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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 33677 | . . . . 5 ⊢ (𝐸/FldExt𝐹 → 𝐸 ∈ Field) | |
3 | fldextfld2 33678 | . . . . 5 ⊢ (𝐸/FldExt𝐹 → 𝐹 ∈ Field) | |
4 | brfldext 33675 | . . . . 5 ⊢ ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) → (𝐸/FldExt𝐹 ↔ (𝐹 = (𝐸 ↾s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸)))) | |
5 | 2, 3, 4 | syl2anc 584 | . . . 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 495 | . 2 ⊢ (𝐸/FldExt𝐹 → (Base‘𝐹) ∈ (SubRing‘𝐸)) |
8 | 1, 7 | eqeltrid 2843 | 1 ⊢ (𝐸/FldExt𝐹 → 𝑈 ∈ (SubRing‘𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 ↾s cress 17274 SubRingcsubrg 20586 Fieldcfield 20747 /FldExtcfldext 33666 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-xp 5695 df-iota 6516 df-fv 6571 df-ov 7434 df-fldext 33670 |
This theorem is referenced by: fldextsralvec 33683 extdgcl 33684 extdggt0 33685 extdgmul 33689 extdg1id 33691 fldextchr 33694 |
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