Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fldextfld1 | Structured version Visualization version GIF version |
Description: A field extension is only defined if the extension is a field. (Contributed by Thierry Arnoux, 29-Jul-2023.) |
Ref | Expression |
---|---|
fldextfld1 | ⊢ (𝐸/FldExt𝐹 → 𝐸 ∈ Field) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opabssxp 5643 | . . 3 ⊢ {〈𝑒, 𝑓〉 ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 ↾s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))} ⊆ (Field × Field) | |
2 | df-br 5067 | . . . . 5 ⊢ (𝐸/FldExt𝐹 ↔ 〈𝐸, 𝐹〉 ∈ /FldExt) | |
3 | 2 | biimpi 218 | . . . 4 ⊢ (𝐸/FldExt𝐹 → 〈𝐸, 𝐹〉 ∈ /FldExt) |
4 | df-fldext 31032 | . . . 4 ⊢ /FldExt = {〈𝑒, 𝑓〉 ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 ↾s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))} | |
5 | 3, 4 | eleqtrdi 2923 | . . 3 ⊢ (𝐸/FldExt𝐹 → 〈𝐸, 𝐹〉 ∈ {〈𝑒, 𝑓〉 ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 ↾s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))}) |
6 | 1, 5 | sseldi 3965 | . 2 ⊢ (𝐸/FldExt𝐹 → 〈𝐸, 𝐹〉 ∈ (Field × Field)) |
7 | opelxp1 5596 | . 2 ⊢ (〈𝐸, 𝐹〉 ∈ (Field × Field) → 𝐸 ∈ Field) | |
8 | 6, 7 | syl 17 | 1 ⊢ (𝐸/FldExt𝐹 → 𝐸 ∈ Field) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 〈cop 4573 class class class wbr 5066 {copab 5128 × cxp 5553 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 ↾s cress 16484 Fieldcfield 19503 SubRingcsubrg 19531 /FldExtcfldext 31028 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-xp 5561 df-fldext 31032 |
This theorem is referenced by: fldextsubrg 31041 fldextress 31042 brfinext 31043 fldextsralvec 31045 extdgcl 31046 extdggt0 31047 fldexttr 31048 extdgmul 31051 extdg1id 31053 |
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