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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 20728 | . . . 4 ⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ (𝐵 ∖ { 0 }) ⊆ 𝐸)) |
5 | 4 | simprbi 496 | . . 3 ⊢ (𝑅 ∈ Domn → (𝐵 ∖ { 0 }) ⊆ 𝐸) |
6 | 5 | 3ad2ant1 1132 | . 2 ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → (𝐵 ∖ { 0 }) ⊆ 𝐸) |
7 | simp2 1136 | . . 3 ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → 𝑋 ∈ 𝐵) | |
8 | simp3 1137 | . . 3 ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → 𝑋 ≠ 0 ) | |
9 | eldifsn 4791 | . . 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 3996 | 1 ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → 𝑋 ∈ 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∖ cdif 3960 ⊆ wss 3963 {csn 4631 ‘cfv 6563 Basecbs 17245 0gc0g 17486 NzRingcnzr 20529 RLRegcrlreg 20708 Domncdomn 20709 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-rlreg 20711 df-domn 20712 |
This theorem is referenced by: deg1ldgdomn 26148 deg1mul 26169 ply1unit 33580 m1pmeq 33588 r1pid2OLD 33609 assafld 33665 |
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