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Mirrors > Home > MPE Home > Th. List > Mathboxes > dimval | Structured version Visualization version GIF version |
Description: The dimension of a vector space 𝐹 is the cardinality of one of its bases. (Contributed by Thierry Arnoux, 6-May-2023.) |
Ref | Expression |
---|---|
dimval.1 | ⊢ 𝐽 = (LBasis‘𝐹) |
Ref | Expression |
---|---|
dimval | ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) → (dim‘𝐹) = (♯‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3463 | . . . 4 ⊢ (𝐹 ∈ LVec → 𝐹 ∈ V) | |
2 | fveq2 6842 | . . . . . . . 8 ⊢ (𝑓 = 𝐹 → (LBasis‘𝑓) = (LBasis‘𝐹)) | |
3 | dimval.1 | . . . . . . . 8 ⊢ 𝐽 = (LBasis‘𝐹) | |
4 | 2, 3 | eqtr4di 2794 | . . . . . . 7 ⊢ (𝑓 = 𝐹 → (LBasis‘𝑓) = 𝐽) |
5 | 4 | imaeq2d 6013 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (♯ “ (LBasis‘𝑓)) = (♯ “ 𝐽)) |
6 | 5 | unieqd 4879 | . . . . 5 ⊢ (𝑓 = 𝐹 → ∪ (♯ “ (LBasis‘𝑓)) = ∪ (♯ “ 𝐽)) |
7 | df-dim 32299 | . . . . 5 ⊢ dim = (𝑓 ∈ V ↦ ∪ (♯ “ (LBasis‘𝑓))) | |
8 | hashf 14238 | . . . . . . 7 ⊢ ♯:V⟶(ℕ0 ∪ {+∞}) | |
9 | ffun 6671 | . . . . . . 7 ⊢ (♯:V⟶(ℕ0 ∪ {+∞}) → Fun ♯) | |
10 | 3 | fvexi 6856 | . . . . . . . 8 ⊢ 𝐽 ∈ V |
11 | 10 | funimaex 6589 | . . . . . . 7 ⊢ (Fun ♯ → (♯ “ 𝐽) ∈ V) |
12 | 8, 9, 11 | mp2b 10 | . . . . . 6 ⊢ (♯ “ 𝐽) ∈ V |
13 | 12 | uniex 7678 | . . . . 5 ⊢ ∪ (♯ “ 𝐽) ∈ V |
14 | 6, 7, 13 | fvmpt 6948 | . . . 4 ⊢ (𝐹 ∈ V → (dim‘𝐹) = ∪ (♯ “ 𝐽)) |
15 | 1, 14 | syl 17 | . . 3 ⊢ (𝐹 ∈ LVec → (dim‘𝐹) = ∪ (♯ “ 𝐽)) |
16 | 15 | adantr 481 | . 2 ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) → (dim‘𝐹) = ∪ (♯ “ 𝐽)) |
17 | 3 | lvecdim 20618 | . . . . . . . . . 10 ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽 ∧ 𝑡 ∈ 𝐽) → 𝑆 ≈ 𝑡) |
18 | 17 | ad4ant124 1173 | . . . . . . . . 9 ⊢ ((((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ (♯ “ 𝐽)) ∧ 𝑡 ∈ 𝐽) → 𝑆 ≈ 𝑡) |
19 | hasheni 14248 | . . . . . . . . 9 ⊢ (𝑆 ≈ 𝑡 → (♯‘𝑆) = (♯‘𝑡)) | |
20 | 18, 19 | syl 17 | . . . . . . . 8 ⊢ ((((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ (♯ “ 𝐽)) ∧ 𝑡 ∈ 𝐽) → (♯‘𝑆) = (♯‘𝑡)) |
21 | 20 | adantr 481 | . . . . . . 7 ⊢ (((((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ (♯ “ 𝐽)) ∧ 𝑡 ∈ 𝐽) ∧ (♯‘𝑡) = 𝑥) → (♯‘𝑆) = (♯‘𝑡)) |
22 | simpr 485 | . . . . . . 7 ⊢ (((((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ (♯ “ 𝐽)) ∧ 𝑡 ∈ 𝐽) ∧ (♯‘𝑡) = 𝑥) → (♯‘𝑡) = 𝑥) | |
23 | 21, 22 | eqtr2d 2777 | . . . . . 6 ⊢ (((((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ (♯ “ 𝐽)) ∧ 𝑡 ∈ 𝐽) ∧ (♯‘𝑡) = 𝑥) → 𝑥 = (♯‘𝑆)) |
24 | 8, 9 | ax-mp 5 | . . . . . . . 8 ⊢ Fun ♯ |
25 | fvelima 6908 | . . . . . . . 8 ⊢ ((Fun ♯ ∧ 𝑥 ∈ (♯ “ 𝐽)) → ∃𝑡 ∈ 𝐽 (♯‘𝑡) = 𝑥) | |
26 | 24, 25 | mpan 688 | . . . . . . 7 ⊢ (𝑥 ∈ (♯ “ 𝐽) → ∃𝑡 ∈ 𝐽 (♯‘𝑡) = 𝑥) |
27 | 26 | adantl 482 | . . . . . 6 ⊢ (((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ (♯ “ 𝐽)) → ∃𝑡 ∈ 𝐽 (♯‘𝑡) = 𝑥) |
28 | 23, 27 | r19.29a 3159 | . . . . 5 ⊢ (((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ (♯ “ 𝐽)) → 𝑥 = (♯‘𝑆)) |
29 | 28 | ralrimiva 3143 | . . . 4 ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) → ∀𝑥 ∈ (♯ “ 𝐽)𝑥 = (♯‘𝑆)) |
30 | ne0i 4294 | . . . . . . 7 ⊢ (𝑆 ∈ 𝐽 → 𝐽 ≠ ∅) | |
31 | 30 | adantl 482 | . . . . . 6 ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) → 𝐽 ≠ ∅) |
32 | ffn 6668 | . . . . . . . . 9 ⊢ (♯:V⟶(ℕ0 ∪ {+∞}) → ♯ Fn V) | |
33 | 8, 32 | ax-mp 5 | . . . . . . . 8 ⊢ ♯ Fn V |
34 | ssv 3968 | . . . . . . . 8 ⊢ 𝐽 ⊆ V | |
35 | fnimaeq0 6634 | . . . . . . . 8 ⊢ ((♯ Fn V ∧ 𝐽 ⊆ V) → ((♯ “ 𝐽) = ∅ ↔ 𝐽 = ∅)) | |
36 | 33, 34, 35 | mp2an 690 | . . . . . . 7 ⊢ ((♯ “ 𝐽) = ∅ ↔ 𝐽 = ∅) |
37 | 36 | necon3bii 2996 | . . . . . 6 ⊢ ((♯ “ 𝐽) ≠ ∅ ↔ 𝐽 ≠ ∅) |
38 | 31, 37 | sylibr 233 | . . . . 5 ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) → (♯ “ 𝐽) ≠ ∅) |
39 | eqsn 4789 | . . . . 5 ⊢ ((♯ “ 𝐽) ≠ ∅ → ((♯ “ 𝐽) = {(♯‘𝑆)} ↔ ∀𝑥 ∈ (♯ “ 𝐽)𝑥 = (♯‘𝑆))) | |
40 | 38, 39 | syl 17 | . . . 4 ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) → ((♯ “ 𝐽) = {(♯‘𝑆)} ↔ ∀𝑥 ∈ (♯ “ 𝐽)𝑥 = (♯‘𝑆))) |
41 | 29, 40 | mpbird 256 | . . 3 ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) → (♯ “ 𝐽) = {(♯‘𝑆)}) |
42 | 41 | unieqd 4879 | . 2 ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) → ∪ (♯ “ 𝐽) = ∪ {(♯‘𝑆)}) |
43 | fvex 6855 | . . . 4 ⊢ (♯‘𝑆) ∈ V | |
44 | 43 | unisn 4887 | . . 3 ⊢ ∪ {(♯‘𝑆)} = (♯‘𝑆) |
45 | 44 | a1i 11 | . 2 ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) → ∪ {(♯‘𝑆)} = (♯‘𝑆)) |
46 | 16, 42, 45 | 3eqtrd 2780 | 1 ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) → (dim‘𝐹) = (♯‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∃wrex 3073 Vcvv 3445 ∪ cun 3908 ⊆ wss 3910 ∅c0 4282 {csn 4586 ∪ cuni 4865 class class class wbr 5105 “ cima 5636 Fun wfun 6490 Fn wfn 6491 ⟶wf 6492 ‘cfv 6496 ≈ cen 8880 +∞cpnf 11186 ℕ0cn0 12413 ♯chash 14230 LBasisclbs 20535 LVecclvec 20563 dimcldim 32298 |
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 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-reg 9528 ax-inf2 9577 ax-ac2 10399 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 |
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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-tpos 8157 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-map 8767 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-oi 9446 df-r1 9700 df-rank 9701 df-card 9875 df-acn 9878 df-ac 10052 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-xnn0 12486 df-z 12500 df-dec 12619 df-uz 12764 df-fz 13425 df-hash 14231 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-tset 17152 df-ple 17153 df-ocomp 17154 df-0g 17323 df-mre 17466 df-mrc 17467 df-mri 17468 df-acs 17469 df-proset 18184 df-drs 18185 df-poset 18202 df-ipo 18417 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-grp 18751 df-minusg 18752 df-sbg 18753 df-subg 18925 df-cmn 19564 df-abl 19565 df-mgp 19897 df-ur 19914 df-ring 19966 df-oppr 20049 df-dvdsr 20070 df-unit 20071 df-invr 20101 df-drng 20187 df-lmod 20324 df-lss 20393 df-lsp 20433 df-lbs 20536 df-lvec 20564 df-dim 32299 |
This theorem is referenced by: dimcl 32302 lvecdim0i 32303 lvecdim0 32304 lssdimle 32305 dimpropd 32306 rgmoddim 32307 frlmdim 32308 lsatdim 32314 dimkerim 32322 fedgmul 32326 extdg1id 32352 ccfldextdgrr 32356 |
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