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Mirrors > Home > HSE Home > Th. List > spansnss | Structured version Visualization version GIF version |
Description: The span of the singleton of an element of a subspace is included in the subspace. (Contributed by NM, 5-Jun-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
spansnss | ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ 𝐴) → (span‘{𝐵}) ⊆ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | shel 28348 | . . . 4 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ ℋ) | |
2 | elspansn 28705 | . . . 4 ⊢ (𝐵 ∈ ℋ → (𝑥 ∈ (span‘{𝐵}) ↔ ∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·ℎ 𝐵))) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ 𝐴) → (𝑥 ∈ (span‘{𝐵}) ↔ ∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·ℎ 𝐵))) |
4 | shmulcl 28355 | . . . . . . . 8 ⊢ ((𝐴 ∈ Sℋ ∧ 𝑦 ∈ ℂ ∧ 𝐵 ∈ 𝐴) → (𝑦 ·ℎ 𝐵) ∈ 𝐴) | |
5 | eleq1a 2822 | . . . . . . . 8 ⊢ ((𝑦 ·ℎ 𝐵) ∈ 𝐴 → (𝑥 = (𝑦 ·ℎ 𝐵) → 𝑥 ∈ 𝐴)) | |
6 | 4, 5 | syl 17 | . . . . . . 7 ⊢ ((𝐴 ∈ Sℋ ∧ 𝑦 ∈ ℂ ∧ 𝐵 ∈ 𝐴) → (𝑥 = (𝑦 ·ℎ 𝐵) → 𝑥 ∈ 𝐴)) |
7 | 6 | 3exp 1112 | . . . . . 6 ⊢ (𝐴 ∈ Sℋ → (𝑦 ∈ ℂ → (𝐵 ∈ 𝐴 → (𝑥 = (𝑦 ·ℎ 𝐵) → 𝑥 ∈ 𝐴)))) |
8 | 7 | com23 86 | . . . . 5 ⊢ (𝐴 ∈ Sℋ → (𝐵 ∈ 𝐴 → (𝑦 ∈ ℂ → (𝑥 = (𝑦 ·ℎ 𝐵) → 𝑥 ∈ 𝐴)))) |
9 | 8 | imp 444 | . . . 4 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ 𝐴) → (𝑦 ∈ ℂ → (𝑥 = (𝑦 ·ℎ 𝐵) → 𝑥 ∈ 𝐴))) |
10 | 9 | rexlimdv 3156 | . . 3 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ 𝐴) → (∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·ℎ 𝐵) → 𝑥 ∈ 𝐴)) |
11 | 3, 10 | sylbid 230 | . 2 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ 𝐴) → (𝑥 ∈ (span‘{𝐵}) → 𝑥 ∈ 𝐴)) |
12 | 11 | ssrdv 3738 | 1 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ 𝐴) → (span‘{𝐵}) ⊆ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 ∧ w3a 1072 = wceq 1620 ∈ wcel 2127 ∃wrex 3039 ⊆ wss 3703 {csn 4309 ‘cfv 6037 (class class class)co 6801 ℂcc 10097 ℋchil 28056 ·ℎ csm 28058 Sℋ csh 28065 spancspn 28069 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1859 ax-4 1874 ax-5 1976 ax-6 2042 ax-7 2078 ax-8 2129 ax-9 2136 ax-10 2156 ax-11 2171 ax-12 2184 ax-13 2379 ax-ext 2728 ax-rep 4911 ax-sep 4921 ax-nul 4929 ax-pow 4980 ax-pr 5043 ax-un 7102 ax-inf2 8699 ax-cc 9420 ax-cnex 10155 ax-resscn 10156 ax-1cn 10157 ax-icn 10158 ax-addcl 10159 ax-addrcl 10160 ax-mulcl 10161 ax-mulrcl 10162 ax-mulcom 10163 ax-addass 10164 ax-mulass 10165 ax-distr 10166 ax-i2m1 10167 ax-1ne0 10168 ax-1rid 10169 ax-rnegex 10170 ax-rrecex 10171 ax-cnre 10172 ax-pre-lttri 10173 ax-pre-lttrn 10174 ax-pre-ltadd 10175 ax-pre-mulgt0 10176 ax-pre-sup 10177 ax-addf 10178 ax-mulf 10179 ax-hilex 28136 ax-hfvadd 28137 ax-hvcom 28138 ax-hvass 28139 ax-hv0cl 28140 ax-hvaddid 28141 ax-hfvmul 28142 ax-hvmulid 28143 ax-hvmulass 28144 ax-hvdistr1 28145 ax-hvdistr2 28146 ax-hvmul0 28147 ax-hfi 28216 ax-his1 28219 ax-his2 28220 ax-his3 28221 ax-his4 28222 ax-hcompl 28339 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1623 df-fal 1626 df-ex 1842 df-nf 1847 df-sb 2035 df-eu 2599 df-mo 2600 df-clab 2735 df-cleq 2741 df-clel 2744 df-nfc 2879 df-ne 2921 df-nel 3024 df-ral 3043 df-rex 3044 df-reu 3045 df-rmo 3046 df-rab 3047 df-v 3330 df-sbc 3565 df-csb 3663 df-dif 3706 df-un 3708 df-in 3710 df-ss 3717 df-pss 3719 df-nul 4047 df-if 4219 df-pw 4292 df-sn 4310 df-pr 4312 df-tp 4314 df-op 4316 df-uni 4577 df-int 4616 df-iun 4662 df-iin 4663 df-br 4793 df-opab 4853 df-mpt 4870 df-tr 4893 df-id 5162 df-eprel 5167 df-po 5175 df-so 5176 df-fr 5213 df-se 5214 df-we 5215 df-xp 5260 df-rel 5261 df-cnv 5262 df-co 5263 df-dm 5264 df-rn 5265 df-res 5266 df-ima 5267 df-pred 5829 df-ord 5875 df-on 5876 df-lim 5877 df-suc 5878 df-iota 6000 df-fun 6039 df-fn 6040 df-f 6041 df-f1 6042 df-fo 6043 df-f1o 6044 df-fv 6045 df-isom 6046 df-riota 6762 df-ov 6804 df-oprab 6805 df-mpt2 6806 df-of 7050 df-om 7219 df-1st 7321 df-2nd 7322 df-supp 7452 df-wrecs 7564 df-recs 7625 df-rdg 7663 df-1o 7717 df-2o 7718 df-oadd 7721 df-omul 7722 df-er 7899 df-map 8013 df-pm 8014 df-ixp 8063 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-fsupp 8429 df-fi 8470 df-sup 8501 df-inf 8502 df-oi 8568 df-card 8926 df-acn 8929 df-cda 9153 df-pnf 10239 df-mnf 10240 df-xr 10241 df-ltxr 10242 df-le 10243 df-sub 10431 df-neg 10432 df-div 10848 df-nn 11184 df-2 11242 df-3 11243 df-4 11244 df-5 11245 df-6 11246 df-7 11247 df-8 11248 df-9 11249 df-n0 11456 df-z 11541 df-dec 11657 df-uz 11851 df-q 11953 df-rp 11997 df-xneg 12110 df-xadd 12111 df-xmul 12112 df-ioo 12343 df-ico 12345 df-icc 12346 df-fz 12491 df-fzo 12631 df-fl 12758 df-seq 12967 df-exp 13026 df-hash 13283 df-cj 14009 df-re 14010 df-im 14011 df-sqrt 14145 df-abs 14146 df-clim 14389 df-rlim 14390 df-sum 14587 df-struct 16032 df-ndx 16033 df-slot 16034 df-base 16036 df-sets 16037 df-ress 16038 df-plusg 16127 df-mulr 16128 df-starv 16129 df-sca 16130 df-vsca 16131 df-ip 16132 df-tset 16133 df-ple 16134 df-ds 16137 df-unif 16138 df-hom 16139 df-cco 16140 df-rest 16256 df-topn 16257 df-0g 16275 df-gsum 16276 df-topgen 16277 df-pt 16278 df-prds 16281 df-xrs 16335 df-qtop 16340 df-imas 16341 df-xps 16343 df-mre 16419 df-mrc 16420 df-acs 16422 df-mgm 17414 df-sgrp 17456 df-mnd 17467 df-submnd 17508 df-mulg 17713 df-cntz 17921 df-cmn 18366 df-psmet 19911 df-xmet 19912 df-met 19913 df-bl 19914 df-mopn 19915 df-fbas 19916 df-fg 19917 df-cnfld 19920 df-top 20872 df-topon 20889 df-topsp 20910 df-bases 20923 df-cld 20996 df-ntr 20997 df-cls 20998 df-nei 21075 df-cn 21204 df-cnp 21205 df-lm 21206 df-haus 21292 df-tx 21538 df-hmeo 21731 df-fil 21822 df-fm 21914 df-flim 21915 df-flf 21916 df-xms 22297 df-ms 22298 df-tms 22299 df-cfil 23224 df-cau 23225 df-cmet 23226 df-grpo 27627 df-gid 27628 df-ginv 27629 df-gdiv 27630 df-ablo 27679 df-vc 27694 df-nv 27727 df-va 27730 df-ba 27731 df-sm 27732 df-0v 27733 df-vs 27734 df-nmcv 27735 df-ims 27736 df-dip 27836 df-ssp 27857 df-ph 27948 df-cbn 27999 df-hnorm 28105 df-hba 28106 df-hvsub 28108 df-hlim 28109 df-hcau 28110 df-sh 28344 df-ch 28358 df-oc 28389 df-ch0 28390 df-span 28448 |
This theorem is referenced by: elspansn3 28711 spansnss2 28714 spansncvi 28791 sh1dle 29490 shatomistici 29500 |
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