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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 33700 | . . . . 5 ⊢ (𝐸/FldExt𝐹 → 𝐸 ∈ Field) | |
| 3 | fldextfld2 33701 | . . . . 5 ⊢ (𝐸/FldExt𝐹 → 𝐹 ∈ Field) | |
| 4 | brfldext 33698 | . . . . 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 2845 | 1 ⊢ (𝐸/FldExt𝐹 → 𝑈 ∈ (SubRing‘𝐸)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 ↾s cress 17274 SubRingcsubrg 20569 Fieldcfield 20730 /FldExtcfldext 33689 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-iota 6514 df-fv 6569 df-ov 7434 df-fldext 33693 |
| This theorem is referenced by: fldextsralvec 33706 extdgcl 33707 extdggt0 33708 extdgmul 33714 extdg1id 33716 fldextchr 33719 |
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