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Mirrors > Home > MPE Home > Th. List > Mathboxes > flddmn | Structured version Visualization version GIF version |
Description: A field is a domain. (Contributed by Jeff Madsen, 10-Jun-2010.) |
Ref | Expression |
---|---|
flddmn | ⊢ (𝐾 ∈ Fld → 𝐾 ∈ Dmn) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | divrngpr 35325 | . . 3 ⊢ (𝐾 ∈ DivRingOps → 𝐾 ∈ PrRing) | |
2 | 1 | anim1i 616 | . 2 ⊢ ((𝐾 ∈ DivRingOps ∧ 𝐾 ∈ CRingOps) → (𝐾 ∈ PrRing ∧ 𝐾 ∈ CRingOps)) |
3 | isfld2 35277 | . 2 ⊢ (𝐾 ∈ Fld ↔ (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ CRingOps)) | |
4 | isdmn2 35327 | . 2 ⊢ (𝐾 ∈ Dmn ↔ (𝐾 ∈ PrRing ∧ 𝐾 ∈ CRingOps)) | |
5 | 2, 3, 4 | 3imtr4i 294 | 1 ⊢ (𝐾 ∈ Fld → 𝐾 ∈ Dmn) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2110 DivRingOpscdrng 35220 Fldcfld 35263 CRingOpsccring 35265 PrRingcprrng 35318 Dmncdmn 35319 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-om 7575 df-1st 7683 df-2nd 7684 df-1o 8096 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-grpo 28264 df-gid 28265 df-ginv 28266 df-ablo 28316 df-ass 35115 df-exid 35117 df-mgmOLD 35121 df-sgrOLD 35133 df-mndo 35139 df-rngo 35167 df-drngo 35221 df-fld 35264 df-crngo 35266 df-idl 35282 df-pridl 35283 df-prrngo 35320 df-dmn 35321 |
This theorem is referenced by: (None) |
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