![]() |
Mathbox for Jeff Madsen |
< Previous
Next >
Nearby theorems |
|
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 2737 | . . . . 5 ⊢ (1st ‘𝐾) = (1st ‘𝐾) | |
2 | eqid 2737 | . . . . 5 ⊢ (2nd ‘𝐾) = (2nd ‘𝐾) | |
3 | eqid 2737 | . . . . 5 ⊢ ran (1st ‘𝐾) = ran (1st ‘𝐾) | |
4 | eqid 2737 | . . . . 5 ⊢ (GId‘(1st ‘𝐾)) = (GId‘(1st ‘𝐾)) | |
5 | 1, 2, 3, 4 | drngoi 36413 | . . . 4 ⊢ (𝐾 ∈ DivRingOps → (𝐾 ∈ RingOps ∧ ((2nd ‘𝐾) ↾ ((ran (1st ‘𝐾) ∖ {(GId‘(1st ‘𝐾))}) × (ran (1st ‘𝐾) ∖ {(GId‘(1st ‘𝐾))}))) ∈ GrpOp)) |
6 | 5 | simpld 496 | . . 3 ⊢ (𝐾 ∈ DivRingOps → 𝐾 ∈ RingOps) |
7 | 6 | anim1i 616 | . 2 ⊢ ((𝐾 ∈ DivRingOps ∧ 𝐾 ∈ Com2) → (𝐾 ∈ RingOps ∧ 𝐾 ∈ Com2)) |
8 | df-fld 36454 | . . 3 ⊢ Fld = (DivRingOps ∩ Com2) | |
9 | 8 | elin2 4158 | . 2 ⊢ (𝐾 ∈ Fld ↔ (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ Com2)) |
10 | iscrngo 36458 | . 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 397 ∈ wcel 2107 ∖ cdif 3908 {csn 4587 × cxp 5632 ran crn 5635 ↾ cres 5636 ‘cfv 6497 1st c1st 7920 2nd c2nd 7921 GrpOpcgr 29434 GIdcgi 29435 RingOpscrngo 36356 DivRingOpscdrng 36410 Com2ccm2 36451 Fldcfld 36453 CRingOpsccring 36455 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-iota 6449 df-fun 6499 df-fv 6505 df-1st 7922 df-2nd 7923 df-drngo 36411 df-fld 36454 df-crngo 36456 |
This theorem is referenced by: isfld2 36467 isfldidl 36530 |
Copyright terms: Public domain | W3C validator |