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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fldextsubrg | Structured version Visualization version GIF version |
Description: Field extension implies a subring relation. (Contributed by Thierry Arnoux, 29-Jul-2023.) |
Ref | Expression |
---|---|
fldextsubrg.1 | ⊢ 𝑈 = (Base‘𝐹) |
Ref | Expression |
---|---|
fldextsubrg | ⊢ (𝐸/FldExt𝐹 → 𝑈 ∈ (SubRing‘𝐸)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fldextsubrg.1 | . 2 ⊢ 𝑈 = (Base‘𝐹) | |
2 | fldextfld1 32395 | . . . . 5 ⊢ (𝐸/FldExt𝐹 → 𝐸 ∈ Field) | |
3 | fldextfld2 32396 | . . . . 5 ⊢ (𝐸/FldExt𝐹 → 𝐹 ∈ Field) | |
4 | brfldext 32393 | . . . . 5 ⊢ ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) → (𝐸/FldExt𝐹 ↔ (𝐹 = (𝐸 ↾s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸)))) | |
5 | 2, 3, 4 | syl2anc 585 | . . . 4 ⊢ (𝐸/FldExt𝐹 → (𝐸/FldExt𝐹 ↔ (𝐹 = (𝐸 ↾s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸)))) |
6 | 5 | ibi 267 | . . 3 ⊢ (𝐸/FldExt𝐹 → (𝐹 = (𝐸 ↾s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸))) |
7 | 6 | simprd 497 | . 2 ⊢ (𝐸/FldExt𝐹 → (Base‘𝐹) ∈ (SubRing‘𝐸)) |
8 | 1, 7 | eqeltrid 2838 | 1 ⊢ (𝐸/FldExt𝐹 → 𝑈 ∈ (SubRing‘𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 class class class wbr 5106 ‘cfv 6497 (class class class)co 7358 Basecbs 17088 ↾s 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-uni 4867 df-br 5107 df-opab 5169 df-xp 5640 df-iota 6449 df-fv 6505 df-ov 7361 df-fldext 32388 |
This theorem is referenced by: fldextsralvec 32401 extdgcl 32402 extdggt0 32403 extdgmul 32407 extdg1id 32409 fldextchr 32411 |
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