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Mathbox for Jeff Madsen |
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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 37472 | . . 3 ⊢ (𝐾 ∈ Fld → 𝐾 ∈ DivRingOps) | |
2 | fldcrngo 37477 | . . 3 ⊢ (𝐾 ∈ Fld → 𝐾 ∈ CRingOps) | |
3 | 1, 2 | jca 511 | . 2 ⊢ (𝐾 ∈ Fld → (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ CRingOps)) |
4 | iscrngo 37469 | . . . 4 ⊢ (𝐾 ∈ CRingOps ↔ (𝐾 ∈ RingOps ∧ 𝐾 ∈ Com2)) | |
5 | 4 | simprbi 496 | . . 3 ⊢ (𝐾 ∈ CRingOps → 𝐾 ∈ Com2) |
6 | elin 3963 | . . . . 5 ⊢ (𝐾 ∈ (DivRingOps ∩ Com2) ↔ (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ Com2)) | |
7 | 6 | biimpri 227 | . . . 4 ⊢ ((𝐾 ∈ DivRingOps ∧ 𝐾 ∈ Com2) → 𝐾 ∈ (DivRingOps ∩ Com2)) |
8 | df-fld 37465 | . . . 4 ⊢ Fld = (DivRingOps ∩ Com2) | |
9 | 7, 8 | eleqtrrdi 2840 | . . 3 ⊢ ((𝐾 ∈ DivRingOps ∧ 𝐾 ∈ Com2) → 𝐾 ∈ Fld) |
10 | 5, 9 | sylan2 592 | . 2 ⊢ ((𝐾 ∈ DivRingOps ∧ 𝐾 ∈ CRingOps) → 𝐾 ∈ Fld) |
11 | 3, 10 | impbii 208 | 1 ⊢ (𝐾 ∈ Fld ↔ (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ CRingOps)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 ∈ wcel 2099 ∩ cin 3946 RingOpscrngo 37367 DivRingOpscdrng 37421 Com2ccm2 37462 Fldcfld 37464 CRingOpsccring 37466 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-iota 6500 df-fun 6550 df-fv 6556 df-1st 7993 df-2nd 7994 df-drngo 37422 df-fld 37465 df-crngo 37467 |
This theorem is referenced by: flddmn 37531 isfldidl 37541 |
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