Theorem ringadd2 13904

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 13903	. 2 |- ( ( R e. Ring /\ X e. B ) -> E. x e. B ( ( x .x. X ) = X /\ ( X .x. x ) = X ) )
4		oveq12 5976	. . . . . . 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 2213	. . . . 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 13896	. . . . . . 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 1252	. . . . . 6 |- ( ( ( R e. Ring /\ X e. B ) /\ x e. B ) -> ( ( x .+ x ) .x. X ) = ( ( x .x. X ) .+ ( x .x. X ) ) )
13	12	eqeq2d 2219	. . . . 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 2610	. 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 1373   e. wcel 2178   E.wrex 2487   ` cfv 5290  (class class class)co 5967   Basecbs 12947   +g cplusg 13024   .rcmulr 13025   Ringcrg 13873

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-pre-ltirr 8072  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-pnf 8144  df-mnf 8145  df-ltxr 8147  df-inn 9072  df-2 9130  df-3 9131  df-ndx 12950  df-slot 12951  df-base 12953  df-sets 12954  df-plusg 13037  df-mulr 13038  df-0g 13205  df-mgm 13303  df-sgrp 13349  df-mnd 13364  df-mgp 13798  df-ur 13837  df-ring 13875

This theorem is referenced by: (None)