fldcrngo - Mathbox for Jeff Madsen

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Theorem fldcrngo 37385

Description: A field is a commutative ring. (Contributed by Jeff Madsen, 8-Jun-2010.)

**Assertion**
Ref	Expression
fldcrngo	⊢ (𝐾 ∈ Fld → 𝐾 ∈ CRingOps)

**Proof of Theorem fldcrngo**
Step	Hyp	Ref	Expression
1		eqid 2726	. . . . 5 ⊢ (1^st ‘𝐾) = (1^st ‘𝐾)
2		eqid 2726	. . . . 5 ⊢ (2^nd ‘𝐾) = (2^nd ‘𝐾)
3		eqid 2726	. . . . 5 ⊢ ran (1^st ‘𝐾) = ran (1^st ‘𝐾)
4		eqid 2726	. . . . 5 ⊢ (GId‘(1^st ‘𝐾)) = (GId‘(1^st ‘𝐾))
5	1, 2, 3, 4	drngoi 37332	. . . 4 ⊢ (𝐾 ∈ DivRingOps → (𝐾 ∈ RingOps ∧ ((2^nd ‘𝐾) ↾ ((ran (1^st ‘𝐾) ∖ {(GId‘(1^st ‘𝐾))}) × (ran (1^st ‘𝐾) ∖ {(GId‘(1^st ‘𝐾))}))) ∈ GrpOp))
6	5	simpld 494	. . 3 ⊢ (𝐾 ∈ DivRingOps → 𝐾 ∈ RingOps)
7	6	anim1i 614	. 2 ⊢ ((𝐾 ∈ DivRingOps ∧ 𝐾 ∈ Com2) → (𝐾 ∈ RingOps ∧ 𝐾 ∈ Com2))
8		df-fld 37373	. . 3 ⊢ Fld = (DivRingOps ∩ Com2)
9	8	elin2 4192	. 2 ⊢ (𝐾 ∈ Fld ↔ (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ Com2))
10		iscrngo 37377	. 2 ⊢ (𝐾 ∈ CRingOps ↔ (𝐾 ∈ RingOps ∧ 𝐾 ∈ Com2))
11	7, 9, 10	3imtr4i 292	1 ⊢ (𝐾 ∈ Fld → 𝐾 ∈ CRingOps)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2098 ∖ cdif 3940 {csn 4623 × cxp 5667 ran crn 5670 ↾ cres 5671 ‘cfv 6537 1^st c1st 7972 2^nd c2nd 7973 GrpOpcgr 30251 GIdcgi 30252 RingOpscrngo 37275 DivRingOpscdrng 37329 Com2ccm2 37370 Fldcfld 37372 CRingOpsccring 37374

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7722

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-iota 6489 df-fun 6539 df-fv 6545 df-1st 7974 df-2nd 7975 df-drngo 37330 df-fld 37373 df-crngo 37375

This theorem is referenced by: isfld2 37386 isfldidl 37449

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