domnrrg - Metamath Proof Explorer

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Theorem domnrrg 20598

Description: In a domain, a nonzero element is a regular element. (Contributed by Mario Carneiro, 28-Mar-2015.)

**Hypotheses**
Ref	Expression
isdomn2.b	⊢ 𝐵 = (Base‘𝑅)
isdomn2.t	⊢ 𝐸 = (RLReg‘𝑅)
isdomn2.z	⊢ 0 = (0_g‘𝑅)

**Assertion**
Ref	Expression
domnrrg	⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → 𝑋 ∈ 𝐸)

**Proof of Theorem domnrrg**
Step	Hyp	Ref	Expression
1		isdomn2.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑅)
2		isdomn2.t	. . . . 5 ⊢ 𝐸 = (RLReg‘𝑅)
3		isdomn2.z	. . . . 5 ⊢ 0 = (0_g‘𝑅)
4	1, 2, 3	isdomn2 20596	. . . 4 ⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ (𝐵 ∖ { 0 }) ⊆ 𝐸))
5	4	simprbi 496	. . 3 ⊢ (𝑅 ∈ Domn → (𝐵 ∖ { 0 }) ⊆ 𝐸)
6	5	3ad2ant1 1133	. 2 ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → (𝐵 ∖ { 0 }) ⊆ 𝐸)
7		simp2 1137	. . 3 ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → 𝑋 ∈ 𝐵)
8		simp3 1138	. . 3 ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → 𝑋 ≠ 0 )
9		eldifsn 4746	. . 3 ⊢ (𝑋 ∈ (𝐵 ∖ { 0 }) ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ))
10	7, 8, 9	sylanbrc 583	. 2 ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → 𝑋 ∈ (𝐵 ∖ { 0 }))
11	6, 10	sseldd 3944	1 ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → 𝑋 ∈ 𝐸)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∖ cdif 3908 ⊆ wss 3911 {csn 4585 ‘cfv 6499 Basecbs 17155 0_gc0g 17378 NzRingcnzr 20397 RLRegcrlreg 20576 Domncdomn 20577

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5246 ax-nul 5256 ax-pr 5382

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rab 3403 df-v 3446 df-sbc 3751 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-br 5103 df-opab 5165 df-mpt 5184 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-iota 6452 df-fun 6501 df-fv 6507 df-ov 7372 df-rlreg 20579 df-domn 20580

This theorem is referenced by: deg1ldgdomn 25975 deg1mul 25996 ply1unit 33517 m1pmeq 33525 r1pid2OLD 33547 assafld 33606

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