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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 33255 | . . 3 โข (๐ธ โ Field โ ๐ธ/FldExt๐ธ) | |
2 | extdgval 33250 | . . 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 20595 | . . . 4 โข (๐ธ โ Field โ (๐ธ โ DivRing โง ๐ธ โ CRing)) | |
5 | 4 | simplbi 497 | . . 3 โข (๐ธ โ Field โ ๐ธ โ DivRing) |
6 | rlmval 21044 | . . . . 5 โข (ringLModโ๐ธ) = ((subringAlg โ๐ธ)โ(Baseโ๐ธ)) | |
7 | 6 | eqcomi 2735 | . . . 4 โข ((subringAlg โ๐ธ)โ(Baseโ๐ธ)) = (ringLModโ๐ธ) |
8 | 7 | rlmdim 33211 | . . 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 2766 | 1 โข (๐ธ โ Field โ (๐ธ[:]๐ธ) = 1) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1533 โ wcel 2098 class class class wbr 5141 โcfv 6536 (class class class)co 7404 1c1 11110 Basecbs 17150 CRingccrg 20136 DivRingcdr 20584 Fieldcfield 20585 subringAlg csra 21016 ringLModcrglmod 21017 dimcldim 33200 /FldExtcfldext 33234 [:]cextdg 33237 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 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 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-tpos 8209 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 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 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-xnn0 12546 df-z 12560 df-dec 12679 df-uz 12824 df-fz 13488 df-hash 14293 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17151 df-ress 17180 df-plusg 17216 df-mulr 17217 df-sca 17219 df-vsca 17220 df-ip 17221 df-tset 17222 df-ple 17223 df-ocomp 17224 df-0g 17393 df-mre 17536 df-mrc 17537 df-mri 17538 df-acs 17539 df-proset 18257 df-drs 18258 df-poset 18275 df-ipo 18490 df-mgm 18570 df-sgrp 18649 df-mnd 18665 df-submnd 18711 df-grp 18863 df-minusg 18864 df-sbg 18865 df-subg 19047 df-cmn 19699 df-abl 19700 df-mgp 20037 df-rng 20055 df-ur 20084 df-ring 20137 df-oppr 20233 df-dvdsr 20256 df-unit 20257 df-invr 20287 df-subrg 20468 df-drng 20586 df-field 20587 df-lmod 20705 df-lss 20776 df-lsp 20816 df-lbs 20920 df-lvec 20948 df-sra 21018 df-rgmod 21019 df-lidl 21064 df-rsp 21065 df-lindf 21696 df-linds 21697 df-dim 33201 df-fldext 33238 df-extdg 33239 |
This theorem is referenced by: extdg1b 33260 |
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