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Mathbox for Jeff Madsen |
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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 2735 | . . 3 ⊢ (GId‘𝐻) = (GId‘𝐻) | |
6 | 1, 2, 3, 4, 5 | isfldidl 38055 | . 2 ⊢ (𝐾 ∈ Fld ↔ (𝐾 ∈ CRingOps ∧ (GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋})) |
7 | crngorngo 37987 | . . . . . . 7 ⊢ (𝐾 ∈ CRingOps → 𝐾 ∈ RingOps) | |
8 | eqcom 2742 | . . . . . . . 8 ⊢ ((GId‘𝐻) = 𝑍 ↔ 𝑍 = (GId‘𝐻)) | |
9 | 1, 2, 3, 4, 5 | 0rngo 38014 | . . . . . . . 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 2983 | . . . . 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 1537 ∈ wcel 2106 ≠ wne 2938 {csn 4631 {cpr 4633 ran crn 5690 ‘cfv 6563 1st c1st 8011 2nd c2nd 8012 GIdcgi 30519 RingOpscrngo 37881 Fldcfld 37978 CRingOpsccring 37980 Idlcidl 37994 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-1o 8505 df-en 8985 df-grpo 30522 df-gid 30523 df-ginv 30524 df-ablo 30574 df-ass 37830 df-exid 37832 df-mgmOLD 37836 df-sgrOLD 37848 df-mndo 37854 df-rngo 37882 df-drngo 37936 df-com2 37977 df-fld 37979 df-crngo 37981 df-idl 37997 df-igen 38047 |
This theorem is referenced by: (None) |
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