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 2610 | . . 3 ⊢ (GId‘𝐻) = (GId‘𝐻) | |
6 | 1, 2, 3, 4, 5 | isfldidl 33037 | . 2 ⊢ (𝐾 ∈ Fld ↔ (𝐾 ∈ CRingOps ∧ (GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋})) |
7 | crngorngo 32969 | . . . . . . 7 ⊢ (𝐾 ∈ CRingOps → 𝐾 ∈ RingOps) | |
8 | eqcom 2617 | . . . . . . . 8 ⊢ ((GId‘𝐻) = 𝑍 ↔ 𝑍 = (GId‘𝐻)) | |
9 | 1, 2, 3, 4, 5 | 0rngo 32996 | . . . . . . . 8 ⊢ (𝐾 ∈ RingOps → (𝑍 = (GId‘𝐻) ↔ 𝑋 = {𝑍})) |
10 | 8, 9 | syl5bb 271 | . . . . . . 7 ⊢ (𝐾 ∈ RingOps → ((GId‘𝐻) = 𝑍 ↔ 𝑋 = {𝑍})) |
11 | 7, 10 | syl 17 | . . . . . 6 ⊢ (𝐾 ∈ CRingOps → ((GId‘𝐻) = 𝑍 ↔ 𝑋 = {𝑍})) |
12 | 11 | necon3bid 2826 | . . . . 5 ⊢ (𝐾 ∈ CRingOps → ((GId‘𝐻) ≠ 𝑍 ↔ 𝑋 ≠ {𝑍})) |
13 | 12 | anbi1d 737 | . . . 4 ⊢ (𝐾 ∈ CRingOps → (((GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ↔ (𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))) |
14 | 13 | pm5.32i 667 | . . 3 ⊢ ((𝐾 ∈ CRingOps ∧ ((GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋})) ↔ (𝐾 ∈ CRingOps ∧ (𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))) |
15 | 3anass 1035 | . . 3 ⊢ ((𝐾 ∈ CRingOps ∧ (GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ↔ (𝐾 ∈ CRingOps ∧ ((GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))) | |
16 | 3anass 1035 | . . 3 ⊢ ((𝐾 ∈ CRingOps ∧ 𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ↔ (𝐾 ∈ CRingOps ∧ (𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))) | |
17 | 14, 15, 16 | 3bitr4i 291 | . 2 ⊢ ((𝐾 ∈ CRingOps ∧ (GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ↔ (𝐾 ∈ CRingOps ∧ 𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋})) |
18 | 6, 17 | bitri 263 | 1 ⊢ (𝐾 ∈ Fld ↔ (𝐾 ∈ CRingOps ∧ 𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋})) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 {csn 4125 {cpr 4127 ran crn 5039 ‘cfv 5804 1st c1st 7057 2nd c2nd 7058 GIdcgi 26728 RingOpscrngo 32863 Fldcfld 32960 CRingOpsccring 32962 Idlcidl 32976 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-1o 7447 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-grpo 26731 df-gid 26732 df-ginv 26733 df-ablo 26783 df-ass 32812 df-exid 32814 df-mgmOLD 32818 df-sgrOLD 32830 df-mndo 32836 df-rngo 32864 df-drngo 32918 df-com2 32959 df-fld 32961 df-crngo 32963 df-idl 32979 df-igen 33029 |
This theorem is referenced by: (None) |
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