fldcrng - Mathbox for Jeff Madsen

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Theorem fldcrng 36162

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

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

**Proof of Theorem fldcrng**
Step	Hyp	Ref	Expression
1		eqid 2738	. . . . 5 ⊢ (1^st ‘𝐾) = (1^st ‘𝐾)
2		eqid 2738	. . . . 5 ⊢ (2^nd ‘𝐾) = (2^nd ‘𝐾)
3		eqid 2738	. . . . 5 ⊢ ran (1^st ‘𝐾) = ran (1^st ‘𝐾)
4		eqid 2738	. . . . 5 ⊢ (GId‘(1^st ‘𝐾)) = (GId‘(1^st ‘𝐾))
5	1, 2, 3, 4	drngoi 36109	. . . 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 495	. . 3 ⊢ (𝐾 ∈ DivRingOps → 𝐾 ∈ RingOps)
7	6	anim1i 615	. 2 ⊢ ((𝐾 ∈ DivRingOps ∧ 𝐾 ∈ Com2) → (𝐾 ∈ RingOps ∧ 𝐾 ∈ Com2))
8		df-fld 36150	. . 3 ⊢ Fld = (DivRingOps ∩ Com2)
9	8	elin2 4131	. 2 ⊢ (𝐾 ∈ Fld ↔ (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ Com2))
10		iscrngo 36154	. 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 396 ∈ wcel 2106 ∖ cdif 3884 {csn 4561 × cxp 5587 ran crn 5590 ↾ cres 5591 ‘cfv 6433 1^st c1st 7829 2^nd c2nd 7830 GrpOpcgr 28851 GIdcgi 28852 RingOpscrngo 36052 DivRingOpscdrng 36106 Com2ccm2 36147 Fldcfld 36149 CRingOpsccring 36151

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 ax-un 7588

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-iota 6391 df-fun 6435 df-fv 6441 df-1st 7831 df-2nd 7832 df-drngo 36107 df-fld 36150 df-crngo 36152

This theorem is referenced by: isfld2 36163 isfldidl 36226

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