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Mirrors > Home > MPE Home > Th. List > Mathboxes > fldextress | Structured version Visualization version GIF version |
Description: Field extension implies a structure restriction relation. (Contributed by Thierry Arnoux, 29-Jul-2023.) |
Ref | Expression |
---|---|
fldextress | ⊢ (𝐸/FldExt𝐹 → 𝐹 = (𝐸 ↾s (Base‘𝐹))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fldextfld1 31049 | . . . 4 ⊢ (𝐸/FldExt𝐹 → 𝐸 ∈ Field) | |
2 | fldextfld2 31050 | . . . 4 ⊢ (𝐸/FldExt𝐹 → 𝐹 ∈ Field) | |
3 | brfldext 31047 | . . . 4 ⊢ ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) → (𝐸/FldExt𝐹 ↔ (𝐹 = (𝐸 ↾s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸)))) | |
4 | 1, 2, 3 | syl2anc 586 | . . 3 ⊢ (𝐸/FldExt𝐹 → (𝐸/FldExt𝐹 ↔ (𝐹 = (𝐸 ↾s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸)))) |
5 | 4 | ibi 269 | . 2 ⊢ (𝐸/FldExt𝐹 → (𝐹 = (𝐸 ↾s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸))) |
6 | 5 | simpld 497 | 1 ⊢ (𝐸/FldExt𝐹 → 𝐹 = (𝐸 ↾s (Base‘𝐹))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 class class class wbr 5052 ‘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-mo 2622 df-eu 2654 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-uni 4825 df-br 5053 df-opab 5115 df-xp 5547 df-iota 6300 df-fv 6349 df-ov 7145 df-fldext 31042 |
This theorem is referenced by: fldextsralvec 31055 extdgcl 31056 extdggt0 31057 extdg1id 31063 fldextchr 31065 |
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