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Mirrors > Home > HSE Home > Th. List > elspansncl | Structured version Visualization version GIF version |
Description: A member of a span of a singleton is a vector. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
elspansncl | ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ (span‘{𝐴})) → 𝐵 ∈ ℋ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snssi 4839 | . 2 ⊢ (𝐴 ∈ ℋ → {𝐴} ⊆ ℋ) | |
2 | elspancl 31372 | . 2 ⊢ (({𝐴} ⊆ ℋ ∧ 𝐵 ∈ (span‘{𝐴})) → 𝐵 ∈ ℋ) | |
3 | 1, 2 | sylan 579 | 1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ (span‘{𝐴})) → 𝐵 ∈ ℋ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2103 ⊆ wss 3976 {csn 4654 ‘cfv 6579 ℋchba 30954 spancspn 30967 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5313 ax-sep 5327 ax-nul 5334 ax-pow 5393 ax-pr 5457 ax-un 7775 ax-cnex 11245 ax-resscn 11246 ax-1cn 11247 ax-icn 11248 ax-addcl 11249 ax-addrcl 11250 ax-mulcl 11251 ax-mulrcl 11252 ax-mulcom 11253 ax-addass 11254 ax-mulass 11255 ax-distr 11256 ax-i2m1 11257 ax-1ne0 11258 ax-1rid 11259 ax-rnegex 11260 ax-rrecex 11261 ax-cnre 11262 ax-pre-lttri 11263 ax-pre-lttrn 11264 ax-pre-ltadd 11265 ax-pre-mulgt0 11266 ax-pre-sup 11267 ax-addf 11268 ax-mulf 11269 ax-hilex 31034 ax-hfvadd 31035 ax-hvcom 31036 ax-hvass 31037 ax-hv0cl 31038 ax-hvaddid 31039 ax-hfvmul 31040 ax-hvmulid 31041 ax-hvmulass 31042 ax-hvdistr1 31043 ax-hvdistr2 31044 ax-hvmul0 31045 ax-hfi 31114 ax-his1 31117 ax-his2 31118 ax-his3 31119 ax-his4 31120 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3384 df-reu 3385 df-rab 3440 df-v 3486 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4354 df-if 4555 df-pw 4630 df-sn 4655 df-pr 4657 df-op 4661 df-uni 4938 df-int 4979 df-iun 5027 df-br 5177 df-opab 5239 df-mpt 5260 df-tr 5294 df-id 5604 df-eprel 5610 df-po 5618 df-so 5619 df-fr 5661 df-we 5663 df-xp 5712 df-rel 5713 df-cnv 5714 df-co 5715 df-dm 5716 df-rn 5717 df-res 5718 df-ima 5719 df-pred 6338 df-ord 6404 df-on 6405 df-lim 6406 df-suc 6407 df-iota 6531 df-fun 6581 df-fn 6582 df-f 6583 df-f1 6584 df-fo 6585 df-f1o 6586 df-fv 6587 df-riota 7410 df-ov 7457 df-oprab 7458 df-mpo 7459 df-om 7909 df-1st 8035 df-2nd 8036 df-frecs 8327 df-wrecs 8358 df-recs 8432 df-rdg 8471 df-er 8768 df-map 8891 df-pm 8892 df-en 9009 df-dom 9010 df-sdom 9011 df-sup 9516 df-inf 9517 df-pnf 11331 df-mnf 11332 df-xr 11333 df-ltxr 11334 df-le 11335 df-sub 11527 df-neg 11528 df-div 11953 df-nn 12299 df-2 12361 df-3 12362 df-4 12363 df-n0 12559 df-z 12645 df-uz 12909 df-q 13019 df-rp 13063 df-xneg 13180 df-xadd 13181 df-xmul 13182 df-icc 13419 df-seq 14058 df-exp 14118 df-cj 15153 df-re 15154 df-im 15155 df-sqrt 15289 df-abs 15290 df-topgen 17509 df-psmet 21385 df-xmet 21386 df-met 21387 df-bl 21388 df-mopn 21389 df-top 22925 df-topon 22942 df-bases 22978 df-lm 23262 df-haus 23348 df-grpo 30528 df-gid 30529 df-ginv 30530 df-gdiv 30531 df-ablo 30580 df-vc 30594 df-nv 30627 df-va 30630 df-ba 30631 df-sm 30632 df-0v 30633 df-vs 30634 df-nmcv 30635 df-ims 30636 df-hnorm 31003 df-hvsub 31006 df-hlim 31007 df-sh 31242 df-ch 31256 df-ch0 31288 df-span 31344 |
This theorem is referenced by: sumdmdlem 32453 |
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