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Mirrors > Home > MPE Home > Th. List > Mathboxes > dimvalfi | Structured version Visualization version GIF version |
Description: The dimension of a vector space 𝐹 is the cardinality of one of its bases. This version of dimval 31003 does not depend on the axiom of choice, but it is limited to the case where the base 𝑆 is finite. (Contributed by Thierry Arnoux, 24-May-2023.) |
Ref | Expression |
---|---|
dimval.1 | ⊢ 𝐽 = (LBasis‘𝐹) |
Ref | Expression |
---|---|
dimvalfi | ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽 ∧ 𝑆 ∈ Fin) → (dim‘𝐹) = (♯‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3514 | . . . 4 ⊢ (𝐹 ∈ LVec → 𝐹 ∈ V) | |
2 | fveq2 6672 | . . . . . . . 8 ⊢ (𝑓 = 𝐹 → (LBasis‘𝑓) = (LBasis‘𝐹)) | |
3 | dimval.1 | . . . . . . . 8 ⊢ 𝐽 = (LBasis‘𝐹) | |
4 | 2, 3 | syl6eqr 2876 | . . . . . . 7 ⊢ (𝑓 = 𝐹 → (LBasis‘𝑓) = 𝐽) |
5 | 4 | imaeq2d 5931 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (♯ “ (LBasis‘𝑓)) = (♯ “ 𝐽)) |
6 | 5 | unieqd 4854 | . . . . 5 ⊢ (𝑓 = 𝐹 → ∪ (♯ “ (LBasis‘𝑓)) = ∪ (♯ “ 𝐽)) |
7 | df-dim 31002 | . . . . 5 ⊢ dim = (𝑓 ∈ V ↦ ∪ (♯ “ (LBasis‘𝑓))) | |
8 | hashf 13701 | . . . . . . 7 ⊢ ♯:V⟶(ℕ0 ∪ {+∞}) | |
9 | ffun 6519 | . . . . . . 7 ⊢ (♯:V⟶(ℕ0 ∪ {+∞}) → Fun ♯) | |
10 | 3 | fvexi 6686 | . . . . . . . 8 ⊢ 𝐽 ∈ V |
11 | 10 | funimaex 6443 | . . . . . . 7 ⊢ (Fun ♯ → (♯ “ 𝐽) ∈ V) |
12 | 8, 9, 11 | mp2b 10 | . . . . . 6 ⊢ (♯ “ 𝐽) ∈ V |
13 | 12 | uniex 7469 | . . . . 5 ⊢ ∪ (♯ “ 𝐽) ∈ V |
14 | 6, 7, 13 | fvmpt 6770 | . . . 4 ⊢ (𝐹 ∈ V → (dim‘𝐹) = ∪ (♯ “ 𝐽)) |
15 | 1, 14 | syl 17 | . . 3 ⊢ (𝐹 ∈ LVec → (dim‘𝐹) = ∪ (♯ “ 𝐽)) |
16 | 15 | 3ad2ant1 1129 | . 2 ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽 ∧ 𝑆 ∈ Fin) → (dim‘𝐹) = ∪ (♯ “ 𝐽)) |
17 | simpll1 1208 | . . . . . . . . . 10 ⊢ ((((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽 ∧ 𝑆 ∈ Fin) ∧ 𝑥 ∈ (♯ “ 𝐽)) ∧ 𝑡 ∈ 𝐽) → 𝐹 ∈ LVec) | |
18 | simpll2 1209 | . . . . . . . . . 10 ⊢ ((((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽 ∧ 𝑆 ∈ Fin) ∧ 𝑥 ∈ (♯ “ 𝐽)) ∧ 𝑡 ∈ 𝐽) → 𝑆 ∈ 𝐽) | |
19 | simpr 487 | . . . . . . . . . 10 ⊢ ((((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽 ∧ 𝑆 ∈ Fin) ∧ 𝑥 ∈ (♯ “ 𝐽)) ∧ 𝑡 ∈ 𝐽) → 𝑡 ∈ 𝐽) | |
20 | simpll3 1210 | . . . . . . . . . 10 ⊢ ((((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽 ∧ 𝑆 ∈ Fin) ∧ 𝑥 ∈ (♯ “ 𝐽)) ∧ 𝑡 ∈ 𝐽) → 𝑆 ∈ Fin) | |
21 | 3, 17, 18, 19, 20 | lvecdimfi 31000 | . . . . . . . . 9 ⊢ ((((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽 ∧ 𝑆 ∈ Fin) ∧ 𝑥 ∈ (♯ “ 𝐽)) ∧ 𝑡 ∈ 𝐽) → 𝑆 ≈ 𝑡) |
22 | hasheni 13711 | . . . . . . . . 9 ⊢ (𝑆 ≈ 𝑡 → (♯‘𝑆) = (♯‘𝑡)) | |
23 | 21, 22 | syl 17 | . . . . . . . 8 ⊢ ((((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽 ∧ 𝑆 ∈ Fin) ∧ 𝑥 ∈ (♯ “ 𝐽)) ∧ 𝑡 ∈ 𝐽) → (♯‘𝑆) = (♯‘𝑡)) |
24 | 23 | adantr 483 | . . . . . . 7 ⊢ (((((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽 ∧ 𝑆 ∈ Fin) ∧ 𝑥 ∈ (♯ “ 𝐽)) ∧ 𝑡 ∈ 𝐽) ∧ (♯‘𝑡) = 𝑥) → (♯‘𝑆) = (♯‘𝑡)) |
25 | simpr 487 | . . . . . . 7 ⊢ (((((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽 ∧ 𝑆 ∈ Fin) ∧ 𝑥 ∈ (♯ “ 𝐽)) ∧ 𝑡 ∈ 𝐽) ∧ (♯‘𝑡) = 𝑥) → (♯‘𝑡) = 𝑥) | |
26 | 24, 25 | eqtr2d 2859 | . . . . . 6 ⊢ (((((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽 ∧ 𝑆 ∈ Fin) ∧ 𝑥 ∈ (♯ “ 𝐽)) ∧ 𝑡 ∈ 𝐽) ∧ (♯‘𝑡) = 𝑥) → 𝑥 = (♯‘𝑆)) |
27 | 8, 9 | ax-mp 5 | . . . . . . . 8 ⊢ Fun ♯ |
28 | fvelima 6733 | . . . . . . . 8 ⊢ ((Fun ♯ ∧ 𝑥 ∈ (♯ “ 𝐽)) → ∃𝑡 ∈ 𝐽 (♯‘𝑡) = 𝑥) | |
29 | 27, 28 | mpan 688 | . . . . . . 7 ⊢ (𝑥 ∈ (♯ “ 𝐽) → ∃𝑡 ∈ 𝐽 (♯‘𝑡) = 𝑥) |
30 | 29 | adantl 484 | . . . . . 6 ⊢ (((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽 ∧ 𝑆 ∈ Fin) ∧ 𝑥 ∈ (♯ “ 𝐽)) → ∃𝑡 ∈ 𝐽 (♯‘𝑡) = 𝑥) |
31 | 26, 30 | r19.29a 3291 | . . . . 5 ⊢ (((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽 ∧ 𝑆 ∈ Fin) ∧ 𝑥 ∈ (♯ “ 𝐽)) → 𝑥 = (♯‘𝑆)) |
32 | 31 | ralrimiva 3184 | . . . 4 ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽 ∧ 𝑆 ∈ Fin) → ∀𝑥 ∈ (♯ “ 𝐽)𝑥 = (♯‘𝑆)) |
33 | ne0i 4302 | . . . . . . 7 ⊢ (𝑆 ∈ 𝐽 → 𝐽 ≠ ∅) | |
34 | 33 | 3ad2ant2 1130 | . . . . . 6 ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽 ∧ 𝑆 ∈ Fin) → 𝐽 ≠ ∅) |
35 | ffn 6516 | . . . . . . . . 9 ⊢ (♯:V⟶(ℕ0 ∪ {+∞}) → ♯ Fn V) | |
36 | 8, 35 | ax-mp 5 | . . . . . . . 8 ⊢ ♯ Fn V |
37 | ssv 3993 | . . . . . . . 8 ⊢ 𝐽 ⊆ V | |
38 | fnimaeq0 6483 | . . . . . . . 8 ⊢ ((♯ Fn V ∧ 𝐽 ⊆ V) → ((♯ “ 𝐽) = ∅ ↔ 𝐽 = ∅)) | |
39 | 36, 37, 38 | mp2an 690 | . . . . . . 7 ⊢ ((♯ “ 𝐽) = ∅ ↔ 𝐽 = ∅) |
40 | 39 | necon3bii 3070 | . . . . . 6 ⊢ ((♯ “ 𝐽) ≠ ∅ ↔ 𝐽 ≠ ∅) |
41 | 34, 40 | sylibr 236 | . . . . 5 ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽 ∧ 𝑆 ∈ Fin) → (♯ “ 𝐽) ≠ ∅) |
42 | eqsn 4764 | . . . . 5 ⊢ ((♯ “ 𝐽) ≠ ∅ → ((♯ “ 𝐽) = {(♯‘𝑆)} ↔ ∀𝑥 ∈ (♯ “ 𝐽)𝑥 = (♯‘𝑆))) | |
43 | 41, 42 | syl 17 | . . . 4 ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽 ∧ 𝑆 ∈ Fin) → ((♯ “ 𝐽) = {(♯‘𝑆)} ↔ ∀𝑥 ∈ (♯ “ 𝐽)𝑥 = (♯‘𝑆))) |
44 | 32, 43 | mpbird 259 | . . 3 ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽 ∧ 𝑆 ∈ Fin) → (♯ “ 𝐽) = {(♯‘𝑆)}) |
45 | 44 | unieqd 4854 | . 2 ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽 ∧ 𝑆 ∈ Fin) → ∪ (♯ “ 𝐽) = ∪ {(♯‘𝑆)}) |
46 | fvex 6685 | . . . 4 ⊢ (♯‘𝑆) ∈ V | |
47 | 46 | unisn 4860 | . . 3 ⊢ ∪ {(♯‘𝑆)} = (♯‘𝑆) |
48 | 47 | a1i 11 | . 2 ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽 ∧ 𝑆 ∈ Fin) → ∪ {(♯‘𝑆)} = (♯‘𝑆)) |
49 | 16, 45, 48 | 3eqtrd 2862 | 1 ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽 ∧ 𝑆 ∈ Fin) → (dim‘𝐹) = (♯‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∀wral 3140 ∃wrex 3141 Vcvv 3496 ∪ cun 3936 ⊆ wss 3938 ∅c0 4293 {csn 4569 ∪ cuni 4840 class class class wbr 5068 “ cima 5560 Fun wfun 6351 Fn wfn 6352 ⟶wf 6353 ‘cfv 6357 ≈ cen 8508 Fincfn 8511 +∞cpnf 10674 ℕ0cn0 11900 ♯chash 13693 LBasisclbs 19848 LVecclvec 19876 dimcldim 31001 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-tpos 7894 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-xnn0 11971 df-z 11985 df-uz 12247 df-hash 13694 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-0g 16717 df-mre 16859 df-mrc 16860 df-mri 16861 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-grp 18108 df-minusg 18109 df-sbg 18110 df-subg 18278 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-oppr 19375 df-dvdsr 19393 df-unit 19394 df-invr 19424 df-drng 19506 df-lmod 19638 df-lss 19706 df-lsp 19746 df-lbs 19849 df-lvec 19877 df-dim 31002 |
This theorem is referenced by: (None) |
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