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 2818 | . . 3 ⊢ (GId‘𝐻) = (GId‘𝐻) | |
6 | 1, 2, 3, 4, 5 | isfldidl 35227 | . 2 ⊢ (𝐾 ∈ Fld ↔ (𝐾 ∈ CRingOps ∧ (GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋})) |
7 | crngorngo 35159 | . . . . . . 7 ⊢ (𝐾 ∈ CRingOps → 𝐾 ∈ RingOps) | |
8 | eqcom 2825 | . . . . . . . 8 ⊢ ((GId‘𝐻) = 𝑍 ↔ 𝑍 = (GId‘𝐻)) | |
9 | 1, 2, 3, 4, 5 | 0rngo 35186 | . . . . . . . 8 ⊢ (𝐾 ∈ RingOps → (𝑍 = (GId‘𝐻) ↔ 𝑋 = {𝑍})) |
10 | 8, 9 | syl5bb 284 | . . . . . . 7 ⊢ (𝐾 ∈ RingOps → ((GId‘𝐻) = 𝑍 ↔ 𝑋 = {𝑍})) |
11 | 7, 10 | syl 17 | . . . . . 6 ⊢ (𝐾 ∈ CRingOps → ((GId‘𝐻) = 𝑍 ↔ 𝑋 = {𝑍})) |
12 | 11 | necon3bid 3057 | . . . . 5 ⊢ (𝐾 ∈ CRingOps → ((GId‘𝐻) ≠ 𝑍 ↔ 𝑋 ≠ {𝑍})) |
13 | 12 | anbi1d 629 | . . . 4 ⊢ (𝐾 ∈ CRingOps → (((GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ↔ (𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))) |
14 | 13 | pm5.32i 575 | . . 3 ⊢ ((𝐾 ∈ CRingOps ∧ ((GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋})) ↔ (𝐾 ∈ CRingOps ∧ (𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))) |
15 | 3anass 1087 | . . 3 ⊢ ((𝐾 ∈ CRingOps ∧ (GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ↔ (𝐾 ∈ CRingOps ∧ ((GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))) | |
16 | 3anass 1087 | . . 3 ⊢ ((𝐾 ∈ CRingOps ∧ 𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ↔ (𝐾 ∈ CRingOps ∧ (𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))) | |
17 | 14, 15, 16 | 3bitr4i 304 | . 2 ⊢ ((𝐾 ∈ CRingOps ∧ (GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ↔ (𝐾 ∈ CRingOps ∧ 𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋})) |
18 | 6, 17 | bitri 276 | 1 ⊢ (𝐾 ∈ Fld ↔ (𝐾 ∈ CRingOps ∧ 𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋})) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 {csn 4557 {cpr 4559 ran crn 5549 ‘cfv 6348 1st c1st 7676 2nd c2nd 7677 GIdcgi 28194 RingOpscrngo 35053 Fldcfld 35150 CRingOpsccring 35152 Idlcidl 35166 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-1o 8091 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-grpo 28197 df-gid 28198 df-ginv 28199 df-ablo 28249 df-ass 35002 df-exid 35004 df-mgmOLD 35008 df-sgrOLD 35020 df-mndo 35026 df-rngo 35054 df-drngo 35108 df-com2 35149 df-fld 35151 df-crngo 35153 df-idl 35169 df-igen 35219 |
This theorem is referenced by: (None) |
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