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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 33495 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 3482 | . . . 4 ⊢ (𝐹 ∈ LVec → 𝐹 ∈ V) | |
2 | fveq2 6901 | . . . . . . . 8 ⊢ (𝑓 = 𝐹 → (LBasis‘𝑓) = (LBasis‘𝐹)) | |
3 | dimval.1 | . . . . . . . 8 ⊢ 𝐽 = (LBasis‘𝐹) | |
4 | 2, 3 | eqtr4di 2784 | . . . . . . 7 ⊢ (𝑓 = 𝐹 → (LBasis‘𝑓) = 𝐽) |
5 | 4 | imaeq2d 6069 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (♯ “ (LBasis‘𝑓)) = (♯ “ 𝐽)) |
6 | 5 | unieqd 4926 | . . . . 5 ⊢ (𝑓 = 𝐹 → ∪ (♯ “ (LBasis‘𝑓)) = ∪ (♯ “ 𝐽)) |
7 | df-dim 33494 | . . . . 5 ⊢ dim = (𝑓 ∈ V ↦ ∪ (♯ “ (LBasis‘𝑓))) | |
8 | hashf 14355 | . . . . . . 7 ⊢ ♯:V⟶(ℕ0 ∪ {+∞}) | |
9 | ffun 6731 | . . . . . . 7 ⊢ (♯:V⟶(ℕ0 ∪ {+∞}) → Fun ♯) | |
10 | 3 | fvexi 6915 | . . . . . . . 8 ⊢ 𝐽 ∈ V |
11 | 10 | funimaex 6647 | . . . . . . 7 ⊢ (Fun ♯ → (♯ “ 𝐽) ∈ V) |
12 | 8, 9, 11 | mp2b 10 | . . . . . 6 ⊢ (♯ “ 𝐽) ∈ V |
13 | 12 | uniex 7752 | . . . . 5 ⊢ ∪ (♯ “ 𝐽) ∈ V |
14 | 6, 7, 13 | fvmpt 7009 | . . . 4 ⊢ (𝐹 ∈ V → (dim‘𝐹) = ∪ (♯ “ 𝐽)) |
15 | 1, 14 | syl 17 | . . 3 ⊢ (𝐹 ∈ LVec → (dim‘𝐹) = ∪ (♯ “ 𝐽)) |
16 | 15 | 3ad2ant1 1130 | . 2 ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽 ∧ 𝑆 ∈ Fin) → (dim‘𝐹) = ∪ (♯ “ 𝐽)) |
17 | simpll1 1209 | . . . . . . . . . 10 ⊢ ((((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽 ∧ 𝑆 ∈ Fin) ∧ 𝑥 ∈ (♯ “ 𝐽)) ∧ 𝑡 ∈ 𝐽) → 𝐹 ∈ LVec) | |
18 | simpll2 1210 | . . . . . . . . . 10 ⊢ ((((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽 ∧ 𝑆 ∈ Fin) ∧ 𝑥 ∈ (♯ “ 𝐽)) ∧ 𝑡 ∈ 𝐽) → 𝑆 ∈ 𝐽) | |
19 | simpr 483 | . . . . . . . . . 10 ⊢ ((((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽 ∧ 𝑆 ∈ Fin) ∧ 𝑥 ∈ (♯ “ 𝐽)) ∧ 𝑡 ∈ 𝐽) → 𝑡 ∈ 𝐽) | |
20 | simpll3 1211 | . . . . . . . . . 10 ⊢ ((((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽 ∧ 𝑆 ∈ Fin) ∧ 𝑥 ∈ (♯ “ 𝐽)) ∧ 𝑡 ∈ 𝐽) → 𝑆 ∈ Fin) | |
21 | 3, 17, 18, 19, 20 | lvecdimfi 33492 | . . . . . . . . 9 ⊢ ((((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽 ∧ 𝑆 ∈ Fin) ∧ 𝑥 ∈ (♯ “ 𝐽)) ∧ 𝑡 ∈ 𝐽) → 𝑆 ≈ 𝑡) |
22 | hasheni 14365 | . . . . . . . . 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 479 | . . . . . . 7 ⊢ (((((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽 ∧ 𝑆 ∈ Fin) ∧ 𝑥 ∈ (♯ “ 𝐽)) ∧ 𝑡 ∈ 𝐽) ∧ (♯‘𝑡) = 𝑥) → (♯‘𝑆) = (♯‘𝑡)) |
25 | simpr 483 | . . . . . . 7 ⊢ (((((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽 ∧ 𝑆 ∈ Fin) ∧ 𝑥 ∈ (♯ “ 𝐽)) ∧ 𝑡 ∈ 𝐽) ∧ (♯‘𝑡) = 𝑥) → (♯‘𝑡) = 𝑥) | |
26 | 24, 25 | eqtr2d 2767 | . . . . . 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 6968 | . . . . . . . 8 ⊢ ((Fun ♯ ∧ 𝑥 ∈ (♯ “ 𝐽)) → ∃𝑡 ∈ 𝐽 (♯‘𝑡) = 𝑥) | |
29 | 27, 28 | mpan 688 | . . . . . . 7 ⊢ (𝑥 ∈ (♯ “ 𝐽) → ∃𝑡 ∈ 𝐽 (♯‘𝑡) = 𝑥) |
30 | 29 | adantl 480 | . . . . . 6 ⊢ (((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽 ∧ 𝑆 ∈ Fin) ∧ 𝑥 ∈ (♯ “ 𝐽)) → ∃𝑡 ∈ 𝐽 (♯‘𝑡) = 𝑥) |
31 | 26, 30 | r19.29a 3152 | . . . . 5 ⊢ (((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽 ∧ 𝑆 ∈ Fin) ∧ 𝑥 ∈ (♯ “ 𝐽)) → 𝑥 = (♯‘𝑆)) |
32 | 31 | ralrimiva 3136 | . . . 4 ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽 ∧ 𝑆 ∈ Fin) → ∀𝑥 ∈ (♯ “ 𝐽)𝑥 = (♯‘𝑆)) |
33 | ne0i 4337 | . . . . . . 7 ⊢ (𝑆 ∈ 𝐽 → 𝐽 ≠ ∅) | |
34 | 33 | 3ad2ant2 1131 | . . . . . 6 ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽 ∧ 𝑆 ∈ Fin) → 𝐽 ≠ ∅) |
35 | ffn 6728 | . . . . . . . . 9 ⊢ (♯:V⟶(ℕ0 ∪ {+∞}) → ♯ Fn V) | |
36 | 8, 35 | ax-mp 5 | . . . . . . . 8 ⊢ ♯ Fn V |
37 | ssv 4004 | . . . . . . . 8 ⊢ 𝐽 ⊆ V | |
38 | fnimaeq0 6694 | . . . . . . . 8 ⊢ ((♯ Fn V ∧ 𝐽 ⊆ V) → ((♯ “ 𝐽) = ∅ ↔ 𝐽 = ∅)) | |
39 | 36, 37, 38 | mp2an 690 | . . . . . . 7 ⊢ ((♯ “ 𝐽) = ∅ ↔ 𝐽 = ∅) |
40 | 39 | necon3bii 2983 | . . . . . 6 ⊢ ((♯ “ 𝐽) ≠ ∅ ↔ 𝐽 ≠ ∅) |
41 | 34, 40 | sylibr 233 | . . . . 5 ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽 ∧ 𝑆 ∈ Fin) → (♯ “ 𝐽) ≠ ∅) |
42 | eqsn 4838 | . . . . 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 256 | . . 3 ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽 ∧ 𝑆 ∈ Fin) → (♯ “ 𝐽) = {(♯‘𝑆)}) |
45 | 44 | unieqd 4926 | . 2 ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽 ∧ 𝑆 ∈ Fin) → ∪ (♯ “ 𝐽) = ∪ {(♯‘𝑆)}) |
46 | fvex 6914 | . . . 4 ⊢ (♯‘𝑆) ∈ V | |
47 | 46 | unisn 4934 | . . 3 ⊢ ∪ {(♯‘𝑆)} = (♯‘𝑆) |
48 | 47 | a1i 11 | . 2 ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽 ∧ 𝑆 ∈ Fin) → ∪ {(♯‘𝑆)} = (♯‘𝑆)) |
49 | 16, 45, 48 | 3eqtrd 2770 | 1 ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽 ∧ 𝑆 ∈ Fin) → (dim‘𝐹) = (♯‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∀wral 3051 ∃wrex 3060 Vcvv 3462 ∪ cun 3945 ⊆ wss 3947 ∅c0 4325 {csn 4633 ∪ cuni 4913 class class class wbr 5153 “ cima 5685 Fun wfun 6548 Fn wfn 6549 ⟶wf 6550 ‘cfv 6554 ≈ cen 8971 Fincfn 8974 +∞cpnf 11295 ℕ0cn0 12524 ♯chash 14347 LBasisclbs 21052 LVecclvec 21080 dimcldim 33493 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-iin 5004 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-tpos 8241 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-2o 8497 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-card 9982 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-xnn0 12597 df-z 12611 df-uz 12875 df-hash 14348 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-ress 17243 df-plusg 17279 df-mulr 17280 df-0g 17456 df-mre 17599 df-mrc 17600 df-mri 17601 df-acs 17602 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-submnd 18774 df-grp 18931 df-minusg 18932 df-sbg 18933 df-subg 19117 df-cmn 19780 df-abl 19781 df-mgp 20118 df-rng 20136 df-ur 20165 df-ring 20218 df-oppr 20316 df-dvdsr 20339 df-unit 20340 df-invr 20370 df-drng 20709 df-lmod 20838 df-lss 20909 df-lsp 20949 df-lbs 21053 df-lvec 21081 df-dim 33494 |
This theorem is referenced by: ply1degltdim 33518 |
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