|   | Mathbox for Jeff Madsen | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > Mathboxes > isfldidl2 | Structured version Visualization version GIF version | ||
| Description: Determine if a ring is a field based on its ideals. (Contributed by Jeff Madsen, 6-Jan-2011.) | 
| Ref | Expression | 
|---|---|
| isfldidl2.1 | ⊢ 𝐺 = (1st ‘𝐾) | 
| isfldidl2.2 | ⊢ 𝐻 = (2nd ‘𝐾) | 
| isfldidl2.3 | ⊢ 𝑋 = ran 𝐺 | 
| isfldidl2.4 | ⊢ 𝑍 = (GId‘𝐺) | 
| Ref | Expression | 
|---|---|
| isfldidl2 | ⊢ (𝐾 ∈ Fld ↔ (𝐾 ∈ CRingOps ∧ 𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋})) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | isfldidl2.1 | . . 3 ⊢ 𝐺 = (1st ‘𝐾) | |
| 2 | isfldidl2.2 | . . 3 ⊢ 𝐻 = (2nd ‘𝐾) | |
| 3 | isfldidl2.3 | . . 3 ⊢ 𝑋 = ran 𝐺 | |
| 4 | isfldidl2.4 | . . 3 ⊢ 𝑍 = (GId‘𝐺) | |
| 5 | eqid 2736 | . . 3 ⊢ (GId‘𝐻) = (GId‘𝐻) | |
| 6 | 1, 2, 3, 4, 5 | isfldidl 38076 | . 2 ⊢ (𝐾 ∈ Fld ↔ (𝐾 ∈ CRingOps ∧ (GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋})) | 
| 7 | crngorngo 38008 | . . . . . . 7 ⊢ (𝐾 ∈ CRingOps → 𝐾 ∈ RingOps) | |
| 8 | eqcom 2743 | . . . . . . . 8 ⊢ ((GId‘𝐻) = 𝑍 ↔ 𝑍 = (GId‘𝐻)) | |
| 9 | 1, 2, 3, 4, 5 | 0rngo 38035 | . . . . . . . 8 ⊢ (𝐾 ∈ RingOps → (𝑍 = (GId‘𝐻) ↔ 𝑋 = {𝑍})) | 
| 10 | 8, 9 | bitrid 283 | . . . . . . 7 ⊢ (𝐾 ∈ RingOps → ((GId‘𝐻) = 𝑍 ↔ 𝑋 = {𝑍})) | 
| 11 | 7, 10 | syl 17 | . . . . . 6 ⊢ (𝐾 ∈ CRingOps → ((GId‘𝐻) = 𝑍 ↔ 𝑋 = {𝑍})) | 
| 12 | 11 | necon3bid 2984 | . . . . 5 ⊢ (𝐾 ∈ CRingOps → ((GId‘𝐻) ≠ 𝑍 ↔ 𝑋 ≠ {𝑍})) | 
| 13 | 12 | anbi1d 631 | . . . 4 ⊢ (𝐾 ∈ CRingOps → (((GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ↔ (𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))) | 
| 14 | 13 | pm5.32i 574 | . . 3 ⊢ ((𝐾 ∈ CRingOps ∧ ((GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋})) ↔ (𝐾 ∈ CRingOps ∧ (𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))) | 
| 15 | 3anass 1094 | . . 3 ⊢ ((𝐾 ∈ CRingOps ∧ (GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ↔ (𝐾 ∈ CRingOps ∧ ((GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))) | |
| 16 | 3anass 1094 | . . 3 ⊢ ((𝐾 ∈ CRingOps ∧ 𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ↔ (𝐾 ∈ CRingOps ∧ (𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))) | |
| 17 | 14, 15, 16 | 3bitr4i 303 | . 2 ⊢ ((𝐾 ∈ CRingOps ∧ (GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ↔ (𝐾 ∈ CRingOps ∧ 𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋})) | 
| 18 | 6, 17 | bitri 275 | 1 ⊢ (𝐾 ∈ Fld ↔ (𝐾 ∈ CRingOps ∧ 𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋})) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 {csn 4625 {cpr 4627 ran crn 5685 ‘cfv 6560 1st c1st 8013 2nd c2nd 8014 GIdcgi 30510 RingOpscrngo 37902 Fldcfld 37999 CRingOpsccring 38001 Idlcidl 38015 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-1st 8015 df-2nd 8016 df-1o 8507 df-en 8987 df-grpo 30513 df-gid 30514 df-ginv 30515 df-ablo 30565 df-ass 37851 df-exid 37853 df-mgmOLD 37857 df-sgrOLD 37869 df-mndo 37875 df-rngo 37903 df-drngo 37957 df-com2 37998 df-fld 38000 df-crngo 38002 df-idl 38018 df-igen 38068 | 
| This theorem is referenced by: (None) | 
| Copyright terms: Public domain | W3C validator |