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Mathbox for Jeff Madsen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fldcrngo | Structured version Visualization version GIF version |
Description: A field is a commutative ring. (Contributed by Jeff Madsen, 8-Jun-2010.) |
Ref | Expression |
---|---|
fldcrngo | ⊢ (𝐾 ∈ Fld → 𝐾 ∈ CRingOps) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2728 | . . . . 5 ⊢ (1st ‘𝐾) = (1st ‘𝐾) | |
2 | eqid 2728 | . . . . 5 ⊢ (2nd ‘𝐾) = (2nd ‘𝐾) | |
3 | eqid 2728 | . . . . 5 ⊢ ran (1st ‘𝐾) = ran (1st ‘𝐾) | |
4 | eqid 2728 | . . . . 5 ⊢ (GId‘(1st ‘𝐾)) = (GId‘(1st ‘𝐾)) | |
5 | 1, 2, 3, 4 | drngoi 37465 | . . . 4 ⊢ (𝐾 ∈ DivRingOps → (𝐾 ∈ RingOps ∧ ((2nd ‘𝐾) ↾ ((ran (1st ‘𝐾) ∖ {(GId‘(1st ‘𝐾))}) × (ran (1st ‘𝐾) ∖ {(GId‘(1st ‘𝐾))}))) ∈ GrpOp)) |
6 | 5 | simpld 493 | . . 3 ⊢ (𝐾 ∈ DivRingOps → 𝐾 ∈ RingOps) |
7 | 6 | anim1i 613 | . 2 ⊢ ((𝐾 ∈ DivRingOps ∧ 𝐾 ∈ Com2) → (𝐾 ∈ RingOps ∧ 𝐾 ∈ Com2)) |
8 | df-fld 37506 | . . 3 ⊢ Fld = (DivRingOps ∩ Com2) | |
9 | 8 | elin2 4199 | . 2 ⊢ (𝐾 ∈ Fld ↔ (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ Com2)) |
10 | iscrngo 37510 | . 2 ⊢ (𝐾 ∈ CRingOps ↔ (𝐾 ∈ RingOps ∧ 𝐾 ∈ Com2)) | |
11 | 7, 9, 10 | 3imtr4i 291 | 1 ⊢ (𝐾 ∈ Fld → 𝐾 ∈ CRingOps) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2098 ∖ cdif 3946 {csn 4632 × cxp 5680 ran crn 5683 ↾ cres 5684 ‘cfv 6553 1st c1st 7999 2nd c2nd 8000 GrpOpcgr 30327 GIdcgi 30328 RingOpscrngo 37408 DivRingOpscdrng 37462 Com2ccm2 37503 Fldcfld 37505 CRingOpsccring 37507 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 ax-un 7748 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-iota 6505 df-fun 6555 df-fv 6561 df-1st 8001 df-2nd 8002 df-drngo 37463 df-fld 37506 df-crngo 37508 |
This theorem is referenced by: isfld2 37519 isfldidl 37582 |
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