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Mirrors > Home > MPE Home > Th. List > Mathboxes > fldcrng | Structured version Visualization version GIF version |
Description: A field is a commutative ring. (Contributed by Jeff Madsen, 8-Jun-2010.) |
Ref | Expression |
---|---|
fldcrng | ⊢ (𝐾 ∈ Fld → 𝐾 ∈ CRingOps) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2818 | . . . . 5 ⊢ (1st ‘𝐾) = (1st ‘𝐾) | |
2 | eqid 2818 | . . . . 5 ⊢ (2nd ‘𝐾) = (2nd ‘𝐾) | |
3 | eqid 2818 | . . . . 5 ⊢ ran (1st ‘𝐾) = ran (1st ‘𝐾) | |
4 | eqid 2818 | . . . . 5 ⊢ (GId‘(1st ‘𝐾)) = (GId‘(1st ‘𝐾)) | |
5 | 1, 2, 3, 4 | drngoi 35110 | . . . 4 ⊢ (𝐾 ∈ DivRingOps → (𝐾 ∈ RingOps ∧ ((2nd ‘𝐾) ↾ ((ran (1st ‘𝐾) ∖ {(GId‘(1st ‘𝐾))}) × (ran (1st ‘𝐾) ∖ {(GId‘(1st ‘𝐾))}))) ∈ GrpOp)) |
6 | 5 | simpld 495 | . . 3 ⊢ (𝐾 ∈ DivRingOps → 𝐾 ∈ RingOps) |
7 | 6 | anim1i 614 | . 2 ⊢ ((𝐾 ∈ DivRingOps ∧ 𝐾 ∈ Com2) → (𝐾 ∈ RingOps ∧ 𝐾 ∈ Com2)) |
8 | df-fld 35151 | . . 3 ⊢ Fld = (DivRingOps ∩ Com2) | |
9 | 8 | elin2 4171 | . 2 ⊢ (𝐾 ∈ Fld ↔ (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ Com2)) |
10 | iscrngo 35155 | . 2 ⊢ (𝐾 ∈ CRingOps ↔ (𝐾 ∈ RingOps ∧ 𝐾 ∈ Com2)) | |
11 | 7, 9, 10 | 3imtr4i 293 | 1 ⊢ (𝐾 ∈ Fld → 𝐾 ∈ CRingOps) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2105 ∖ cdif 3930 {csn 4557 × cxp 5546 ran crn 5549 ↾ cres 5550 ‘cfv 6348 1st c1st 7676 2nd c2nd 7677 GrpOpcgr 28193 GIdcgi 28194 RingOpscrngo 35053 DivRingOpscdrng 35107 Com2ccm2 35148 Fldcfld 35150 CRingOpsccring 35152 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-iota 6307 df-fun 6350 df-fv 6356 df-1st 7678 df-2nd 7679 df-drngo 35108 df-fld 35151 df-crngo 35153 |
This theorem is referenced by: isfld2 35164 isfldidl 35227 |
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