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Mirrors > Home > MPE Home > Th. List > domnrrg | Structured version Visualization version GIF version |
Description: In a domain, any nonzero element is a nonzero-divisor. (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 20204 | . . . 4 ⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ (𝐵 ∖ { 0 }) ⊆ 𝐸)) |
5 | 4 | simprbi 500 | . . 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 4685 | . . 3 ⊢ (𝑋 ∈ (𝐵 ∖ { 0 }) ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 )) | |
10 | 7, 8, 9 | sylanbrc 586 | . 2 ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → 𝑋 ∈ (𝐵 ∖ { 0 })) |
11 | 6, 10 | sseldd 3888 | 1 ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → 𝑋 ∈ 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ≠ wne 2935 ∖ cdif 3850 ⊆ wss 3853 {csn 4526 ‘cfv 6350 Basecbs 16599 0gc0g 16829 NzRingcnzr 20162 RLRegcrlreg 20184 Domncdomn 20185 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pr 5306 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3402 df-sbc 3686 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-br 5041 df-opab 5103 df-mpt 5121 df-id 5439 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-iota 6308 df-fun 6352 df-fv 6358 df-ov 7186 df-rlreg 20188 df-domn 20189 |
This theorem is referenced by: deg1ldgdomn 24860 |
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