dimval - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > dimval		Structured version Visualization version GIF version

Theorem dimval 34215

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 3472	. . . 4 ⊢ (𝐹 ∈ LVec → 𝐹 ∈ V)
2		fveq2 6877	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (LBasis‘𝑓) = (LBasis‘𝐹))
3		dimval.1	. . . . . . . 8 ⊢ 𝐽 = (LBasis‘𝐹)
4	2, 3	eqtr4di 2814	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (LBasis‘𝑓) = 𝐽)
5	4	imaeq2d 6054	. . . . . 6 ⊢ (𝑓 = 𝐹 → (♯ “ (LBasis‘𝑓)) = (♯ “ 𝐽))
6	5	unieqd 4880	. . . . 5 ⊢ (𝑓 = 𝐹 → ∪ (♯ “ (LBasis‘𝑓)) = ∪ (♯ “ 𝐽))
7		df-dim 34214	. . . . 5 ⊢ dim = (𝑓 ∈ V ↦ ∪ (♯ “ (LBasis‘𝑓)))
8		hashf 14462	. . . . . . 7 ⊢ ♯:V⟶(ℕ₀ ∪ {+∞})
9		ffun 6704	. . . . . . 7 ⊢ (♯:V⟶(ℕ₀ ∪ {+∞}) → Fun ♯)
10	3	fvexi 6891	. . . . . . . 8 ⊢ 𝐽 ∈ V
11	10	funimaex 6619	. . . . . . 7 ⊢ (Fun ♯ → (♯ “ 𝐽) ∈ V)
12	8, 9, 11	mp2b 10	. . . . . 6 ⊢ (♯ “ 𝐽) ∈ V
13	12	uniex 7747	. . . . 5 ⊢ ∪ (♯ “ 𝐽) ∈ V
14	6, 7, 13	fvmpt 6985	. . . 4 ⊢ (𝐹 ∈ V → (dim‘𝐹) = ∪ (♯ “ 𝐽))
15	1, 14	syl 18	. . 3 ⊢ (𝐹 ∈ LVec → (dim‘𝐹) = ∪ (♯ “ 𝐽))
16	15	adantr 486	. 2 ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) → (dim‘𝐹) = ∪ (♯ “ 𝐽))
17	3	lvecdim 21415	. . . . . . . . . 10 ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽 ∧ 𝑡 ∈ 𝐽) → 𝑆 ≈ 𝑡)
18	17	ad4ant124 1192	. . . . . . . . 9 ⊢ ((((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ (♯ “ 𝐽)) ∧ 𝑡 ∈ 𝐽) → 𝑆 ≈ 𝑡)
19		hasheni 14472	. . . . . . . . 9 ⊢ (𝑆 ≈ 𝑡 → (♯‘𝑆) = (♯‘𝑡))
20	18, 19	syl 18	. . . . . . . 8 ⊢ ((((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ (♯ “ 𝐽)) ∧ 𝑡 ∈ 𝐽) → (♯‘𝑆) = (♯‘𝑡))
21	20	adantr 486	. . . . . . 7 ⊢ (((((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ (♯ “ 𝐽)) ∧ 𝑡 ∈ 𝐽) ∧ (♯‘𝑡) = 𝑥) → (♯‘𝑆) = (♯‘𝑡))
22		simpr 490	. . . . . . 7 ⊢ (((((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ (♯ “ 𝐽)) ∧ 𝑡 ∈ 𝐽) ∧ (♯‘𝑡) = 𝑥) → (♯‘𝑡) = 𝑥)
23	21, 22	eqtr2d 2797	. . . . . 6 ⊢ (((((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ (♯ “ 𝐽)) ∧ 𝑡 ∈ 𝐽) ∧ (♯‘𝑡) = 𝑥) → 𝑥 = (♯‘𝑆))
24	8, 9	ax-mp 5	. . . . . . . 8 ⊢ Fun ♯
25		fvelima 6942	. . . . . . . 8 ⊢ ((Fun ♯ ∧ 𝑥 ∈ (♯ “ 𝐽)) → ∃𝑡 ∈ 𝐽 (♯‘𝑡) = 𝑥)
26	24, 25	mpan 703	. . . . . . 7 ⊢ (𝑥 ∈ (♯ “ 𝐽) → ∃𝑡 ∈ 𝐽 (♯‘𝑡) = 𝑥)
27	26	adantl 487	. . . . . 6 ⊢ (((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ (♯ “ 𝐽)) → ∃𝑡 ∈ 𝐽 (♯‘𝑡) = 𝑥)
28	23, 27	r19.29a 3171	. . . . 5 ⊢ (((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ (♯ “ 𝐽)) → 𝑥 = (♯‘𝑆))
29	28	ralrimiva 3155	. . . 4 ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) → ∀𝑥 ∈ (♯ “ 𝐽)𝑥 = (♯‘𝑆))
30		ne0i 4287	. . . . . . 7 ⊢ (𝑆 ∈ 𝐽 → 𝐽 ≠ ∅)
31	30	adantl 487	. . . . . 6 ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) → 𝐽 ≠ ∅)
32		ffn 6701	. . . . . . . . 9 ⊢ (♯:V⟶(ℕ₀ ∪ {+∞}) → ♯ Fn V)
33	8, 32	ax-mp 5	. . . . . . . 8 ⊢ ♯ Fn V
34		ssv 3955	. . . . . . . 8 ⊢ 𝐽 ⊆ V
35		fnimaeq0 6664	. . . . . . . 8 ⊢ ((♯ Fn V ∧ 𝐽 ⊆ V) → ((♯ “ 𝐽) = ∅ ↔ 𝐽 = ∅))
36	33, 34, 35	mp2an 705	. . . . . . 7 ⊢ ((♯ “ 𝐽) = ∅ ↔ 𝐽 = ∅)
37	36	necon3bii 3008	. . . . . 6 ⊢ ((♯ “ 𝐽) ≠ ∅ ↔ 𝐽 ≠ ∅)
38	31, 37	sylibr 237	. . . . 5 ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) → (♯ “ 𝐽) ≠ ∅)
39		eqsn 4790	. . . . 5 ⊢ ((♯ “ 𝐽) ≠ ∅ → ((♯ “ 𝐽) = {(♯‘𝑆)} ↔ ∀𝑥 ∈ (♯ “ 𝐽)𝑥 = (♯‘𝑆)))
40	38, 39	syl 18	. . . 4 ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) → ((♯ “ 𝐽) = {(♯‘𝑆)} ↔ ∀𝑥 ∈ (♯ “ 𝐽)𝑥 = (♯‘𝑆)))
41	29, 40	mpbird 260	. . 3 ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) → (♯ “ 𝐽) = {(♯‘𝑆)})
42	41	unieqd 4880	. 2 ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) → ∪ (♯ “ 𝐽) = ∪ {(♯‘𝑆)})
43		fvex 6890	. . . 4 ⊢ (♯‘𝑆) ∈ V
44	43	unisn 4886	. . 3 ⊢ ∪ {(♯‘𝑆)} = (♯‘𝑆)
45	44	a1i 11	. 2 ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) → ∪ {(♯‘𝑆)} = (♯‘𝑆))
46	16, 42, 45	3eqtrd 2800	1 ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) → (dim‘𝐹) = (♯‘𝑆))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ≠ wne 2956 ∀wral 3077 ∃wrex 3087 Vcvv 3451 ∪ cun 3897 ⊆ wss 3899 ∅c0 4279 {csn 4584 ∪ cuni 4867 class class class wbr 5103 “ cima 5654 Fun wfun 6525 Fn wfn 6526 ⟶wf 6527 ‘cfv 6531 ≈ cen 8954 +∞cpnf 11321 ℕ₀cn0 12587 ♯chash 14454 LBasisclbs 21329 LVecclvec 21357 dimcldim 34213

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7740 ax-reg 9570 ax-inf2 9626 ax-ac2 10522 ax-cnex 11237 ax-resscn 11238 ax-1cn 11239 ax-icn 11240 ax-addcl 11241 ax-addrcl 11242 ax-mulcl 11243 ax-mulrcl 11244 ax-mulcom 11245 ax-addass 11246 ax-mulass 11247 ax-distr 11248 ax-i2m1 11249 ax-1ne0 11250 ax-1rid 11251 ax-rnegex 11252 ax-rrecex 11253 ax-cnre 11254 ax-pre-lttri 11255 ax-pre-lttrn 11256 ax-pre-ltadd 11257 ax-pre-mulgt0 11258

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6297 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6487 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7867 df-1st 7990 df-2nd 7991 df-tpos 8227 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8460 df-2o 8461 df-er 8701 df-map 8833 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-oi 9488 df-r1 9752 df-rank 9753 df-scott 9910 df-card 10001 df-acn 10004 df-ac 10176 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11524 df-neg 11525 df-nn 12317 df-2 12386 df-3 12387 df-4 12388 df-5 12389 df-6 12390 df-7 12391 df-8 12392 df-9 12393 df-n0 12588 df-xnn0 12661 df-z 12675 df-dec 12796 df-uz 12947 df-fz 13621 df-hash 14455 df-struct 17305 df-sets 17322 df-slot 17340 df-ndx 17352 df-base 17368 df-ress 17389 df-plusg 17421 df-mulr 17422 df-tset 17427 df-ple 17428 df-ocomp 17429 df-0g 17592 df-mre 17736 df-mrc 17737 df-mri 17738 df-acs 17739 df-proset 18448 df-drs 18449 df-poset 18467 df-ipo 18682 df-mgm 18796 df-sgrp 18888 df-mnd 18904 df-submnd 18959 df-grp 19127 df-minusg 19128 df-sbg 19129 df-subg 19313 df-cmn 19976 df-abl 19977 df-mgp 20341 df-rng 20355 df-ur 20388 df-ring 20441 df-oppr 20547 df-dvdsr 20567 df-unit 20568 df-invr 20598 df-drng 20962 df-lmod 21117 df-lss 21187 df-lsp 21227 df-lbs 21330 df-lvec 21358 df-dim 34214

This theorem is used by: dimcl 34217 lmimdim 34218 lvecdim0i 34220 lvecdim0 34221 lssdimle 34222 dimpropd 34223 rlmdim 34224 frlmdim 34225 lsatdim 34231 dimkerim 34241 fedgmul 34245 dimlssid 34246 extdg1id 34280 ccfldextdgrr 34286 fldextrspunlem1 34289 extdgfialglem1 34306

W3C validator