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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 31061 | . . . 4 ⊢ (𝐸/FldExt𝐹 → 𝐸 ∈ Field) | |
2 | fldextfld2 31062 | . . . 4 ⊢ (𝐸/FldExt𝐹 → 𝐹 ∈ Field) | |
3 | brfldext 31059 | . . . 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 1536 ∈ wcel 2113 class class class wbr 5059 ‘cfv 6348 (class class class)co 7149 Basecbs 16476 ↾s cress 16477 Fieldcfield 19496 SubRingcsubrg 19524 /FldExtcfldext 31050 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-xp 5554 df-iota 6307 df-fv 6356 df-ov 7152 df-fldext 31054 |
This theorem is referenced by: fldextsralvec 31067 extdgcl 31068 extdggt0 31069 extdg1id 31075 fldextchr 31077 |
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