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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 5761 | . . 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 5142 | . . . . 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 33239 | . . . 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 2837 | . . 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 3975 | . 2 β’ (πΈ/FldExtπΉ β β¨πΈ, πΉβ© β (Field Γ Field)) |
7 | opelxp1 5711 | . 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 395 = wceq 1533 β wcel 2098 β¨cop 4629 class class class wbr 5141 {copab 5203 Γ cxp 5667 βcfv 6537 (class class class)co 7405 Basecbs 17153 βΎs cress 17182 SubRingcsubrg 20469 Fieldcfield 20588 /FldExtcfldext 33235 |
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 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-br 5142 df-opab 5204 df-xp 5675 df-fldext 33239 |
This theorem is referenced by: fldextsubrg 33248 fldextress 33249 brfinext 33250 fldextsralvec 33252 extdgcl 33253 extdggt0 33254 fldexttr 33255 extdgmul 33258 extdg1id 33260 |
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