domnrrg - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > domnrrg		Structured version Visualization version GIF version

Theorem domnrrg 20735

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 20733	. . . 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 4811	. . 3 ⊢ (𝑋 ∈ (𝐵 ∖ { 0 }) ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ))
10	7, 8, 9	sylanbrc 582	. 2 ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → 𝑋 ∈ (𝐵 ∖ { 0 }))
11	6, 10	sseldd 4009	1 ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → 𝑋 ∈ 𝐸)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ∖ cdif 3973 ⊆ wss 3976 {csn 4648 ‘cfv 6573 Basecbs 17258 0_gc0g 17499 NzRingcnzr 20538 RLRegcrlreg 20713 Domncdomn 20714

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-sbc 3805 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-iota 6525 df-fun 6575 df-fv 6581 df-ov 7451 df-rlreg 20716 df-domn 20717

This theorem is referenced by: deg1ldgdomn 26153 deg1mul 26174 ply1unit 33565 m1pmeq 33573 r1pid2OLD 33594 assafld 33650

W3C validator