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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 |
---|---|
rgmoddim | ⊢ (𝐹 ∈ Field → (dim‘𝑉) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfld 20149 | . . . . 5 ⊢ (𝐹 ∈ Field ↔ (𝐹 ∈ DivRing ∧ 𝐹 ∈ CRing)) | |
2 | 1 | simplbi 498 | . . . 4 ⊢ (𝐹 ∈ Field → 𝐹 ∈ DivRing) |
3 | eqid 2737 | . . . . . 6 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
4 | 3 | ressid 17085 | . . . . 5 ⊢ (𝐹 ∈ Field → (𝐹 ↾s (Base‘𝐹)) = 𝐹) |
5 | 4, 2 | eqeltrd 2838 | . . . 4 ⊢ (𝐹 ∈ Field → (𝐹 ↾s (Base‘𝐹)) ∈ DivRing) |
6 | drngring 20145 | . . . . 5 ⊢ (𝐹 ∈ DivRing → 𝐹 ∈ Ring) | |
7 | 3 | subrgid 20177 | . . . . 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 20613 | . . . . . 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 32109 | . . . 4 ⊢ ((𝐹 ∈ DivRing ∧ (𝐹 ↾s (Base‘𝐹)) ∈ DivRing ∧ (Base‘𝐹) ∈ (SubRing‘𝐹)) → 𝑉 ∈ LVec) |
14 | 2, 5, 8, 13 | syl3anc 1371 | . . 3 ⊢ (𝐹 ∈ Field → 𝑉 ∈ LVec) |
15 | 2, 6 | syl 17 | . . . . . . 7 ⊢ (𝐹 ∈ Field → 𝐹 ∈ Ring) |
16 | ssidd 3965 | . . . . . . 7 ⊢ (𝐹 ∈ Field → (Base‘𝐹) ⊆ (Base‘𝐹)) | |
17 | 11, 3 | sraring 32106 | . . . . . . 7 ⊢ ((𝐹 ∈ Ring ∧ (Base‘𝐹) ⊆ (Base‘𝐹)) → 𝑉 ∈ Ring) |
18 | 15, 16, 17 | syl2anc 584 | . . . . . 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 19943 | . . . . . 6 ⊢ (𝑉 ∈ Ring → (1r‘𝑉) ∈ (Base‘𝑉)) |
22 | 18, 21 | syl 17 | . . . . 5 ⊢ (𝐹 ∈ Field → (1r‘𝑉) ∈ (Base‘𝑉)) |
23 | 11, 3 | sradrng 32107 | . . . . . . 7 ⊢ ((𝐹 ∈ DivRing ∧ (Base‘𝐹) ⊆ (Base‘𝐹)) → 𝑉 ∈ DivRing) |
24 | 2, 16, 23 | syl2anc 584 | . . . . . 6 ⊢ (𝐹 ∈ Field → 𝑉 ∈ DivRing) |
25 | eqid 2737 | . . . . . . 7 ⊢ (0g‘𝑉) = (0g‘𝑉) | |
26 | 25, 20 | drngunz 20156 | . . . . . 6 ⊢ (𝑉 ∈ DivRing → (1r‘𝑉) ≠ (0g‘𝑉)) |
27 | 24, 26 | syl 17 | . . . . 5 ⊢ (𝐹 ∈ Field → (1r‘𝑉) ≠ (0g‘𝑉)) |
28 | 19, 25 | lindssn 31992 | . . . . 5 ⊢ ((𝑉 ∈ LVec ∧ (1r‘𝑉) ∈ (Base‘𝑉) ∧ (1r‘𝑉) ≠ (0g‘𝑉)) → {(1r‘𝑉)} ∈ (LIndS‘𝑉)) |
29 | 14, 22, 27, 28 | syl3anc 1371 | . . . 4 ⊢ (𝐹 ∈ Field → {(1r‘𝑉)} ∈ (LIndS‘𝑉)) |
30 | rspval 20615 | . . . . . . . . 9 ⊢ (RSpan‘𝐹) = (LSpan‘(ringLMod‘𝐹)) | |
31 | 9 | fveq2i 6842 | . . . . . . . . 9 ⊢ (LSpan‘𝑉) = (LSpan‘(ringLMod‘𝐹)) |
32 | 30, 31 | eqtr4i 2768 | . . . . . . . 8 ⊢ (RSpan‘𝐹) = (LSpan‘𝑉) |
33 | 32 | fveq1i 6840 | . . . . . . 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 20647 | . . . . . . 7 ⊢ (𝐹 ∈ Ring → ((RSpan‘𝐹)‘{(1r‘𝐹)}) = (Base‘𝐹)) |
37 | 33, 36 | eqtr3id 2791 | . . . . . 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 32105 | . . . . . . 7 ⊢ (𝐹 ∈ Field → (1r‘𝐹) = (1r‘𝑉)) |
42 | 41 | sneqd 4596 | . . . . . 6 ⊢ (𝐹 ∈ Field → {(1r‘𝐹)} = {(1r‘𝑉)}) |
43 | 42 | fveq2d 6843 | . . . . 5 ⊢ (𝐹 ∈ Field → ((LSpan‘𝑉)‘{(1r‘𝐹)}) = ((LSpan‘𝑉)‘{(1r‘𝑉)})) |
44 | 39, 16 | srabase 20593 | . . . . 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 21191 | . . . 4 ⊢ ({(1r‘𝑉)} ∈ (LBasis‘𝑉) ↔ ({(1r‘𝑉)} ∈ (LIndS‘𝑉) ∧ ((LSpan‘𝑉)‘{(1r‘𝑉)}) = (Base‘𝑉))) |
49 | 29, 45, 48 | sylanbrc 583 | . . 3 ⊢ (𝐹 ∈ Field → {(1r‘𝑉)} ∈ (LBasis‘𝑉)) |
50 | 46 | dimval 32120 | . . 3 ⊢ ((𝑉 ∈ LVec ∧ {(1r‘𝑉)} ∈ (LBasis‘𝑉)) → (dim‘𝑉) = (♯‘{(1r‘𝑉)})) |
51 | 14, 49, 50 | syl2anc 584 | . 2 ⊢ (𝐹 ∈ Field → (dim‘𝑉) = (♯‘{(1r‘𝑉)})) |
52 | fvex 6852 | . . 3 ⊢ (1r‘𝑉) ∈ V | |
53 | hashsng 14223 | . . 3 ⊢ ((1r‘𝑉) ∈ V → (♯‘{(1r‘𝑉)}) = 1) | |
54 | 52, 53 | ax-mp 5 | . 2 ⊢ (♯‘{(1r‘𝑉)}) = 1 |
55 | 51, 54 | eqtrdi 2793 | 1 ⊢ (𝐹 ∈ Field → (dim‘𝑉) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ≠ wne 2941 Vcvv 3443 ⊆ wss 3908 {csn 4584 ‘cfv 6493 (class class class)co 7351 1c1 11010 ♯chash 14184 Basecbs 17043 ↾s cress 17072 0gc0g 17281 1rcur 19872 Ringcrg 19918 CRingccrg 19919 DivRingcdr 20138 Fieldcfield 20139 SubRingcsubrg 20171 LSpanclspn 20385 LBasisclbs 20488 LVecclvec 20516 subringAlg csra 20582 ringLModcrglmod 20583 RSpancrsp 20585 LIndSclinds 21164 dimcldim 32118 |
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 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-reg 9486 ax-inf2 9535 ax-ac2 10357 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-iin 4955 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-1st 7913 df-2nd 7914 df-tpos 8149 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-er 8606 df-map 8725 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-oi 9404 df-r1 9658 df-rank 9659 df-card 9833 df-acn 9836 df-ac 10010 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-nn 12112 df-2 12174 df-3 12175 df-4 12176 df-5 12177 df-6 12178 df-7 12179 df-8 12180 df-9 12181 df-n0 12372 df-xnn0 12444 df-z 12458 df-dec 12577 df-uz 12722 df-fz 13379 df-hash 14185 df-struct 16979 df-sets 16996 df-slot 17014 df-ndx 17026 df-base 17044 df-ress 17073 df-plusg 17106 df-mulr 17107 df-sca 17109 df-vsca 17110 df-ip 17111 df-tset 17112 df-ple 17113 df-ocomp 17114 df-0g 17283 df-mre 17426 df-mrc 17427 df-mri 17428 df-acs 17429 df-proset 18144 df-drs 18145 df-poset 18162 df-ipo 18377 df-mgm 18457 df-sgrp 18506 df-mnd 18517 df-submnd 18562 df-grp 18711 df-minusg 18712 df-sbg 18713 df-subg 18884 df-cmn 19523 df-abl 19524 df-mgp 19856 df-ur 19873 df-ring 19920 df-oppr 20002 df-dvdsr 20023 df-unit 20024 df-invr 20054 df-drng 20140 df-field 20141 df-subrg 20173 df-lmod 20277 df-lss 20346 df-lsp 20386 df-lbs 20489 df-lvec 20517 df-sra 20586 df-rgmod 20587 df-lidl 20588 df-rsp 20589 df-lindf 21165 df-linds 21166 df-dim 32119 |
This theorem is referenced by: extdgid 32169 |
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