dimval - Mathbox for Thierry Arnoux

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Theorem dimval 31822

Description: The dimension of a vector space 𝐹 is the cardinality of one of its bases. (Contributed by Thierry Arnoux, 6-May-2023.)

**Hypothesis**
Ref	Expression
dimval.1	⊢ 𝐽 = (LBasis‘𝐹)

**Assertion**
Ref	Expression
dimval	⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) → (dim‘𝐹) = (♯‘𝑆))

Proof of Theorem dimval Dummy variables 𝑡 𝑥 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3458	. . . 4 ⊢ (𝐹 ∈ LVec → 𝐹 ∈ V)
2		fveq2 6811	. . . . . . . 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 5986	. . . . . 6 ⊢ (𝑓 = 𝐹 → (♯ “ (LBasis‘𝑓)) = (♯ “ 𝐽))
6	5	unieqd 4863	. . . . 5 ⊢ (𝑓 = 𝐹 → ∪ (♯ “ (LBasis‘𝑓)) = ∪ (♯ “ 𝐽))
7		df-dim 31821	. . . . 5 ⊢ dim = (𝑓 ∈ V ↦ ∪ (♯ “ (LBasis‘𝑓)))
8		hashf 14131	. . . . . . 7 ⊢ ♯:V⟶(ℕ₀ ∪ {+∞})
9		ffun 6640	. . . . . . 7 ⊢ (♯:V⟶(ℕ₀ ∪ {+∞}) → Fun ♯)
10	3	fvexi 6825	. . . . . . . 8 ⊢ 𝐽 ∈ V
11	10	funimaex 6558	. . . . . . 7 ⊢ (Fun ♯ → (♯ “ 𝐽) ∈ V)
12	8, 9, 11	mp2b 10	. . . . . 6 ⊢ (♯ “ 𝐽) ∈ V
13	12	uniex 7635	. . . . 5 ⊢ ∪ (♯ “ 𝐽) ∈ V
14	6, 7, 13	fvmpt 6914	. . . 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 20499	. . . . . . . . . 10 ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽 ∧ 𝑡 ∈ 𝐽) → 𝑆 ≈ 𝑡)
18	17	ad4ant124 1172	. . . . . . . . 9 ⊢ ((((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ (♯ “ 𝐽)) ∧ 𝑡 ∈ 𝐽) → 𝑆 ≈ 𝑡)
19		hasheni 14141	. . . . . . . . 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 6874	. . . . . . . 8 ⊢ ((Fun ♯ ∧ 𝑥 ∈ (♯ “ 𝐽)) → ∃𝑡 ∈ 𝐽 (♯‘𝑡) = 𝑥)
26	24, 25	mpan 687	. . . . . . 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 3155	. . . . 5 ⊢ (((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ (♯ “ 𝐽)) → 𝑥 = (♯‘𝑆))
29	28	ralrimiva 3139	. . . 4 ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) → ∀𝑥 ∈ (♯ “ 𝐽)𝑥 = (♯‘𝑆))
30		ne0i 4278	. . . . . . 7 ⊢ (𝑆 ∈ 𝐽 → 𝐽 ≠ ∅)
31	30	adantl 482	. . . . . 6 ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) → 𝐽 ≠ ∅)
32		ffn 6637	. . . . . . . . 9 ⊢ (♯:V⟶(ℕ₀ ∪ {+∞}) → ♯ Fn V)
33	8, 32	ax-mp 5	. . . . . . . 8 ⊢ ♯ Fn V
34		ssv 3954	. . . . . . . 8 ⊢ 𝐽 ⊆ V
35		fnimaeq0 6603	. . . . . . . 8 ⊢ ((♯ Fn V ∧ 𝐽 ⊆ V) → ((♯ “ 𝐽) = ∅ ↔ 𝐽 = ∅))
36	33, 34, 35	mp2an 689	. . . . . . 7 ⊢ ((♯ “ 𝐽) = ∅ ↔ 𝐽 = ∅)
37	36	necon3bii 2993	. . . . . 6 ⊢ ((♯ “ 𝐽) ≠ ∅ ↔ 𝐽 ≠ ∅)
38	31, 37	sylibr 233	. . . . 5 ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) → (♯ “ 𝐽) ≠ ∅)
39		eqsn 4773	. . . . 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 4863	. 2 ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) → ∪ (♯ “ 𝐽) = ∪ {(♯‘𝑆)})
43		fvex 6824	. . . 4 ⊢ (♯‘𝑆) ∈ V
44	43	unisn 4871	. . 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 1540 ∈ wcel 2105 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 Vcvv 3440 ∪ cun 3894 ⊆ wss 3896 ∅c0 4266 {csn 4570 ∪ cuni 4849 class class class wbr 5086 “ cima 5610 Fun wfun 6459 Fn wfn 6460 ⟶wf 6461 ‘cfv 6465 ≈ cen 8779 +∞cpnf 11085 ℕ₀cn0 12312 ♯chash 14123 LBasisclbs 20416 LVecclvec 20444 dimcldim 31820

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 2707 ax-rep 5223 ax-sep 5237 ax-nul 5244 ax-pow 5302 ax-pr 5366 ax-un 7629 ax-reg 9427 ax-inf2 9476 ax-ac2 10298 ax-cnex 11006 ax-resscn 11007 ax-1cn 11008 ax-icn 11009 ax-addcl 11010 ax-addrcl 11011 ax-mulcl 11012 ax-mulrcl 11013 ax-mulcom 11014 ax-addass 11015 ax-mulass 11016 ax-distr 11017 ax-i2m1 11018 ax-1ne0 11019 ax-1rid 11020 ax-rnegex 11021 ax-rrecex 11022 ax-cnre 11023 ax-pre-lttri 11024 ax-pre-lttrn 11025 ax-pre-ltadd 11026 ax-pre-mulgt0 11027

This theorem depends on definitions: df-bi 206 df-an 397 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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3442 df-sbc 3726 df-csb 3842 df-dif 3899 df-un 3901 df-in 3903 df-ss 3913 df-pss 3915 df-nul 4267 df-if 4471 df-pw 4546 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4850 df-int 4892 df-iun 4938 df-iin 4939 df-br 5087 df-opab 5149 df-mpt 5170 df-tr 5204 df-id 5506 df-eprel 5512 df-po 5520 df-so 5521 df-fr 5562 df-se 5563 df-we 5564 df-xp 5613 df-rel 5614 df-cnv 5615 df-co 5616 df-dm 5617 df-rn 5618 df-res 5619 df-ima 5620 df-pred 6224 df-ord 6291 df-on 6292 df-lim 6293 df-suc 6294 df-iota 6417 df-fun 6467 df-fn 6468 df-f 6469 df-f1 6470 df-fo 6471 df-f1o 6472 df-fv 6473 df-isom 6474 df-riota 7273 df-ov 7319 df-oprab 7320 df-mpo 7321 df-om 7759 df-1st 7877 df-2nd 7878 df-tpos 8090 df-frecs 8145 df-wrecs 8176 df-recs 8250 df-rdg 8289 df-1o 8345 df-er 8547 df-map 8666 df-en 8783 df-dom 8784 df-sdom 8785 df-fin 8786 df-oi 9345 df-r1 9599 df-rank 9600 df-card 9774 df-acn 9777 df-ac 9951 df-pnf 11090 df-mnf 11091 df-xr 11092 df-ltxr 11093 df-le 11094 df-sub 11286 df-neg 11287 df-nn 12053 df-2 12115 df-3 12116 df-4 12117 df-5 12118 df-6 12119 df-7 12120 df-8 12121 df-9 12122 df-n0 12313 df-xnn0 12385 df-z 12399 df-dec 12517 df-uz 12662 df-fz 13319 df-hash 14124 df-struct 16922 df-sets 16939 df-slot 16957 df-ndx 16969 df-base 16987 df-ress 17016 df-plusg 17049 df-mulr 17050 df-tset 17055 df-ple 17056 df-ocomp 17057 df-0g 17226 df-mre 17369 df-mrc 17370 df-mri 17371 df-acs 17372 df-proset 18087 df-drs 18088 df-poset 18105 df-ipo 18320 df-mgm 18400 df-sgrp 18449 df-mnd 18460 df-submnd 18505 df-grp 18653 df-minusg 18654 df-sbg 18655 df-subg 18825 df-cmn 19460 df-abl 19461 df-mgp 19793 df-ur 19810 df-ring 19857 df-oppr 19934 df-dvdsr 19955 df-unit 19956 df-invr 19986 df-drng 20069 df-lmod 20205 df-lss 20274 df-lsp 20314 df-lbs 20417 df-lvec 20445 df-dim 31821

This theorem is referenced by: dimcl 31824 lvecdim0i 31825 lvecdim0 31826 lssdimle 31827 dimpropd 31828 rgmoddim 31829 frlmdim 31830 lsatdim 31836 dimkerim 31844 fedgmul 31848 extdg1id 31874 ccfldextdgrr 31878

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