 2632	. 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 1395   e. wcel 2200   E.wrex 2509   ` cfv 5322  (class class class)co 6011   Basecbs 13069   +g cplusg 13147   .rcmulr 13148   Ringcrg 13996

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4203  ax-pow 4260  ax-pr 4295  ax-un 4526  ax-setind 4631  ax-cnex 8111  ax-resscn 8112  ax-1cn 8113  ax-1re 8114  ax-icn 8115  ax-addcl 8116  ax-addrcl 8117  ax-mulcl 8118  ax-addcom 8120  ax-addass 8122  ax-i2m1 8125  ax-0lt1 8126  ax-0id 8128  ax-rnegex 8129  ax-pre-ltirr 8132  ax-pre-ltadd 8136

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3890  df-int 3925  df-br 4085  df-opab 4147  df-mpt 4148  df-id 4386  df-xp 4727  df-rel 4728  df-cnv 4729  df-co 4730  df-dm 4731  df-rn 4732  df-res 4733  df-ima 4734  df-iota 5282  df-fun 5324  df-fn 5325  df-fv 5330  df-riota 5964  df-ov 6014  df-oprab 6015  df-mpo 6016  df-pnf 8204  df-mnf 8205  df-ltxr 8207  df-inn 9132  df-2 9190  df-3 9191  df-ndx 13072  df-slot 13073  df-base 13075  df-sets 13076  df-plusg 13160  df-mulr 13161  df-0g 13328  df-mgm 13426  df-sgrp 13472  df-mnd 13487  df-mgp 13921  df-ur 13960  df-ring 13998

This theorem is referenced by: (None)