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 5629 | . . 3 ⊢ {〈𝑒, 𝑓〉 ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 ↾s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))} ⊆ (Field × Field) | |
2 | df-br 5044 | . . . . 5 ⊢ (𝐸/FldExt𝐹 ↔ 〈𝐸, 𝐹〉 ∈ /FldExt) | |
3 | 2 | biimpi 219 | . . . 4 ⊢ (𝐸/FldExt𝐹 → 〈𝐸, 𝐹〉 ∈ /FldExt) |
4 | df-fldext 31403 | . . . 4 ⊢ /FldExt = {〈𝑒, 𝑓〉 ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 ↾s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))} | |
5 | 3, 4 | eleqtrdi 2844 | . . 3 ⊢ (𝐸/FldExt𝐹 → 〈𝐸, 𝐹〉 ∈ {〈𝑒, 𝑓〉 ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 ↾s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))}) |
6 | 1, 5 | sseldi 3889 | . 2 ⊢ (𝐸/FldExt𝐹 → 〈𝐸, 𝐹〉 ∈ (Field × Field)) |
7 | opelxp2 5582 | . 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 399 = wceq 1543 ∈ wcel 2110 〈cop 4537 class class class wbr 5043 {copab 5105 × cxp 5538 ‘cfv 6369 (class class class)co 7202 Basecbs 16684 ↾s cress 16685 Fieldcfield 19740 SubRingcsubrg 19768 /FldExtcfldext 31399 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pr 5311 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2713 df-cleq 2726 df-clel 2812 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-sn 4532 df-pr 4534 df-op 4538 df-br 5044 df-opab 5106 df-xp 5546 df-fldext 31403 |
This theorem is referenced by: fldextsubrg 31412 fldextress 31413 brfinext 31414 fldextsralvec 31416 extdgcl 31417 extdggt0 31418 fldexttr 31419 extdgmul 31422 extdg1id 31424 extdg1b 31425 |
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