Theorem ringadd2 13003

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 13002	. 2 |- ( ( R e. Ring /\ X e. B ) -> E. x e. B ( ( x .x. X ) = X /\ ( X .x. x ) = X ) )
4		oveq12 5874	. . . . . . 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 2181	. . . . 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 12995	. . . . . . 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 2187	. . . . 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	syl5ibr 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 2577	. 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 2146   E.wrex 2454   ` cfv 5208  (class class class)co 5865   Basecbs 12427   +g cplusg 12491   .rcmulr 12492   Ringcrg 12972

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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-addcom 7886  ax-addass 7888  ax-i2m1 7891  ax-0lt1 7892  ax-0id 7894  ax-rnegex 7895  ax-pre-ltirr 7898  ax-pre-ltadd 7902

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-pnf 7968  df-mnf 7969  df-ltxr 7971  df-inn 8891  df-2 8949  df-3 8950  df-ndx 12430  df-slot 12431  df-base 12433  df-sets 12434  df-plusg 12504  df-mulr 12505  df-0g 12627  df-mgm 12639  df-sgrp 12672  df-mnd 12682  df-mgp 12926  df-ur 12936  df-ring 12974

This theorem is referenced by: (None)