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| Mirrors > Home > MPE Home > Th. List > domnrrg | Structured version Visualization version GIF version | ||
| Description: In a domain, a nonzero element is a regular element. (Contributed by Mario Carneiro, 28-Mar-2015.) | 
| Ref | Expression | 
|---|---|
| isdomn2.b | ⊢ 𝐵 = (Base‘𝑅) | 
| isdomn2.t | ⊢ 𝐸 = (RLReg‘𝑅) | 
| isdomn2.z | ⊢ 0 = (0g‘𝑅) | 
| Ref | Expression | 
|---|---|
| domnrrg | ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → 𝑋 ∈ 𝐸) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | isdomn2.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
| 2 | isdomn2.t | . . . . 5 ⊢ 𝐸 = (RLReg‘𝑅) | |
| 3 | isdomn2.z | . . . . 5 ⊢ 0 = (0g‘𝑅) | |
| 4 | 1, 2, 3 | isdomn2 20712 | . . . 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 4785 | . . 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 3983 | 1 ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → 𝑋 ∈ 𝐸) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 ∖ cdif 3947 ⊆ wss 3950 {csn 4625 ‘cfv 6560 Basecbs 17248 0gc0g 17485 NzRingcnzr 20513 RLRegcrlreg 20692 Domncdomn 20693 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-iota 6513 df-fun 6562 df-fv 6568 df-ov 7435 df-rlreg 20695 df-domn 20696 | 
| This theorem is referenced by: deg1ldgdomn 26134 deg1mul 26155 ply1unit 33601 m1pmeq 33609 r1pid2OLD 33630 assafld 33689 | 
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