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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fldextfld1 | Structured version Visualization version GIF version |
Description: A field extension is only defined if the extension is a field. (Contributed by Thierry Arnoux, 29-Jul-2023.) |
Ref | Expression |
---|---|
fldextfld1 | β’ (πΈ/FldExtπΉ β πΈ β Field) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opabssxp 5764 | . . 3 β’ {β¨π, πβ© β£ ((π β Field β§ π β Field) β§ (π = (π βΎs (Baseβπ)) β§ (Baseβπ) β (SubRingβπ)))} β (Field Γ Field) | |
2 | df-br 5144 | . . . . 5 β’ (πΈ/FldExtπΉ β β¨πΈ, πΉβ© β /FldExt) | |
3 | 2 | biimpi 215 | . . . 4 β’ (πΈ/FldExtπΉ β β¨πΈ, πΉβ© β /FldExt) |
4 | df-fldext 33391 | . . . 4 β’ /FldExt = {β¨π, πβ© β£ ((π β Field β§ π β Field) β§ (π = (π βΎs (Baseβπ)) β§ (Baseβπ) β (SubRingβπ)))} | |
5 | 3, 4 | eleqtrdi 2835 | . . 3 β’ (πΈ/FldExtπΉ β β¨πΈ, πΉβ© β {β¨π, πβ© β£ ((π β Field β§ π β Field) β§ (π = (π βΎs (Baseβπ)) β§ (Baseβπ) β (SubRingβπ)))}) |
6 | 1, 5 | sselid 3970 | . 2 β’ (πΈ/FldExtπΉ β β¨πΈ, πΉβ© β (Field Γ Field)) |
7 | opelxp1 5714 | . 2 β’ (β¨πΈ, πΉβ© β (Field Γ Field) β πΈ β Field) | |
8 | 6, 7 | syl 17 | 1 β’ (πΈ/FldExtπΉ β πΈ β Field) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 β¨cop 4630 class class class wbr 5143 {copab 5205 Γ cxp 5670 βcfv 6543 (class class class)co 7416 Basecbs 17179 βΎs cress 17208 SubRingcsubrg 20510 Fieldcfield 20629 /FldExtcfldext 33387 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3942 df-un 3944 df-ss 3956 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5144 df-opab 5206 df-xp 5678 df-fldext 33391 |
This theorem is referenced by: fldextsubrg 33400 fldextress 33401 brfinext 33402 fldextsralvec 33404 extdgcl 33405 extdggt0 33406 fldexttr 33407 extdgmul 33410 extdg1id 33412 |
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