Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rgmoddim | Structured version Visualization version GIF version |
Description: The left vector space induced by a ring over itself has dimension 1. (Contributed by Thierry Arnoux, 5-Aug-2023.) |
Ref | Expression |
---|---|
rgmoddim.1 | ⊢ 𝑉 = (ringLMod‘𝐹) |
Ref | Expression |
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rgmoddim | ⊢ (𝐹 ∈ Field → (dim‘𝑉) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfld 19776 | . . . . 5 ⊢ (𝐹 ∈ Field ↔ (𝐹 ∈ DivRing ∧ 𝐹 ∈ CRing)) | |
2 | 1 | simplbi 501 | . . . 4 ⊢ (𝐹 ∈ Field → 𝐹 ∈ DivRing) |
3 | eqid 2737 | . . . . . 6 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
4 | 3 | ressid 16796 | . . . . 5 ⊢ (𝐹 ∈ Field → (𝐹 ↾s (Base‘𝐹)) = 𝐹) |
5 | 4, 2 | eqeltrd 2838 | . . . 4 ⊢ (𝐹 ∈ Field → (𝐹 ↾s (Base‘𝐹)) ∈ DivRing) |
6 | drngring 19774 | . . . . 5 ⊢ (𝐹 ∈ DivRing → 𝐹 ∈ Ring) | |
7 | 3 | subrgid 19802 | . . . . 5 ⊢ (𝐹 ∈ Ring → (Base‘𝐹) ∈ (SubRing‘𝐹)) |
8 | 2, 6, 7 | 3syl 18 | . . . 4 ⊢ (𝐹 ∈ Field → (Base‘𝐹) ∈ (SubRing‘𝐹)) |
9 | rgmoddim.1 | . . . . . 6 ⊢ 𝑉 = (ringLMod‘𝐹) | |
10 | rlmval 20228 | . . . . . 6 ⊢ (ringLMod‘𝐹) = ((subringAlg ‘𝐹)‘(Base‘𝐹)) | |
11 | 9, 10 | eqtri 2765 | . . . . 5 ⊢ 𝑉 = ((subringAlg ‘𝐹)‘(Base‘𝐹)) |
12 | eqid 2737 | . . . . 5 ⊢ (𝐹 ↾s (Base‘𝐹)) = (𝐹 ↾s (Base‘𝐹)) | |
13 | 11, 12 | sralvec 31389 | . . . 4 ⊢ ((𝐹 ∈ DivRing ∧ (𝐹 ↾s (Base‘𝐹)) ∈ DivRing ∧ (Base‘𝐹) ∈ (SubRing‘𝐹)) → 𝑉 ∈ LVec) |
14 | 2, 5, 8, 13 | syl3anc 1373 | . . 3 ⊢ (𝐹 ∈ Field → 𝑉 ∈ LVec) |
15 | 2, 6 | syl 17 | . . . . . . 7 ⊢ (𝐹 ∈ Field → 𝐹 ∈ Ring) |
16 | ssidd 3924 | . . . . . . 7 ⊢ (𝐹 ∈ Field → (Base‘𝐹) ⊆ (Base‘𝐹)) | |
17 | 11, 3 | sraring 31386 | . . . . . . 7 ⊢ ((𝐹 ∈ Ring ∧ (Base‘𝐹) ⊆ (Base‘𝐹)) → 𝑉 ∈ Ring) |
18 | 15, 16, 17 | syl2anc 587 | . . . . . 6 ⊢ (𝐹 ∈ Field → 𝑉 ∈ Ring) |
19 | eqid 2737 | . . . . . . 7 ⊢ (Base‘𝑉) = (Base‘𝑉) | |
20 | eqid 2737 | . . . . . . 7 ⊢ (1r‘𝑉) = (1r‘𝑉) | |
21 | 19, 20 | ringidcl 19586 | . . . . . 6 ⊢ (𝑉 ∈ Ring → (1r‘𝑉) ∈ (Base‘𝑉)) |
22 | 18, 21 | syl 17 | . . . . 5 ⊢ (𝐹 ∈ Field → (1r‘𝑉) ∈ (Base‘𝑉)) |
23 | 11, 3 | sradrng 31387 | . . . . . . 7 ⊢ ((𝐹 ∈ DivRing ∧ (Base‘𝐹) ⊆ (Base‘𝐹)) → 𝑉 ∈ DivRing) |
24 | 2, 16, 23 | syl2anc 587 | . . . . . 6 ⊢ (𝐹 ∈ Field → 𝑉 ∈ DivRing) |
25 | eqid 2737 | . . . . . . 7 ⊢ (0g‘𝑉) = (0g‘𝑉) | |
26 | 25, 20 | drngunz 19782 | . . . . . 6 ⊢ (𝑉 ∈ DivRing → (1r‘𝑉) ≠ (0g‘𝑉)) |
27 | 24, 26 | syl 17 | . . . . 5 ⊢ (𝐹 ∈ Field → (1r‘𝑉) ≠ (0g‘𝑉)) |
28 | 19, 25 | lindssn 31287 | . . . . 5 ⊢ ((𝑉 ∈ LVec ∧ (1r‘𝑉) ∈ (Base‘𝑉) ∧ (1r‘𝑉) ≠ (0g‘𝑉)) → {(1r‘𝑉)} ∈ (LIndS‘𝑉)) |
29 | 14, 22, 27, 28 | syl3anc 1373 | . . . 4 ⊢ (𝐹 ∈ Field → {(1r‘𝑉)} ∈ (LIndS‘𝑉)) |
30 | rspval 20230 | . . . . . . . . 9 ⊢ (RSpan‘𝐹) = (LSpan‘(ringLMod‘𝐹)) | |
31 | 9 | fveq2i 6720 | . . . . . . . . 9 ⊢ (LSpan‘𝑉) = (LSpan‘(ringLMod‘𝐹)) |
32 | 30, 31 | eqtr4i 2768 | . . . . . . . 8 ⊢ (RSpan‘𝐹) = (LSpan‘𝑉) |
33 | 32 | fveq1i 6718 | . . . . . . 7 ⊢ ((RSpan‘𝐹)‘{(1r‘𝐹)}) = ((LSpan‘𝑉)‘{(1r‘𝐹)}) |
34 | eqid 2737 | . . . . . . . 8 ⊢ (RSpan‘𝐹) = (RSpan‘𝐹) | |
35 | eqid 2737 | . . . . . . . 8 ⊢ (1r‘𝐹) = (1r‘𝐹) | |
36 | 34, 3, 35 | rsp1 20262 | . . . . . . 7 ⊢ (𝐹 ∈ Ring → ((RSpan‘𝐹)‘{(1r‘𝐹)}) = (Base‘𝐹)) |
37 | 33, 36 | eqtr3id 2792 | . . . . . 6 ⊢ (𝐹 ∈ Ring → ((LSpan‘𝑉)‘{(1r‘𝐹)}) = (Base‘𝐹)) |
38 | 2, 6, 37 | 3syl 18 | . . . . 5 ⊢ (𝐹 ∈ Field → ((LSpan‘𝑉)‘{(1r‘𝐹)}) = (Base‘𝐹)) |
39 | 11 | a1i 11 | . . . . . . . 8 ⊢ (𝐹 ∈ Field → 𝑉 = ((subringAlg ‘𝐹)‘(Base‘𝐹))) |
40 | eqidd 2738 | . . . . . . . 8 ⊢ (𝐹 ∈ Field → (1r‘𝐹) = (1r‘𝐹)) | |
41 | 39, 40, 16 | sra1r 31385 | . . . . . . 7 ⊢ (𝐹 ∈ Field → (1r‘𝐹) = (1r‘𝑉)) |
42 | 41 | sneqd 4553 | . . . . . 6 ⊢ (𝐹 ∈ Field → {(1r‘𝐹)} = {(1r‘𝑉)}) |
43 | 42 | fveq2d 6721 | . . . . 5 ⊢ (𝐹 ∈ Field → ((LSpan‘𝑉)‘{(1r‘𝐹)}) = ((LSpan‘𝑉)‘{(1r‘𝑉)})) |
44 | 39, 16 | srabase 20215 | . . . . 5 ⊢ (𝐹 ∈ Field → (Base‘𝐹) = (Base‘𝑉)) |
45 | 38, 43, 44 | 3eqtr3d 2785 | . . . 4 ⊢ (𝐹 ∈ Field → ((LSpan‘𝑉)‘{(1r‘𝑉)}) = (Base‘𝑉)) |
46 | eqid 2737 | . . . . 5 ⊢ (LBasis‘𝑉) = (LBasis‘𝑉) | |
47 | eqid 2737 | . . . . 5 ⊢ (LSpan‘𝑉) = (LSpan‘𝑉) | |
48 | 19, 46, 47 | islbs4 20794 | . . . 4 ⊢ ({(1r‘𝑉)} ∈ (LBasis‘𝑉) ↔ ({(1r‘𝑉)} ∈ (LIndS‘𝑉) ∧ ((LSpan‘𝑉)‘{(1r‘𝑉)}) = (Base‘𝑉))) |
49 | 29, 45, 48 | sylanbrc 586 | . . 3 ⊢ (𝐹 ∈ Field → {(1r‘𝑉)} ∈ (LBasis‘𝑉)) |
50 | 46 | dimval 31400 | . . 3 ⊢ ((𝑉 ∈ LVec ∧ {(1r‘𝑉)} ∈ (LBasis‘𝑉)) → (dim‘𝑉) = (♯‘{(1r‘𝑉)})) |
51 | 14, 49, 50 | syl2anc 587 | . 2 ⊢ (𝐹 ∈ Field → (dim‘𝑉) = (♯‘{(1r‘𝑉)})) |
52 | fvex 6730 | . . 3 ⊢ (1r‘𝑉) ∈ V | |
53 | hashsng 13936 | . . 3 ⊢ ((1r‘𝑉) ∈ V → (♯‘{(1r‘𝑉)}) = 1) | |
54 | 52, 53 | ax-mp 5 | . 2 ⊢ (♯‘{(1r‘𝑉)}) = 1 |
55 | 51, 54 | eqtrdi 2794 | 1 ⊢ (𝐹 ∈ Field → (dim‘𝑉) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 ≠ wne 2940 Vcvv 3408 ⊆ wss 3866 {csn 4541 ‘cfv 6380 (class class class)co 7213 1c1 10730 ♯chash 13896 Basecbs 16760 ↾s cress 16784 0gc0g 16944 1rcur 19516 Ringcrg 19562 CRingccrg 19563 DivRingcdr 19767 Fieldcfield 19768 SubRingcsubrg 19796 LSpanclspn 20008 LBasisclbs 20111 LVecclvec 20139 subringAlg csra 20205 ringLModcrglmod 20206 RSpancrsp 20208 LIndSclinds 20767 dimcldim 31398 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-reg 9208 ax-inf2 9256 ax-ac2 10077 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-iin 4907 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-tpos 7968 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-map 8510 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-oi 9126 df-r1 9380 df-rank 9381 df-card 9555 df-acn 9558 df-ac 9730 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-xnn0 12163 df-z 12177 df-dec 12294 df-uz 12439 df-fz 13096 df-hash 13897 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-sca 16818 df-vsca 16819 df-ip 16820 df-tset 16821 df-ple 16822 df-ocomp 16823 df-0g 16946 df-mre 17089 df-mrc 17090 df-mri 17091 df-acs 17092 df-proset 17802 df-drs 17803 df-poset 17820 df-ipo 18034 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-submnd 18219 df-grp 18368 df-minusg 18369 df-sbg 18370 df-subg 18540 df-cmn 19172 df-abl 19173 df-mgp 19505 df-ur 19517 df-ring 19564 df-oppr 19641 df-dvdsr 19659 df-unit 19660 df-invr 19690 df-drng 19769 df-field 19770 df-subrg 19798 df-lmod 19901 df-lss 19969 df-lsp 20009 df-lbs 20112 df-lvec 20140 df-sra 20209 df-rgmod 20210 df-lidl 20211 df-rsp 20212 df-lindf 20768 df-linds 20769 df-dim 31399 |
This theorem is referenced by: extdgid 31449 |
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