fldcrngo - Mathbox for Jeff Madsen

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Theorem fldcrngo 39204

Description: Obsolete theorem, use fldcrngd 21046 instead. A field is a commutative ring. (Contributed by Jeff Madsen, 8-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)

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

**Proof of Theorem fldcrngo**
Step	Hyp	Ref	Expression
1		eqid 2763	. . . . 5 ⊢ (1^st ‘𝐾) = (1^st ‘𝐾)
2		eqid 2763	. . . . 5 ⊢ (2^nd ‘𝐾) = (2^nd ‘𝐾)
3		eqid 2763	. . . . 5 ⊢ ran (1^st ‘𝐾) = ran (1^st ‘𝐾)
4		eqid 2763	. . . . 5 ⊢ (GId‘(1^st ‘𝐾)) = (GId‘(1^st ‘𝐾))
5	1, 2, 3, 4	drngoi 39151	. . . 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 500	. . 3 ⊢ (𝐾 ∈ DivRingOps → 𝐾 ∈ RingOps)
7	6	anim1i 627	. 2 ⊢ ((𝐾 ∈ DivRingOps ∧ 𝐾 ∈ Com2) → (𝐾 ∈ RingOps ∧ 𝐾 ∈ Com2))
8		df-fld 39192	. . 3 ⊢ Fld = (DivRingOps ∩ Com2)
9	8	elin2 4103	. 2 ⊢ (𝐾 ∈ Fld ↔ (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ Com2))
10		iscrngo 39196	. 2 ⊢ (𝐾 ∈ CRingOps ↔ (𝐾 ∈ RingOps ∧ 𝐾 ∈ Com2))
11	7, 9, 10	3imtr4i 295	1 ⊢ (𝐾 ∈ Fld → 𝐾 ∈ CRingOps)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∈ wcel 2147 ∖ cdif 4048 {csn 4588 × cxp 5656 ran crn 5659 ↾ cres 5660 ‘cfv 6566 1^st c1st 8033 2^nd c2nd 8034 GrpOpcgr 31148 GIdcgi 31149 RingOpscrngo 39094 DivRingOpscdrng 39148 Com2ccm2 39189 Fldcfld 39191 CRingOpsccring 39193

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5255 ax-nul 5266 ax-pr 5398 ax-un 7783

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-rab 3417 df-v 3456 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-br 5109 df-opab 5174 df-mpt 5193 df-id 5553 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-iota 6520 df-fun 6568 df-fv 6574 df-1st 8035 df-2nd 8036 df-drngo 39149 df-fld 39192 df-crngo 39194

This theorem is used by: isfld2 39205 isfldidl 39268

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