ringadd2 - Metamath Proof Explorer

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Theorem ringadd2 19795

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	⊢ · = (._r‘𝑅)

**Assertion**
Ref	Expression
ringadd2	⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ∃𝑥 ∈ 𝐵 (𝑋 + 𝑋) = ((𝑥 + 𝑥) · 𝑋))

Distinct variable groups: 𝑥,𝐵 𝑥,𝑅 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 ⊢ · = (._r‘𝑅)
3	1, 2	ringid 19794	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ∃𝑥 ∈ 𝐵 ((𝑥 · 𝑋) = 𝑋 ∧ (𝑋 · 𝑥) = 𝑋))
4		oveq12 7277	. . . . . . 7 ⊢ (((𝑥 · 𝑋) = 𝑋 ∧ (𝑥 · 𝑋) = 𝑋) → ((𝑥 · 𝑋) + (𝑥 · 𝑋)) = (𝑋 + 𝑋))
5	4	anidms 566	. . . . . 6 ⊢ ((𝑥 · 𝑋) = 𝑋 → ((𝑥 · 𝑋) + (𝑥 · 𝑋)) = (𝑋 + 𝑋))
6	5	eqcomd 2745	. . . . 5 ⊢ ((𝑥 · 𝑋) = 𝑋 → (𝑋 + 𝑋) = ((𝑥 · 𝑋) + (𝑥 · 𝑋)))
7		simpll 763	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑅 ∈ Ring)
8		simpr 484	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
9		simplr 765	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑋 ∈ 𝐵)
10		ringadd2.p	. . . . . . . 8 ⊢ + = (+_g‘𝑅)
11	1, 10, 2	ringdir 19787	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((𝑥 + 𝑥) · 𝑋) = ((𝑥 · 𝑋) + (𝑥 · 𝑋)))
12	7, 8, 8, 9, 11	syl13anc 1370	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → ((𝑥 + 𝑥) · 𝑋) = ((𝑥 · 𝑋) + (𝑥 · 𝑋)))
13	12	eqeq2d 2750	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → ((𝑋 + 𝑋) = ((𝑥 + 𝑥) · 𝑋) ↔ (𝑋 + 𝑋) = ((𝑥 · 𝑋) + (𝑥 · 𝑋))))
14	6, 13	syl5ibr 245	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → ((𝑥 · 𝑋) = 𝑋 → (𝑋 + 𝑋) = ((𝑥 + 𝑥) · 𝑋)))
15	14	adantrd 491	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → (((𝑥 · 𝑋) = 𝑋 ∧ (𝑋 · 𝑥) = 𝑋) → (𝑋 + 𝑋) = ((𝑥 + 𝑥) · 𝑋)))
16	15	reximdva 3204	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (∃𝑥 ∈ 𝐵 ((𝑥 · 𝑋) = 𝑋 ∧ (𝑋 · 𝑥) = 𝑋) → ∃𝑥 ∈ 𝐵 (𝑋 + 𝑋) = ((𝑥 + 𝑥) · 𝑋)))
17	3, 16	mpd 15	1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ∃𝑥 ∈ 𝐵 (𝑋 + 𝑋) = ((𝑥 + 𝑥) · 𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2109 ∃wrex 3066 ‘cfv 6430 (class class class)co 7268 Basecbs 16893 +_gcplusg 16943 ._rcmulr 16944 Ringcrg 19764

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7701 df-2nd 7818 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-er 8472 df-en 8708 df-dom 8709 df-sdom 8710 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-nn 11957 df-2 12019 df-sets 16846 df-slot 16864 df-ndx 16876 df-base 16894 df-plusg 16956 df-0g 17133 df-mgm 18307 df-sgrp 18356 df-mnd 18367 df-mgp 19702 df-ur 19719 df-ring 19766

This theorem is referenced by: (None)

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