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Mirrors > Home > MPE Home > Th. List > Mathboxes > fldextid | Structured version Visualization version GIF version |
Description: The field extension relation is reflexive. (Contributed by Thierry Arnoux, 30-Jul-2023.) |
Ref | Expression |
---|---|
fldextid | ⊢ (𝐹 ∈ Field → 𝐹/FldExt𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2739 | . . . 4 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
2 | 1 | ressid 16675 | . . 3 ⊢ (𝐹 ∈ Field → (𝐹 ↾s (Base‘𝐹)) = 𝐹) |
3 | 2 | eqcomd 2745 | . 2 ⊢ (𝐹 ∈ Field → 𝐹 = (𝐹 ↾s (Base‘𝐹))) |
4 | isfld 19643 | . . . 4 ⊢ (𝐹 ∈ Field ↔ (𝐹 ∈ DivRing ∧ 𝐹 ∈ CRing)) | |
5 | 4 | simplbi 501 | . . 3 ⊢ (𝐹 ∈ Field → 𝐹 ∈ DivRing) |
6 | drngring 19641 | . . 3 ⊢ (𝐹 ∈ DivRing → 𝐹 ∈ Ring) | |
7 | 1 | subrgid 19669 | . . 3 ⊢ (𝐹 ∈ Ring → (Base‘𝐹) ∈ (SubRing‘𝐹)) |
8 | 5, 6, 7 | 3syl 18 | . 2 ⊢ (𝐹 ∈ Field → (Base‘𝐹) ∈ (SubRing‘𝐹)) |
9 | brfldext 31307 | . . 3 ⊢ ((𝐹 ∈ Field ∧ 𝐹 ∈ Field) → (𝐹/FldExt𝐹 ↔ (𝐹 = (𝐹 ↾s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐹)))) | |
10 | 9 | anidms 570 | . 2 ⊢ (𝐹 ∈ Field → (𝐹/FldExt𝐹 ↔ (𝐹 = (𝐹 ↾s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐹)))) |
11 | 3, 8, 10 | mpbir2and 713 | 1 ⊢ (𝐹 ∈ Field → 𝐹/FldExt𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 class class class wbr 5040 ‘cfv 6350 (class class class)co 7183 Basecbs 16599 ↾s cress 16600 Ringcrg 19429 CRingccrg 19430 DivRingcdr 19634 Fieldcfield 19635 SubRingcsubrg 19663 /FldExtcfldext 31298 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7492 ax-cnex 10684 ax-resscn 10685 ax-1cn 10686 ax-icn 10687 ax-addcl 10688 ax-addrcl 10689 ax-mulcl 10690 ax-mulrcl 10691 ax-mulcom 10692 ax-addass 10693 ax-mulass 10694 ax-distr 10695 ax-i2m1 10696 ax-1ne0 10697 ax-1rid 10698 ax-rnegex 10699 ax-rrecex 10700 ax-cnre 10701 ax-pre-lttri 10702 ax-pre-lttrn 10703 ax-pre-ltadd 10704 ax-pre-mulgt0 10705 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6186 df-on 6187 df-lim 6188 df-suc 6189 df-iota 6308 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7140 df-ov 7186 df-oprab 7187 df-mpo 7188 df-om 7613 df-wrecs 7989 df-recs 8050 df-rdg 8088 df-er 8333 df-en 8569 df-dom 8570 df-sdom 8571 df-pnf 10768 df-mnf 10769 df-xr 10770 df-ltxr 10771 df-le 10772 df-sub 10963 df-neg 10964 df-nn 11730 df-2 11792 df-ndx 16602 df-slot 16603 df-base 16605 df-sets 16606 df-ress 16607 df-plusg 16694 df-0g 16831 df-mgm 17981 df-sgrp 18030 df-mnd 18041 df-mgp 19372 df-ur 19384 df-ring 19431 df-drng 19636 df-field 19637 df-subrg 19665 df-fldext 31302 |
This theorem is referenced by: extdgid 31320 |
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