domnrrg - Metamath Proof Explorer

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Theorem domnrrg 20690

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 20688	. . . 4 ⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ (𝐵 ∖ { 0 }) ⊆ 𝐸))
5	4	simprbi 497	. . 3 ⊢ (𝑅 ∈ Domn → (𝐵 ∖ { 0 }) ⊆ 𝐸)
6	5	3ad2ant1 1134	. 2 ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → (𝐵 ∖ { 0 }) ⊆ 𝐸)
7		simp2 1138	. . 3 ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → 𝑋 ∈ 𝐵)
8		simp3 1139	. . 3 ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → 𝑋 ≠ 0 )
9		eldifsn 4731	. . 3 ⊢ (𝑋 ∈ (𝐵 ∖ { 0 }) ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ))
10	7, 8, 9	sylanbrc 584	. 2 ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → 𝑋 ∈ (𝐵 ∖ { 0 }))
11	6, 10	sseldd 3922	1 ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → 𝑋 ∈ 𝐸)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2932 ∖ cdif 3886 ⊆ wss 3889 {csn 4567 ‘cfv 6498 Basecbs 17179 0_gc0g 17402 NzRingcnzr 20489 RLRegcrlreg 20668 Domncdomn 20669

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-sep 5231 ax-nul 5241 ax-pr 5375

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rab 3390 df-v 3431 df-sbc 3729 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-br 5086 df-opab 5148 df-mpt 5167 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-iota 6454 df-fun 6500 df-fv 6506 df-ov 7370 df-rlreg 20671 df-domn 20672

This theorem is referenced by: deg1ldgdomn 26059 deg1mul 26080 ply1unit 33635 m1pmeq 33645 r1pid2OLD 33669 assafld 33781

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