fldcrngo - Mathbox for Jeff Madsen

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Theorem fldcrngo 38938

Description: Obsolete theorem, use fldcrngd 20995 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 2761	. . . . 5 ⊢ (1^st ‘𝐾) = (1^st ‘𝐾)
2		eqid 2761	. . . . 5 ⊢ (2^nd ‘𝐾) = (2^nd ‘𝐾)
3		eqid 2761	. . . . 5 ⊢ ran (1^st ‘𝐾) = ran (1^st ‘𝐾)
4		eqid 2761	. . . . 5 ⊢ (GId‘(1^st ‘𝐾)) = (GId‘(1^st ‘𝐾))
5	1, 2, 3, 4	drngoi 38885	. . . 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 38926	. . 3 ⊢ Fld = (DivRingOps ∩ Com2)
9	8	elin2 4149	. 2 ⊢ (𝐾 ∈ Fld ↔ (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ Com2))
10		iscrngo 38930	. 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 2145 ∖ cdif 3896 {csn 4584 × cxp 5649 ran crn 5652 ↾ cres 5653 ‘cfv 6538 1^st c1st 7999 2^nd c2nd 8000 GrpOpcgr 31091 GIdcgi 31092 RingOpscrngo 38828 DivRingOpscdrng 38882 Com2ccm2 38923 Fldcfld 38925 CRingOpsccring 38927

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 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-sep 5249 ax-nul 5260 ax-pr 5391 ax-un 7751

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 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3078 df-rex 3088 df-rab 3414 df-v 3453 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5546 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-iota 6494 df-fun 6540 df-fv 6546 df-1st 8001 df-2nd 8002 df-drngo 38883 df-fld 38926 df-crngo 38928

This theorem is used by: isfld2 38939 isfldidl 39002

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