| Mathbox for Thierry Arnoux |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > fldextfld2 | Structured version Visualization version GIF version | ||
| Description: A field extension is only defined if the subfield is a field. (Contributed by Thierry Arnoux, 29-Jul-2023.) |
| Ref | Expression |
|---|---|
| fldextfld2 | ⊢ (𝐸/FldExt𝐹 → 𝐹 ∈ Field) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | opabssxp 5778 | . . 3 ⊢ {〈𝑒, 𝑓〉 ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 ↾s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))} ⊆ (Field × Field) | |
| 2 | df-br 5144 | . . . . 5 ⊢ (𝐸/FldExt𝐹 ↔ 〈𝐸, 𝐹〉 ∈ /FldExt) | |
| 3 | 2 | biimpi 216 | . . . 4 ⊢ (𝐸/FldExt𝐹 → 〈𝐸, 𝐹〉 ∈ /FldExt) |
| 4 | df-fldext 33693 | . . . 4 ⊢ /FldExt = {〈𝑒, 𝑓〉 ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 ↾s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))} | |
| 5 | 3, 4 | eleqtrdi 2851 | . . 3 ⊢ (𝐸/FldExt𝐹 → 〈𝐸, 𝐹〉 ∈ {〈𝑒, 𝑓〉 ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 ↾s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))}) |
| 6 | 1, 5 | sselid 3981 | . 2 ⊢ (𝐸/FldExt𝐹 → 〈𝐸, 𝐹〉 ∈ (Field × Field)) |
| 7 | opelxp2 5728 | . 2 ⊢ (〈𝐸, 𝐹〉 ∈ (Field × Field) → 𝐹 ∈ Field) | |
| 8 | 6, 7 | syl 17 | 1 ⊢ (𝐸/FldExt𝐹 → 𝐹 ∈ Field) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 〈cop 4632 class class class wbr 5143 {copab 5205 × cxp 5683 ‘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-br 5144 df-opab 5206 df-xp 5691 df-fldext 33693 |
| This theorem is referenced by: fldextsubrg 33702 fldextress 33703 brfinext 33704 fldextsralvec 33706 extdgcl 33707 extdggt0 33708 fldexttr 33709 extdgmul 33714 extdg1id 33716 extdg1b 33717 |
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