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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > extdgid | Structured version Visualization version GIF version |
Description: A trivial field extension has degree one. (Contributed by Thierry Arnoux, 4-Aug-2023.) |
Ref | Expression |
---|---|
extdgid | โข (๐ธ โ Field โ (๐ธ[:]๐ธ) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fldextid 32733 | . . 3 โข (๐ธ โ Field โ ๐ธ/FldExt๐ธ) | |
2 | extdgval 32728 | . . 3 โข (๐ธ/FldExt๐ธ โ (๐ธ[:]๐ธ) = (dimโ((subringAlg โ๐ธ)โ(Baseโ๐ธ)))) | |
3 | 1, 2 | syl 17 | . 2 โข (๐ธ โ Field โ (๐ธ[:]๐ธ) = (dimโ((subringAlg โ๐ธ)โ(Baseโ๐ธ)))) |
4 | isfld 20367 | . . . 4 โข (๐ธ โ Field โ (๐ธ โ DivRing โง ๐ธ โ CRing)) | |
5 | 4 | simplbi 498 | . . 3 โข (๐ธ โ Field โ ๐ธ โ DivRing) |
6 | rlmval 20812 | . . . . 5 โข (ringLModโ๐ธ) = ((subringAlg โ๐ธ)โ(Baseโ๐ธ)) | |
7 | 6 | eqcomi 2741 | . . . 4 โข ((subringAlg โ๐ธ)โ(Baseโ๐ธ)) = (ringLModโ๐ธ) |
8 | 7 | rlmdim 32689 | . . 3 โข (๐ธ โ DivRing โ (dimโ((subringAlg โ๐ธ)โ(Baseโ๐ธ))) = 1) |
9 | 5, 8 | syl 17 | . 2 โข (๐ธ โ Field โ (dimโ((subringAlg โ๐ธ)โ(Baseโ๐ธ))) = 1) |
10 | 3, 9 | eqtrd 2772 | 1 โข (๐ธ โ Field โ (๐ธ[:]๐ธ) = 1) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1541 โ wcel 2106 class class class wbr 5148 โcfv 6543 (class class class)co 7408 1c1 11110 Basecbs 17143 CRingccrg 20056 DivRingcdr 20356 Fieldcfield 20357 subringAlg csra 20780 ringLModcrglmod 20781 dimcldim 32679 /FldExtcfldext 32712 [:]cextdg 32715 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-reg 9586 ax-inf2 9635 ax-ac2 10457 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-tpos 8210 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-oi 9504 df-r1 9758 df-rank 9759 df-card 9933 df-acn 9936 df-ac 10110 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-xnn0 12544 df-z 12558 df-dec 12677 df-uz 12822 df-fz 13484 df-hash 14290 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ocomp 17217 df-0g 17386 df-mre 17529 df-mrc 17530 df-mri 17531 df-acs 17532 df-proset 18247 df-drs 18248 df-poset 18265 df-ipo 18480 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-submnd 18671 df-grp 18821 df-minusg 18822 df-sbg 18823 df-subg 19002 df-cmn 19649 df-abl 19650 df-mgp 19987 df-ur 20004 df-ring 20057 df-oppr 20149 df-dvdsr 20170 df-unit 20171 df-invr 20201 df-subrg 20316 df-drng 20358 df-field 20359 df-lmod 20472 df-lss 20542 df-lsp 20582 df-lbs 20685 df-lvec 20713 df-sra 20784 df-rgmod 20785 df-lidl 20786 df-rsp 20787 df-lindf 21360 df-linds 21361 df-dim 32680 df-fldext 32716 df-extdg 32717 |
This theorem is referenced by: extdg1b 32738 |
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