dimval - Mathbox for Thierry Arnoux

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Theorem dimval 31001

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 3512	. . . 4 ⊢ (𝐹 ∈ LVec → 𝐹 ∈ V)
2		fveq2 6670	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (LBasis‘𝑓) = (LBasis‘𝐹))
3		dimval.1	. . . . . . . 8 ⊢ 𝐽 = (LBasis‘𝐹)
4	2, 3	syl6eqr 2874	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (LBasis‘𝑓) = 𝐽)
5	4	imaeq2d 5929	. . . . . 6 ⊢ (𝑓 = 𝐹 → (♯ “ (LBasis‘𝑓)) = (♯ “ 𝐽))
6	5	unieqd 4852	. . . . 5 ⊢ (𝑓 = 𝐹 → ∪ (♯ “ (LBasis‘𝑓)) = ∪ (♯ “ 𝐽))
7		df-dim 31000	. . . . 5 ⊢ dim = (𝑓 ∈ V ↦ ∪ (♯ “ (LBasis‘𝑓)))
8		hashf 13699	. . . . . . 7 ⊢ ♯:V⟶(ℕ₀ ∪ {+∞})
9		ffun 6517	. . . . . . 7 ⊢ (♯:V⟶(ℕ₀ ∪ {+∞}) → Fun ♯)
10	3	fvexi 6684	. . . . . . . 8 ⊢ 𝐽 ∈ V
11	10	funimaex 6441	. . . . . . 7 ⊢ (Fun ♯ → (♯ “ 𝐽) ∈ V)
12	8, 9, 11	mp2b 10	. . . . . 6 ⊢ (♯ “ 𝐽) ∈ V
13	12	uniex 7467	. . . . 5 ⊢ ∪ (♯ “ 𝐽) ∈ V
14	6, 7, 13	fvmpt 6768	. . . 4 ⊢ (𝐹 ∈ V → (dim‘𝐹) = ∪ (♯ “ 𝐽))
15	1, 14	syl 17	. . 3 ⊢ (𝐹 ∈ LVec → (dim‘𝐹) = ∪ (♯ “ 𝐽))
16	15	adantr 483	. 2 ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) → (dim‘𝐹) = ∪ (♯ “ 𝐽))
17	3	lvecdim 19929	. . . . . . . . . 10 ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽 ∧ 𝑡 ∈ 𝐽) → 𝑆 ≈ 𝑡)
18	17	ad4ant124 1169	. . . . . . . . 9 ⊢ ((((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ (♯ “ 𝐽)) ∧ 𝑡 ∈ 𝐽) → 𝑆 ≈ 𝑡)
19		hasheni 13709	. . . . . . . . 9 ⊢ (𝑆 ≈ 𝑡 → (♯‘𝑆) = (♯‘𝑡))
20	18, 19	syl 17	. . . . . . . 8 ⊢ ((((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ (♯ “ 𝐽)) ∧ 𝑡 ∈ 𝐽) → (♯‘𝑆) = (♯‘𝑡))
21	20	adantr 483	. . . . . . 7 ⊢ (((((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ (♯ “ 𝐽)) ∧ 𝑡 ∈ 𝐽) ∧ (♯‘𝑡) = 𝑥) → (♯‘𝑆) = (♯‘𝑡))
22		simpr 487	. . . . . . 7 ⊢ (((((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ (♯ “ 𝐽)) ∧ 𝑡 ∈ 𝐽) ∧ (♯‘𝑡) = 𝑥) → (♯‘𝑡) = 𝑥)
23	21, 22	eqtr2d 2857	. . . . . 6 ⊢ (((((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ (♯ “ 𝐽)) ∧ 𝑡 ∈ 𝐽) ∧ (♯‘𝑡) = 𝑥) → 𝑥 = (♯‘𝑆))
24	8, 9	ax-mp 5	. . . . . . . 8 ⊢ Fun ♯
25		fvelima 6731	. . . . . . . 8 ⊢ ((Fun ♯ ∧ 𝑥 ∈ (♯ “ 𝐽)) → ∃𝑡 ∈ 𝐽 (♯‘𝑡) = 𝑥)
26	24, 25	mpan 688	. . . . . . 7 ⊢ (𝑥 ∈ (♯ “ 𝐽) → ∃𝑡 ∈ 𝐽 (♯‘𝑡) = 𝑥)
27	26	adantl 484	. . . . . 6 ⊢ (((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ (♯ “ 𝐽)) → ∃𝑡 ∈ 𝐽 (♯‘𝑡) = 𝑥)
28	23, 27	r19.29a 3289	. . . . 5 ⊢ (((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ (♯ “ 𝐽)) → 𝑥 = (♯‘𝑆))
29	28	ralrimiva 3182	. . . 4 ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) → ∀𝑥 ∈ (♯ “ 𝐽)𝑥 = (♯‘𝑆))
30		ne0i 4300	. . . . . . 7 ⊢ (𝑆 ∈ 𝐽 → 𝐽 ≠ ∅)
31	30	adantl 484	. . . . . 6 ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) → 𝐽 ≠ ∅)
32		ffn 6514	. . . . . . . . 9 ⊢ (♯:V⟶(ℕ₀ ∪ {+∞}) → ♯ Fn V)
33	8, 32	ax-mp 5	. . . . . . . 8 ⊢ ♯ Fn V
34		ssv 3991	. . . . . . . 8 ⊢ 𝐽 ⊆ V
35		fnimaeq0 6481	. . . . . . . 8 ⊢ ((♯ Fn V ∧ 𝐽 ⊆ V) → ((♯ “ 𝐽) = ∅ ↔ 𝐽 = ∅))
36	33, 34, 35	mp2an 690	. . . . . . 7 ⊢ ((♯ “ 𝐽) = ∅ ↔ 𝐽 = ∅)
37	36	necon3bii 3068	. . . . . 6 ⊢ ((♯ “ 𝐽) ≠ ∅ ↔ 𝐽 ≠ ∅)
38	31, 37	sylibr 236	. . . . 5 ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) → (♯ “ 𝐽) ≠ ∅)
39		eqsn 4762	. . . . 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 259	. . 3 ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) → (♯ “ 𝐽) = {(♯‘𝑆)})
42	41	unieqd 4852	. 2 ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) → ∪ (♯ “ 𝐽) = ∪ {(♯‘𝑆)})
43		fvex 6683	. . . 4 ⊢ (♯‘𝑆) ∈ V
44	43	unisn 4858	. . 3 ⊢ ∪ {(♯‘𝑆)} = (♯‘𝑆)
45	44	a1i 11	. 2 ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) → ∪ {(♯‘𝑆)} = (♯‘𝑆))
46	16, 42, 45	3eqtrd 2860	1 ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) → (dim‘𝐹) = (♯‘𝑆))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 Vcvv 3494 ∪ cun 3934 ⊆ wss 3936 ∅c0 4291 {csn 4567 ∪ cuni 4838 class class class wbr 5066 “ cima 5558 Fun wfun 6349 Fn wfn 6350 ⟶wf 6351 ‘cfv 6355 ≈ cen 8506 +∞cpnf 10672 ℕ₀cn0 11898 ♯chash 13691 LBasisclbs 19846 LVecclvec 19874 dimcldim 30999

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 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-reg 9056 ax-inf2 9104 ax-ac2 9885 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614

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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-tpos 7892 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-oi 8974 df-r1 9193 df-rank 9194 df-card 9368 df-acn 9371 df-ac 9542 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-xnn0 11969 df-z 11983 df-dec 12100 df-uz 12245 df-fz 12894 df-hash 13692 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-tset 16584 df-ple 16585 df-ocomp 16586 df-0g 16715 df-mre 16857 df-mrc 16858 df-mri 16859 df-acs 16860 df-proset 17538 df-drs 17539 df-poset 17556 df-ipo 17762 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-grp 18106 df-minusg 18107 df-sbg 18108 df-subg 18276 df-cmn 18908 df-abl 18909 df-mgp 19240 df-ur 19252 df-ring 19299 df-oppr 19373 df-dvdsr 19391 df-unit 19392 df-invr 19422 df-drng 19504 df-lmod 19636 df-lss 19704 df-lsp 19744 df-lbs 19847 df-lvec 19875 df-dim 31000

This theorem is referenced by: dimcl 31003 lvecdim0i 31004 lvecdim0 31005 lssdimle 31006 dimpropd 31007 rgmoddim 31008 frlmdim 31009 lsatdim 31015 dimkerim 31023 fedgmul 31027 extdg1id 31053 ccfldextdgrr 31057

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