| Mathbox for Jeff Madsen |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > Mathboxes > flddmn | Structured version Visualization version GIF version | ||
| Description: A field is a domain. (Contributed by Jeff Madsen, 10-Jun-2010.) |
| Ref | Expression |
|---|---|
| flddmn | ⊢ (𝐾 ∈ Fld → 𝐾 ∈ Dmn) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | divrngpr 38038 | . . 3 ⊢ (𝐾 ∈ DivRingOps → 𝐾 ∈ PrRing) | |
| 2 | 1 | anim1i 615 | . 2 ⊢ ((𝐾 ∈ DivRingOps ∧ 𝐾 ∈ CRingOps) → (𝐾 ∈ PrRing ∧ 𝐾 ∈ CRingOps)) |
| 3 | isfld2 37990 | . 2 ⊢ (𝐾 ∈ Fld ↔ (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ CRingOps)) | |
| 4 | isdmn2 38040 | . 2 ⊢ (𝐾 ∈ Dmn ↔ (𝐾 ∈ PrRing ∧ 𝐾 ∈ CRingOps)) | |
| 5 | 2, 3, 4 | 3imtr4i 292 | 1 ⊢ (𝐾 ∈ Fld → 𝐾 ∈ Dmn) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 DivRingOpscdrng 37933 Fldcfld 37976 CRingOpsccring 37978 PrRingcprrng 38031 Dmncdmn 38032 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5277 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5224 df-id 5576 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-suc 6388 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-riota 7386 df-ov 7432 df-1st 8010 df-2nd 8011 df-1o 8502 df-en 8982 df-grpo 30502 df-gid 30503 df-ginv 30504 df-ablo 30554 df-ass 37828 df-exid 37830 df-mgmOLD 37834 df-sgrOLD 37846 df-mndo 37852 df-rngo 37880 df-drngo 37934 df-fld 37977 df-crngo 37979 df-idl 37995 df-pridl 37996 df-prrngo 38033 df-dmn 38034 |
| This theorem is referenced by: (None) |
| Copyright terms: Public domain | W3C validator |