Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 18574 | . . . 4 ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ 𝑈 = (𝐵 ∖ { 0 }))) |
5 | 4 | simprbi 479 | . . 3 ⊢ (𝑅 ∈ DivRing → 𝑈 = (𝐵 ∖ { 0 })) |
6 | 5 | eleq2d 2673 | . 2 ⊢ (𝑅 ∈ DivRing → (𝑋 ∈ 𝑈 ↔ 𝑋 ∈ (𝐵 ∖ { 0 }))) |
7 | eldifsn 4260 | . 2 ⊢ (𝑋 ∈ (𝐵 ∖ { 0 }) ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 )) | |
8 | 6, 7 | syl6bb 275 | 1 ⊢ (𝑅 ∈ DivRing → (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∖ cdif 3537 {csn 4125 ‘cfv 5804 Basecbs 15695 0gc0g 15923 Ringcrg 18370 Unitcui 18462 DivRingcdr 18570 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-drng 18572 |
This theorem is referenced by: drngunz 18585 drnginvrcl 18587 drnginvrn0 18588 drnginvrl 18589 drnginvrr 18590 issubdrg 18628 abvdiv 18660 qsssubdrg 19624 redvr 19782 drnguc1p 23734 lgseisenlem3 24902 ornglmullt 29138 orngrmullt 29139 isarchiofld 29148 qqhval2lem 29353 qqhf 29358 matunitlindf 32577 lincreslvec3 42065 isldepslvec2 42068 |
Copyright terms: Public domain | W3C validator |