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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 36479 | . . 3 ⊢ (𝐾 ∈ DivRingOps → 𝐾 ∈ PrRing) | |
2 | 1 | anim1i 615 | . 2 ⊢ ((𝐾 ∈ DivRingOps ∧ 𝐾 ∈ CRingOps) → (𝐾 ∈ PrRing ∧ 𝐾 ∈ CRingOps)) |
3 | isfld2 36431 | . 2 ⊢ (𝐾 ∈ Fld ↔ (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ CRingOps)) | |
4 | isdmn2 36481 | . 2 ⊢ (𝐾 ∈ Dmn ↔ (𝐾 ∈ PrRing ∧ 𝐾 ∈ CRingOps)) | |
5 | 2, 3, 4 | 3imtr4i 291 | 1 ⊢ (𝐾 ∈ Fld → 𝐾 ∈ Dmn) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2106 DivRingOpscdrng 36374 Fldcfld 36417 CRingOpsccring 36419 PrRingcprrng 36472 Dmncdmn 36473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7309 df-ov 7356 df-1st 7917 df-2nd 7918 df-1o 8408 df-en 8880 df-grpo 29321 df-gid 29322 df-ginv 29323 df-ablo 29373 df-ass 36269 df-exid 36271 df-mgmOLD 36275 df-sgrOLD 36287 df-mndo 36293 df-rngo 36321 df-drngo 36375 df-fld 36418 df-crngo 36420 df-idl 36436 df-pridl 36437 df-prrngo 36474 df-dmn 36475 |
This theorem is referenced by: (None) |
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