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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 5725 | . . 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 5107 | . . . . 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 32388 | . . . 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 2844 | . . 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 3943 | . 2 β’ (πΈ/FldExtπΉ β β¨πΈ, πΉβ© β (Field Γ Field)) |
7 | opelxp1 5675 | . 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 397 = wceq 1542 β wcel 2107 β¨cop 4593 class class class wbr 5106 {copab 5168 Γ cxp 5632 βcfv 6497 (class class class)co 7358 Basecbs 17088 βΎs cress 17117 Fieldcfield 20198 SubRingcsubrg 20232 /FldExtcfldext 32384 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-xp 5640 df-fldext 32388 |
This theorem is referenced by: fldextsubrg 32397 fldextress 32398 brfinext 32399 fldextsralvec 32401 extdgcl 32402 extdggt0 32403 fldexttr 32404 extdgmul 32407 extdg1id 32409 |
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