Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > extdggt0 | Structured version Visualization version GIF version |
Description: Degrees of field extension are greater than zero. (Contributed by Thierry Arnoux, 30-Jul-2023.) |
Ref | Expression |
---|---|
extdggt0 | ⊢ (𝐸/FldExt𝐹 → 0 < (𝐸[:]𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fldextfld1 31049 | . . . . 5 ⊢ (𝐸/FldExt𝐹 → 𝐸 ∈ Field) | |
2 | isfld 19494 | . . . . . 6 ⊢ (𝐸 ∈ Field ↔ (𝐸 ∈ DivRing ∧ 𝐸 ∈ CRing)) | |
3 | 2 | simplbi 500 | . . . . 5 ⊢ (𝐸 ∈ Field → 𝐸 ∈ DivRing) |
4 | 1, 3 | syl 17 | . . . 4 ⊢ (𝐸/FldExt𝐹 → 𝐸 ∈ DivRing) |
5 | fldextress 31052 | . . . . 5 ⊢ (𝐸/FldExt𝐹 → 𝐹 = (𝐸 ↾s (Base‘𝐹))) | |
6 | fldextfld2 31050 | . . . . . 6 ⊢ (𝐸/FldExt𝐹 → 𝐹 ∈ Field) | |
7 | isfld 19494 | . . . . . . 7 ⊢ (𝐹 ∈ Field ↔ (𝐹 ∈ DivRing ∧ 𝐹 ∈ CRing)) | |
8 | 7 | simplbi 500 | . . . . . 6 ⊢ (𝐹 ∈ Field → 𝐹 ∈ DivRing) |
9 | 6, 8 | syl 17 | . . . . 5 ⊢ (𝐸/FldExt𝐹 → 𝐹 ∈ DivRing) |
10 | 5, 9 | eqeltrrd 2914 | . . . 4 ⊢ (𝐸/FldExt𝐹 → (𝐸 ↾s (Base‘𝐹)) ∈ DivRing) |
11 | eqid 2821 | . . . . 5 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
12 | 11 | fldextsubrg 31051 | . . . 4 ⊢ (𝐸/FldExt𝐹 → (Base‘𝐹) ∈ (SubRing‘𝐸)) |
13 | eqid 2821 | . . . . 5 ⊢ ((subringAlg ‘𝐸)‘(Base‘𝐹)) = ((subringAlg ‘𝐸)‘(Base‘𝐹)) | |
14 | eqid 2821 | . . . . 5 ⊢ (𝐸 ↾s (Base‘𝐹)) = (𝐸 ↾s (Base‘𝐹)) | |
15 | 13, 14 | sralvec 31000 | . . . 4 ⊢ ((𝐸 ∈ DivRing ∧ (𝐸 ↾s (Base‘𝐹)) ∈ DivRing ∧ (Base‘𝐹) ∈ (SubRing‘𝐸)) → ((subringAlg ‘𝐸)‘(Base‘𝐹)) ∈ LVec) |
16 | 4, 10, 12, 15 | syl3anc 1367 | . . 3 ⊢ (𝐸/FldExt𝐹 → ((subringAlg ‘𝐸)‘(Base‘𝐹)) ∈ LVec) |
17 | eqid 2821 | . . . . . 6 ⊢ (Base‘𝐸) = (Base‘𝐸) | |
18 | 17 | subrgss 19519 | . . . . 5 ⊢ ((Base‘𝐹) ∈ (SubRing‘𝐸) → (Base‘𝐹) ⊆ (Base‘𝐸)) |
19 | 12, 18 | syl 17 | . . . 4 ⊢ (𝐸/FldExt𝐹 → (Base‘𝐹) ⊆ (Base‘𝐸)) |
20 | 13, 17 | sradrng 30998 | . . . 4 ⊢ ((𝐸 ∈ DivRing ∧ (Base‘𝐹) ⊆ (Base‘𝐸)) → ((subringAlg ‘𝐸)‘(Base‘𝐹)) ∈ DivRing) |
21 | 4, 19, 20 | syl2anc 586 | . . 3 ⊢ (𝐸/FldExt𝐹 → ((subringAlg ‘𝐸)‘(Base‘𝐹)) ∈ DivRing) |
22 | drngdimgt0 31026 | . . 3 ⊢ ((((subringAlg ‘𝐸)‘(Base‘𝐹)) ∈ LVec ∧ ((subringAlg ‘𝐸)‘(Base‘𝐹)) ∈ DivRing) → 0 < (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) | |
23 | 16, 21, 22 | syl2anc 586 | . 2 ⊢ (𝐸/FldExt𝐹 → 0 < (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) |
24 | extdgval 31054 | . 2 ⊢ (𝐸/FldExt𝐹 → (𝐸[:]𝐹) = (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) | |
25 | 23, 24 | breqtrrd 5080 | 1 ⊢ (𝐸/FldExt𝐹 → 0 < (𝐸[:]𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ⊆ wss 3924 class class class wbr 5052 ‘cfv 6341 (class class class)co 7142 0cc0 10523 < clt 10661 Basecbs 16466 ↾s cress 16467 CRingccrg 19281 DivRingcdr 19485 Fieldcfield 19486 SubRingcsubrg 19514 LVecclvec 19857 subringAlg csra 19923 dimcldim 31009 /FldExtcfldext 31038 [:]cextdg 31041 |
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-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-reg 9042 ax-inf2 9090 ax-ac2 9871 ax-cnex 10579 ax-resscn 10580 ax-1cn 10581 ax-icn 10582 ax-addcl 10583 ax-addrcl 10584 ax-mulcl 10585 ax-mulrcl 10586 ax-mulcom 10587 ax-addass 10588 ax-mulass 10589 ax-distr 10590 ax-i2m1 10591 ax-1ne0 10592 ax-1rid 10593 ax-rnegex 10594 ax-rrecex 10595 ax-cnre 10596 ax-pre-lttri 10597 ax-pre-lttrn 10598 ax-pre-ltadd 10599 ax-pre-mulgt0 10600 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 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-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-int 4863 df-iun 4907 df-iin 4908 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-se 5501 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-isom 6350 df-riota 7100 df-ov 7145 df-oprab 7146 df-mpo 7147 df-rpss 7435 df-om 7567 df-1st 7675 df-2nd 7676 df-tpos 7878 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-1o 8088 df-oadd 8092 df-er 8275 df-map 8394 df-en 8496 df-dom 8497 df-sdom 8498 df-fin 8499 df-oi 8960 df-r1 9179 df-rank 9180 df-dju 9316 df-card 9354 df-acn 9357 df-ac 9528 df-pnf 10663 df-mnf 10664 df-xr 10665 df-ltxr 10666 df-le 10667 df-sub 10858 df-neg 10859 df-nn 11625 df-2 11687 df-3 11688 df-4 11689 df-5 11690 df-6 11691 df-7 11692 df-8 11693 df-9 11694 df-n0 11885 df-xnn0 11955 df-z 11969 df-dec 12086 df-uz 12231 df-fz 12883 df-hash 13681 df-struct 16468 df-ndx 16469 df-slot 16470 df-base 16472 df-sets 16473 df-ress 16474 df-plusg 16561 df-mulr 16562 df-sca 16564 df-vsca 16565 df-ip 16566 df-tset 16567 df-ple 16568 df-ocomp 16569 df-0g 16698 df-mre 16840 df-mrc 16841 df-mri 16842 df-acs 16843 df-proset 17521 df-drs 17522 df-poset 17539 df-ipo 17745 df-mgm 17835 df-sgrp 17884 df-mnd 17895 df-submnd 17940 df-grp 18089 df-minusg 18090 df-sbg 18091 df-subg 18259 df-cmn 18891 df-abl 18892 df-mgp 19223 df-ur 19235 df-ring 19282 df-oppr 19356 df-dvdsr 19374 df-unit 19375 df-invr 19405 df-drng 19487 df-field 19488 df-subrg 19516 df-lmod 19619 df-lss 19687 df-lsp 19727 df-lbs 19830 df-lvec 19858 df-sra 19927 df-nzr 20014 df-lindf 20933 df-linds 20934 df-dim 31010 df-fldext 31042 df-extdg 31043 |
This theorem is referenced by: finexttrb 31062 |
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