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Mirrors > Home > HSE Home > Th. List > span0 | Structured version Visualization version GIF version |
Description: The span of the empty set is the zero subspace. Remark 11.6.e of [Schechter] p. 276. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
span0 | ⊢ (span‘∅) = 0ℋ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | h0elsh 28664 | . . . . 5 ⊢ 0ℋ ∈ Sℋ | |
2 | 1 | shssii 28621 | . . . 4 ⊢ 0ℋ ⊆ ℋ |
3 | 0ss 4199 | . . . 4 ⊢ ∅ ⊆ 0ℋ | |
4 | spanss 28758 | . . . 4 ⊢ ((0ℋ ⊆ ℋ ∧ ∅ ⊆ 0ℋ) → (span‘∅) ⊆ (span‘0ℋ)) | |
5 | 2, 3, 4 | mp2an 683 | . . 3 ⊢ (span‘∅) ⊆ (span‘0ℋ) |
6 | spanid 28757 | . . . 4 ⊢ (0ℋ ∈ Sℋ → (span‘0ℋ) = 0ℋ) | |
7 | 1, 6 | ax-mp 5 | . . 3 ⊢ (span‘0ℋ) = 0ℋ |
8 | 5, 7 | sseqtri 3862 | . 2 ⊢ (span‘∅) ⊆ 0ℋ |
9 | 0ss 4199 | . . . 4 ⊢ ∅ ⊆ ℋ | |
10 | spancl 28746 | . . . 4 ⊢ (∅ ⊆ ℋ → (span‘∅) ∈ Sℋ ) | |
11 | 9, 10 | ax-mp 5 | . . 3 ⊢ (span‘∅) ∈ Sℋ |
12 | sh0le 28850 | . . 3 ⊢ ((span‘∅) ∈ Sℋ → 0ℋ ⊆ (span‘∅)) | |
13 | 11, 12 | ax-mp 5 | . 2 ⊢ 0ℋ ⊆ (span‘∅) |
14 | 8, 13 | eqssi 3843 | 1 ⊢ (span‘∅) = 0ℋ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1656 ∈ wcel 2164 ⊆ wss 3798 ∅c0 4146 ‘cfv 6127 ℋchba 28327 Sℋ csh 28336 spancspn 28340 0ℋc0h 28343 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 ax-pre-sup 10337 ax-addf 10338 ax-mulf 10339 ax-hilex 28407 ax-hfvadd 28408 ax-hvcom 28409 ax-hvass 28410 ax-hv0cl 28411 ax-hvaddid 28412 ax-hfvmul 28413 ax-hvmulid 28414 ax-hvmulass 28415 ax-hvdistr1 28416 ax-hvdistr2 28417 ax-hvmul0 28418 ax-hfi 28487 ax-his1 28490 ax-his2 28491 ax-his3 28492 ax-his4 28493 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-er 8014 df-map 8129 df-pm 8130 df-en 8229 df-dom 8230 df-sdom 8231 df-sup 8623 df-inf 8624 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-div 11017 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-n0 11626 df-z 11712 df-uz 11976 df-q 12079 df-rp 12120 df-xneg 12239 df-xadd 12240 df-xmul 12241 df-icc 12477 df-seq 13103 df-exp 13162 df-cj 14223 df-re 14224 df-im 14225 df-sqrt 14359 df-abs 14360 df-topgen 16464 df-psmet 20105 df-xmet 20106 df-met 20107 df-bl 20108 df-mopn 20109 df-top 21076 df-topon 21093 df-bases 21128 df-lm 21411 df-haus 21497 df-grpo 27899 df-gid 27900 df-ginv 27901 df-gdiv 27902 df-ablo 27951 df-vc 27965 df-nv 27998 df-va 28001 df-ba 28002 df-sm 28003 df-0v 28004 df-vs 28005 df-nmcv 28006 df-ims 28007 df-hnorm 28376 df-hvsub 28379 df-hlim 28380 df-sh 28615 df-ch 28629 df-ch0 28661 df-span 28719 |
This theorem is referenced by: (None) |
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