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Mirrors > Home > MPE Home > Th. List > drngunit | Structured version Visualization version GIF version |
Description: Elementhood in the set of units when 𝑅 is a division ring. (Contributed by Mario Carneiro, 2-Dec-2014.) |
Ref | Expression |
---|---|
isdrng.b | ⊢ 𝐵 = (Base‘𝑅) |
isdrng.u | ⊢ 𝑈 = (Unit‘𝑅) |
isdrng.z | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
drngunit | ⊢ (𝑅 ∈ DivRing → (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isdrng.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
2 | isdrng.u | . . . . 5 ⊢ 𝑈 = (Unit‘𝑅) | |
3 | isdrng.z | . . . . 5 ⊢ 0 = (0g‘𝑅) | |
4 | 1, 2, 3 | isdrng 18961 | . . . 4 ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ 𝑈 = (𝐵 ∖ { 0 }))) |
5 | 4 | simprbi 484 | . . 3 ⊢ (𝑅 ∈ DivRing → 𝑈 = (𝐵 ∖ { 0 })) |
6 | 5 | eleq2d 2836 | . 2 ⊢ (𝑅 ∈ DivRing → (𝑋 ∈ 𝑈 ↔ 𝑋 ∈ (𝐵 ∖ { 0 }))) |
7 | eldifsn 4453 | . 2 ⊢ (𝑋 ∈ (𝐵 ∖ { 0 }) ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 )) | |
8 | 6, 7 | syl6bb 276 | 1 ⊢ (𝑅 ∈ DivRing → (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 ∖ cdif 3720 {csn 4316 ‘cfv 6031 Basecbs 16064 0gc0g 16308 Ringcrg 18755 Unitcui 18847 DivRingcdr 18957 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-iota 5994 df-fv 6039 df-drng 18959 |
This theorem is referenced by: drngunz 18972 drnginvrcl 18974 drnginvrn0 18975 drnginvrl 18976 drnginvrr 18977 issubdrg 19015 abvdiv 19047 qsssubdrg 20020 redvr 20180 drnguc1p 24150 lgseisenlem3 25323 ornglmullt 30147 orngrmullt 30148 isarchiofld 30157 qqhval2lem 30365 qqhf 30370 matunitlindf 33740 lincreslvec3 42799 isldepslvec2 42802 |
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