fldcrngo - Mathbox for Jeff Madsen

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Theorem fldcrngo 38503

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 2762	. . . . 5 ⊢ (1^st ‘𝐾) = (1^st ‘𝐾)
2		eqid 2762	. . . . 5 ⊢ (2^nd ‘𝐾) = (2^nd ‘𝐾)
3		eqid 2762	. . . . 5 ⊢ ran (1^st ‘𝐾) = ran (1^st ‘𝐾)
4		eqid 2762	. . . . 5 ⊢ (GId‘(1^st ‘𝐾)) = (GId‘(1^st ‘𝐾))
5	1, 2, 3, 4	drngoi 38450	. . . 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 498	. . 3 ⊢ (𝐾 ∈ DivRingOps → 𝐾 ∈ RingOps)
7	6	anim1i 624	. 2 ⊢ ((𝐾 ∈ DivRingOps ∧ 𝐾 ∈ Com2) → (𝐾 ∈ RingOps ∧ 𝐾 ∈ Com2))
8		df-fld 38491	. . 3 ⊢ Fld = (DivRingOps ∩ Com2)
9	8	elin2 4155	. 2 ⊢ (𝐾 ∈ Fld ↔ (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ Com2))
10		iscrngo 38495	. 2 ⊢ (𝐾 ∈ CRingOps ↔ (𝐾 ∈ RingOps ∧ 𝐾 ∈ Com2))
11	7, 9, 10	3imtr4i 294	1 ⊢ (𝐾 ∈ Fld → 𝐾 ∈ CRingOps)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2142 ∖ cdif 3901 {csn 4582 × cxp 5645 ran crn 5648 ↾ cres 5649 ‘cfv 6521 1^st c1st 7968 2^nd c2nd 7969 GrpOpcgr 30692 GIdcgi 30693 RingOpscrngo 38393 DivRingOpscdrng 38447 Com2ccm2 38488 Fldcfld 38490 CRingOpsccring 38492

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-sep 5246 ax-nul 5256 ax-pr 5390 ax-un 7718

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-ral 3077 df-rex 3087 df-rab 3415 df-v 3456 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4481 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5542 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-iota 6477 df-fun 6523 df-fv 6529 df-1st 7970 df-2nd 7971 df-drngo 38448 df-fld 38491 df-crngo 38493

This theorem is referenced by: isfld2 38504 isfldidl 38567

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