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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rlmdim | 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.) Generalize to division rings. (Revised by SN, 22-Mar-2025.) |
Ref | Expression |
---|---|
rlmdim.1 | ⊢ 𝑉 = (ringLMod‘𝐹) |
Ref | Expression |
---|---|
rlmdim | ⊢ (𝐹 ∈ DivRing → (dim‘𝑉) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rlmdim.1 | . . . 4 ⊢ 𝑉 = (ringLMod‘𝐹) | |
2 | rlmlvec 21056 | . . . 4 ⊢ (𝐹 ∈ DivRing → (ringLMod‘𝐹) ∈ LVec) | |
3 | 1, 2 | eqeltrid 2836 | . . 3 ⊢ (𝐹 ∈ DivRing → 𝑉 ∈ LVec) |
4 | ssid 4004 | . . . . . . . 8 ⊢ (Base‘𝐹) ⊆ (Base‘𝐹) | |
5 | rlmval 21043 | . . . . . . . . . 10 ⊢ (ringLMod‘𝐹) = ((subringAlg ‘𝐹)‘(Base‘𝐹)) | |
6 | 1, 5 | eqtri 2759 | . . . . . . . . 9 ⊢ 𝑉 = ((subringAlg ‘𝐹)‘(Base‘𝐹)) |
7 | eqid 2731 | . . . . . . . . 9 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
8 | 6, 7 | sradrng 33126 | . . . . . . . 8 ⊢ ((𝐹 ∈ DivRing ∧ (Base‘𝐹) ⊆ (Base‘𝐹)) → 𝑉 ∈ DivRing) |
9 | 4, 8 | mpan2 688 | . . . . . . 7 ⊢ (𝐹 ∈ DivRing → 𝑉 ∈ DivRing) |
10 | 9 | drngringd 20591 | . . . . . 6 ⊢ (𝐹 ∈ DivRing → 𝑉 ∈ Ring) |
11 | eqid 2731 | . . . . . . 7 ⊢ (Base‘𝑉) = (Base‘𝑉) | |
12 | eqid 2731 | . . . . . . 7 ⊢ (1r‘𝑉) = (1r‘𝑉) | |
13 | 11, 12 | ringidcl 20161 | . . . . . 6 ⊢ (𝑉 ∈ Ring → (1r‘𝑉) ∈ (Base‘𝑉)) |
14 | 10, 13 | syl 17 | . . . . 5 ⊢ (𝐹 ∈ DivRing → (1r‘𝑉) ∈ (Base‘𝑉)) |
15 | eqid 2731 | . . . . . . 7 ⊢ (0g‘𝑉) = (0g‘𝑉) | |
16 | 15, 12 | drngunz 20602 | . . . . . 6 ⊢ (𝑉 ∈ DivRing → (1r‘𝑉) ≠ (0g‘𝑉)) |
17 | 9, 16 | syl 17 | . . . . 5 ⊢ (𝐹 ∈ DivRing → (1r‘𝑉) ≠ (0g‘𝑉)) |
18 | 11, 15 | lindssn 32936 | . . . . 5 ⊢ ((𝑉 ∈ LVec ∧ (1r‘𝑉) ∈ (Base‘𝑉) ∧ (1r‘𝑉) ≠ (0g‘𝑉)) → {(1r‘𝑉)} ∈ (LIndS‘𝑉)) |
19 | 3, 14, 17, 18 | syl3anc 1370 | . . . 4 ⊢ (𝐹 ∈ DivRing → {(1r‘𝑉)} ∈ (LIndS‘𝑉)) |
20 | drngring 20590 | . . . . . 6 ⊢ (𝐹 ∈ DivRing → 𝐹 ∈ Ring) | |
21 | 1 | fveq2i 6894 | . . . . . . . 8 ⊢ (LSpan‘𝑉) = (LSpan‘(ringLMod‘𝐹)) |
22 | rspval 21066 | . . . . . . . 8 ⊢ (RSpan‘𝐹) = (LSpan‘(ringLMod‘𝐹)) | |
23 | 21, 22 | eqtr4i 2762 | . . . . . . 7 ⊢ (LSpan‘𝑉) = (RSpan‘𝐹) |
24 | eqid 2731 | . . . . . . 7 ⊢ (1r‘𝐹) = (1r‘𝐹) | |
25 | 23, 7, 24 | rsp1 21092 | . . . . . 6 ⊢ (𝐹 ∈ Ring → ((LSpan‘𝑉)‘{(1r‘𝐹)}) = (Base‘𝐹)) |
26 | 20, 25 | syl 17 | . . . . 5 ⊢ (𝐹 ∈ DivRing → ((LSpan‘𝑉)‘{(1r‘𝐹)}) = (Base‘𝐹)) |
27 | 6 | a1i 11 | . . . . . . . 8 ⊢ (𝐹 ∈ DivRing → 𝑉 = ((subringAlg ‘𝐹)‘(Base‘𝐹))) |
28 | eqidd 2732 | . . . . . . . 8 ⊢ (𝐹 ∈ DivRing → (1r‘𝐹) = (1r‘𝐹)) | |
29 | ssidd 4005 | . . . . . . . 8 ⊢ (𝐹 ∈ DivRing → (Base‘𝐹) ⊆ (Base‘𝐹)) | |
30 | 27, 28, 29 | sra1r 33125 | . . . . . . 7 ⊢ (𝐹 ∈ DivRing → (1r‘𝐹) = (1r‘𝑉)) |
31 | 30 | sneqd 4640 | . . . . . 6 ⊢ (𝐹 ∈ DivRing → {(1r‘𝐹)} = {(1r‘𝑉)}) |
32 | 31 | fveq2d 6895 | . . . . 5 ⊢ (𝐹 ∈ DivRing → ((LSpan‘𝑉)‘{(1r‘𝐹)}) = ((LSpan‘𝑉)‘{(1r‘𝑉)})) |
33 | 27, 29 | srabase 21022 | . . . . 5 ⊢ (𝐹 ∈ DivRing → (Base‘𝐹) = (Base‘𝑉)) |
34 | 26, 32, 33 | 3eqtr3d 2779 | . . . 4 ⊢ (𝐹 ∈ DivRing → ((LSpan‘𝑉)‘{(1r‘𝑉)}) = (Base‘𝑉)) |
35 | eqid 2731 | . . . . 5 ⊢ (LBasis‘𝑉) = (LBasis‘𝑉) | |
36 | eqid 2731 | . . . . 5 ⊢ (LSpan‘𝑉) = (LSpan‘𝑉) | |
37 | 11, 35, 36 | islbs4 21698 | . . . 4 ⊢ ({(1r‘𝑉)} ∈ (LBasis‘𝑉) ↔ ({(1r‘𝑉)} ∈ (LIndS‘𝑉) ∧ ((LSpan‘𝑉)‘{(1r‘𝑉)}) = (Base‘𝑉))) |
38 | 19, 34, 37 | sylanbrc 582 | . . 3 ⊢ (𝐹 ∈ DivRing → {(1r‘𝑉)} ∈ (LBasis‘𝑉)) |
39 | 35 | dimval 33141 | . . 3 ⊢ ((𝑉 ∈ LVec ∧ {(1r‘𝑉)} ∈ (LBasis‘𝑉)) → (dim‘𝑉) = (♯‘{(1r‘𝑉)})) |
40 | 3, 38, 39 | syl2anc 583 | . 2 ⊢ (𝐹 ∈ DivRing → (dim‘𝑉) = (♯‘{(1r‘𝑉)})) |
41 | fvex 6904 | . . 3 ⊢ (1r‘𝑉) ∈ V | |
42 | hashsng 14336 | . . 3 ⊢ ((1r‘𝑉) ∈ V → (♯‘{(1r‘𝑉)}) = 1) | |
43 | 41, 42 | ax-mp 5 | . 2 ⊢ (♯‘{(1r‘𝑉)}) = 1 |
44 | 40, 43 | eqtrdi 2787 | 1 ⊢ (𝐹 ∈ DivRing → (dim‘𝑉) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 Vcvv 3473 ⊆ wss 3948 {csn 4628 ‘cfv 6543 1c1 11117 ♯chash 14297 Basecbs 17151 0gc0g 17392 1rcur 20082 Ringcrg 20134 DivRingcdr 20583 LSpanclspn 20814 LBasisclbs 20918 LVecclvec 20946 subringAlg csra 21015 ringLModcrglmod 21016 RSpancrsp 21062 LIndSclinds 21671 dimcldim 33139 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-reg 9593 ax-inf2 9642 ax-ac2 10464 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 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 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-tpos 8217 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-map 8828 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-oi 9511 df-r1 9765 df-rank 9766 df-card 9940 df-acn 9943 df-ac 10117 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-xnn0 12552 df-z 12566 df-dec 12685 df-uz 12830 df-fz 13492 df-hash 14298 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ocomp 17225 df-0g 17394 df-mre 17537 df-mrc 17538 df-mri 17539 df-acs 17540 df-proset 18258 df-drs 18259 df-poset 18276 df-ipo 18491 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-submnd 18712 df-grp 18864 df-minusg 18865 df-sbg 18866 df-subg 19046 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-oppr 20232 df-dvdsr 20255 df-unit 20256 df-invr 20286 df-subrg 20467 df-drng 20585 df-lmod 20704 df-lss 20775 df-lsp 20815 df-lbs 20919 df-lvec 20947 df-sra 21017 df-rgmod 21018 df-lidl 21063 df-rsp 21064 df-lindf 21672 df-linds 21673 df-dim 33140 |
This theorem is referenced by: extdgid 33195 |
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