Theorem ringadd2 14039

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 14038	. 2 |- ( ( R e. Ring /\ X e. B ) -> E. x e. B ( ( x .x. X ) = X /\ ( X .x. x ) = X ) )
4		oveq12 6026	. . . . . . 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 2237	. . . . 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 529	. . . . . . 7 |- ( ( ( R e. Ring /\ X e. B ) /\ x e. B ) -> X e. B )
10		ringadd2.p	. . . . . . . 8 |- .+ = ( +g ` R )
11	1, 10, 2	ringdir 14031	. . . . . . 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 1275	. . . . . 6 |- ( ( ( R e. Ring /\ X e. B ) /\ x e. B ) -> ( ( x .+ x ) .x. X ) = ( ( x .x. X ) .+ ( x .x. X ) ) )
13	12	eqeq2d 2243	. . . . 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 2634	. 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 1397   e. wcel 2202   E.wrex 2511   ` cfv 5326  (class class class)co 6017   Basecbs 13081   +g cplusg 13159   .rcmulr 13160   Ringcrg 14008

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-pre-ltirr 8143  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-ltxr 8218  df-inn 9143  df-2 9201  df-3 9202  df-ndx 13084  df-slot 13085  df-base 13087  df-sets 13088  df-plusg 13172  df-mulr 13173  df-0g 13340  df-mgm 13438  df-sgrp 13484  df-mnd 13499  df-mgp 13933  df-ur 13972  df-ring 14010

This theorem is referenced by: (None)