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| Mirrors > Home > MPE Home > Th. List > Mathboxes > isfld2 | Structured version Visualization version GIF version | ||
| Description: The predicate "is a field". (Contributed by Jeff Madsen, 10-Jun-2010.) |
| Ref | Expression |
|---|---|
| isfld2 | ⊢ (𝐾 ∈ Fld ↔ (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ CRingOps)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | flddivrng 38006 | . . 3 ⊢ (𝐾 ∈ Fld → 𝐾 ∈ DivRingOps) | |
| 2 | fldcrngo 38011 | . . 3 ⊢ (𝐾 ∈ Fld → 𝐾 ∈ CRingOps) | |
| 3 | 1, 2 | jca 511 | . 2 ⊢ (𝐾 ∈ Fld → (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ CRingOps)) |
| 4 | iscrngo 38003 | . . . 4 ⊢ (𝐾 ∈ CRingOps ↔ (𝐾 ∈ RingOps ∧ 𝐾 ∈ Com2)) | |
| 5 | 4 | simprbi 496 | . . 3 ⊢ (𝐾 ∈ CRingOps → 𝐾 ∈ Com2) |
| 6 | elin 3967 | . . . . 5 ⊢ (𝐾 ∈ (DivRingOps ∩ Com2) ↔ (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ Com2)) | |
| 7 | 6 | biimpri 228 | . . . 4 ⊢ ((𝐾 ∈ DivRingOps ∧ 𝐾 ∈ Com2) → 𝐾 ∈ (DivRingOps ∩ Com2)) |
| 8 | df-fld 37999 | . . . 4 ⊢ Fld = (DivRingOps ∩ Com2) | |
| 9 | 7, 8 | eleqtrrdi 2852 | . . 3 ⊢ ((𝐾 ∈ DivRingOps ∧ 𝐾 ∈ Com2) → 𝐾 ∈ Fld) |
| 10 | 5, 9 | sylan2 593 | . 2 ⊢ ((𝐾 ∈ DivRingOps ∧ 𝐾 ∈ CRingOps) → 𝐾 ∈ Fld) |
| 11 | 3, 10 | impbii 209 | 1 ⊢ (𝐾 ∈ Fld ↔ (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ CRingOps)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 ∈ wcel 2108 ∩ cin 3950 RingOpscrngo 37901 DivRingOpscdrng 37955 Com2ccm2 37996 Fldcfld 37998 CRingOpsccring 38000 |
| 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 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| 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 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-iota 6514 df-fun 6563 df-fv 6569 df-1st 8014 df-2nd 8015 df-drngo 37956 df-fld 37999 df-crngo 38001 |
| This theorem is referenced by: flddmn 38065 isfldidl 38075 |
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