Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
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 5629 | . . 3 ⊢ {〈𝑒, 𝑓〉 ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 ↾s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))} ⊆ (Field × Field) | |
2 | df-br 5053 | . . . . 5 ⊢ (𝐸/FldExt𝐹 ↔ 〈𝐸, 𝐹〉 ∈ /FldExt) | |
3 | 2 | biimpi 218 | . . . 4 ⊢ (𝐸/FldExt𝐹 → 〈𝐸, 𝐹〉 ∈ /FldExt) |
4 | df-fldext 31042 | . . . 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 3953 | . 2 ⊢ (𝐸/FldExt𝐹 → 〈𝐸, 𝐹〉 ∈ (Field × Field)) |
7 | opelxp1 5582 | . 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 4559 class class class wbr 5052 {copab 5114 × cxp 5539 ‘cfv 6341 (class class class)co 7142 Basecbs 16466 ↾s cress 16467 Fieldcfield 19486 SubRingcsubrg 19514 /FldExtcfldext 31038 |
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 5189 ax-nul 5196 ax-pr 5316 |
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 3488 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-sn 4554 df-pr 4556 df-op 4560 df-br 5053 df-opab 5115 df-xp 5547 df-fldext 31042 |
This theorem is referenced by: fldextsubrg 31051 fldextress 31052 brfinext 31053 fldextsralvec 31055 extdgcl 31056 extdggt0 31057 fldexttr 31058 extdgmul 31061 extdg1id 31063 |
Copyright terms: Public domain | W3C validator |