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Mirrors > Home > MPE Home > Th. List > Mathboxes > frlmdim | Structured version Visualization version GIF version |
Description: Dimension of a free left module. (Contributed by Thierry Arnoux, 20-May-2023.) |
Ref | Expression |
---|---|
frlmdim.f | ⊢ 𝐹 = (𝑅 freeLMod 𝐼) |
Ref | Expression |
---|---|
frlmdim | ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑉) → (dim‘𝐹) = (♯‘𝐼)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frlmdim.f | . . . 4 ⊢ 𝐹 = (𝑅 freeLMod 𝐼) | |
2 | 1 | frlmlvec 21249 | . . 3 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑉) → 𝐹 ∈ LVec) |
3 | drngring 20272 | . . . 4 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring) | |
4 | eqid 2731 | . . . . 5 ⊢ (𝑅 unitVec 𝐼) = (𝑅 unitVec 𝐼) | |
5 | eqid 2731 | . . . . 5 ⊢ (LBasis‘𝐹) = (LBasis‘𝐹) | |
6 | 1, 4, 5 | frlmlbs 21285 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → ran (𝑅 unitVec 𝐼) ∈ (LBasis‘𝐹)) |
7 | 3, 6 | sylan 580 | . . 3 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑉) → ran (𝑅 unitVec 𝐼) ∈ (LBasis‘𝐹)) |
8 | 5 | dimval 32525 | . . 3 ⊢ ((𝐹 ∈ LVec ∧ ran (𝑅 unitVec 𝐼) ∈ (LBasis‘𝐹)) → (dim‘𝐹) = (♯‘ran (𝑅 unitVec 𝐼))) |
9 | 2, 7, 8 | syl2anc 584 | . 2 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑉) → (dim‘𝐹) = (♯‘ran (𝑅 unitVec 𝐼))) |
10 | simpr 485 | . . 3 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑉) → 𝐼 ∈ 𝑉) | |
11 | drngnzr 20284 | . . . 4 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ NzRing) | |
12 | eqid 2731 | . . . . 5 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
13 | 4, 1, 12 | uvcf1 21280 | . . . 4 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑉) → (𝑅 unitVec 𝐼):𝐼–1-1→(Base‘𝐹)) |
14 | 11, 13 | sylan 580 | . . 3 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑉) → (𝑅 unitVec 𝐼):𝐼–1-1→(Base‘𝐹)) |
15 | hashf1rn 14294 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑅 unitVec 𝐼):𝐼–1-1→(Base‘𝐹)) → (♯‘(𝑅 unitVec 𝐼)) = (♯‘ran (𝑅 unitVec 𝐼))) | |
16 | 10, 14, 15 | syl2anc 584 | . 2 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑉) → (♯‘(𝑅 unitVec 𝐼)) = (♯‘ran (𝑅 unitVec 𝐼))) |
17 | mptexg 7207 | . . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, (1r‘𝑅), (0g‘𝑅))) ∈ V) | |
18 | 17 | ad2antlr 725 | . . . . . 6 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑉) ∧ 𝑗 ∈ 𝐼) → (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, (1r‘𝑅), (0g‘𝑅))) ∈ V) |
19 | 18 | ralrimiva 3145 | . . . . 5 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑉) → ∀𝑗 ∈ 𝐼 (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, (1r‘𝑅), (0g‘𝑅))) ∈ V) |
20 | eqid 2731 | . . . . . 6 ⊢ (𝑗 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, (1r‘𝑅), (0g‘𝑅)))) = (𝑗 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, (1r‘𝑅), (0g‘𝑅)))) | |
21 | 20 | fnmpt 6677 | . . . . 5 ⊢ (∀𝑗 ∈ 𝐼 (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, (1r‘𝑅), (0g‘𝑅))) ∈ V → (𝑗 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, (1r‘𝑅), (0g‘𝑅)))) Fn 𝐼) |
22 | 19, 21 | syl 17 | . . . 4 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑉) → (𝑗 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, (1r‘𝑅), (0g‘𝑅)))) Fn 𝐼) |
23 | eqid 2731 | . . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
24 | eqid 2731 | . . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
25 | 4, 23, 24 | uvcfval 21272 | . . . . 5 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑉) → (𝑅 unitVec 𝐼) = (𝑗 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, (1r‘𝑅), (0g‘𝑅))))) |
26 | 25 | fneq1d 6631 | . . . 4 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑉) → ((𝑅 unitVec 𝐼) Fn 𝐼 ↔ (𝑗 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, (1r‘𝑅), (0g‘𝑅)))) Fn 𝐼)) |
27 | 22, 26 | mpbird 256 | . . 3 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑉) → (𝑅 unitVec 𝐼) Fn 𝐼) |
28 | hashfn 14317 | . . 3 ⊢ ((𝑅 unitVec 𝐼) Fn 𝐼 → (♯‘(𝑅 unitVec 𝐼)) = (♯‘𝐼)) | |
29 | 27, 28 | syl 17 | . 2 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑉) → (♯‘(𝑅 unitVec 𝐼)) = (♯‘𝐼)) |
30 | 9, 16, 29 | 3eqtr2d 2777 | 1 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑉) → (dim‘𝐹) = (♯‘𝐼)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3060 Vcvv 3473 ifcif 4522 ↦ cmpt 5224 ran crn 5670 Fn wfn 6527 –1-1→wf1 6529 ‘cfv 6532 (class class class)co 7393 ♯chash 14272 Basecbs 17126 0gc0g 17367 1rcur 19963 Ringcrg 20014 NzRingcnzr 20241 DivRingcdr 20265 LBasisclbs 20634 LVecclvec 20662 freeLMod cfrlm 21234 unitVec cuvc 21270 dimcldim 32523 |
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 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-reg 9569 ax-inf2 9618 ax-ac2 10440 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 |
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 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 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-tp 4627 df-op 4629 df-uni 4902 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 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-isom 6541 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-of 7653 df-om 7839 df-1st 7957 df-2nd 7958 df-supp 8129 df-tpos 8193 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-1o 8448 df-er 8686 df-map 8805 df-ixp 8875 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-fsupp 9345 df-sup 9419 df-oi 9487 df-r1 9741 df-rank 9742 df-card 9916 df-acn 9919 df-ac 10093 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-nn 12195 df-2 12257 df-3 12258 df-4 12259 df-5 12260 df-6 12261 df-7 12262 df-8 12263 df-9 12264 df-n0 12455 df-xnn0 12527 df-z 12541 df-dec 12660 df-uz 12805 df-fz 13467 df-fzo 13610 df-seq 13949 df-hash 14273 df-struct 17062 df-sets 17079 df-slot 17097 df-ndx 17109 df-base 17127 df-ress 17156 df-plusg 17192 df-mulr 17193 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ocomp 17200 df-ds 17201 df-hom 17203 df-cco 17204 df-0g 17369 df-gsum 17370 df-prds 17375 df-pws 17377 df-mre 17512 df-mrc 17513 df-mri 17514 df-acs 17515 df-proset 18230 df-drs 18231 df-poset 18248 df-ipo 18463 df-mgm 18543 df-sgrp 18592 df-mnd 18603 df-mhm 18647 df-submnd 18648 df-grp 18797 df-minusg 18798 df-sbg 18799 df-mulg 18923 df-subg 18975 df-ghm 19056 df-cntz 19147 df-cmn 19614 df-abl 19615 df-mgp 19947 df-ur 19964 df-ring 20016 df-oppr 20102 df-dvdsr 20123 df-unit 20124 df-invr 20154 df-nzr 20242 df-drng 20267 df-subrg 20310 df-lmod 20422 df-lss 20492 df-lsp 20532 df-lmhm 20582 df-lbs 20635 df-lvec 20663 df-sra 20734 df-rgmod 20735 df-dsmm 21220 df-frlm 21235 df-uvc 21271 df-dim 32524 |
This theorem is referenced by: rrxdim 32537 matdim 32538 |
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