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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 5765 | . . 3 ⊢ {〈𝑒, 𝑓〉 ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 ↾s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))} ⊆ (Field × Field) | |
2 | df-br 5145 | . . . . 5 ⊢ (𝐸/FldExt𝐹 ↔ 〈𝐸, 𝐹〉 ∈ /FldExt) | |
3 | 2 | biimpi 215 | . . . 4 ⊢ (𝐸/FldExt𝐹 → 〈𝐸, 𝐹〉 ∈ /FldExt) |
4 | df-fldext 33535 | . . . 4 ⊢ /FldExt = {〈𝑒, 𝑓〉 ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 ↾s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))} | |
5 | 3, 4 | eleqtrdi 2836 | . . 3 ⊢ (𝐸/FldExt𝐹 → 〈𝐸, 𝐹〉 ∈ {〈𝑒, 𝑓〉 ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 ↾s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))}) |
6 | 1, 5 | sselid 3977 | . 2 ⊢ (𝐸/FldExt𝐹 → 〈𝐸, 𝐹〉 ∈ (Field × Field)) |
7 | opelxp2 5716 | . 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 394 = wceq 1534 ∈ wcel 2099 〈cop 4630 class class class wbr 5144 {copab 5206 × cxp 5671 ‘cfv 6544 (class class class)co 7414 Basecbs 17206 ↾s cress 17235 SubRingcsubrg 20545 Fieldcfield 20702 /FldExtcfldext 33531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 ax-sep 5295 ax-nul 5302 ax-pr 5424 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ral 3052 df-rex 3061 df-rab 3421 df-v 3465 df-dif 3950 df-un 3952 df-ss 3964 df-nul 4324 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5145 df-opab 5207 df-xp 5679 df-fldext 33535 |
This theorem is referenced by: fldextsubrg 33544 fldextress 33545 brfinext 33546 fldextsralvec 33548 extdgcl 33549 extdggt0 33550 fldexttr 33551 extdgmul 33554 extdg1id 33556 extdg1b 33557 |
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