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Mirrors > Home > HSE Home > Th. List > spanss | Structured version Visualization version GIF version |
Description: Ordering relationship for the spans of subsets of Hilbert space. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
spanss | β’ ((π΅ β β β§ π΄ β π΅) β (spanβπ΄) β (spanβπ΅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sstr2 3952 | . . . . . 6 β’ (π΄ β π΅ β (π΅ β π₯ β π΄ β π₯)) | |
2 | 1 | adantr 482 | . . . . 5 β’ ((π΄ β π΅ β§ π₯ β Sβ ) β (π΅ β π₯ β π΄ β π₯)) |
3 | 2 | ss2rabdv 4034 | . . . 4 β’ (π΄ β π΅ β {π₯ β Sβ β£ π΅ β π₯} β {π₯ β Sβ β£ π΄ β π₯}) |
4 | intss 4931 | . . . 4 β’ ({π₯ β Sβ β£ π΅ β π₯} β {π₯ β Sβ β£ π΄ β π₯} β β© {π₯ β Sβ β£ π΄ β π₯} β β© {π₯ β Sβ β£ π΅ β π₯}) | |
5 | 3, 4 | syl 17 | . . 3 β’ (π΄ β π΅ β β© {π₯ β Sβ β£ π΄ β π₯} β β© {π₯ β Sβ β£ π΅ β π₯}) |
6 | 5 | adantl 483 | . 2 β’ ((π΅ β β β§ π΄ β π΅) β β© {π₯ β Sβ β£ π΄ β π₯} β β© {π₯ β Sβ β£ π΅ β π₯}) |
7 | sstr 3953 | . . . 4 β’ ((π΄ β π΅ β§ π΅ β β) β π΄ β β) | |
8 | 7 | ancoms 460 | . . 3 β’ ((π΅ β β β§ π΄ β π΅) β π΄ β β) |
9 | spanval 30317 | . . 3 β’ (π΄ β β β (spanβπ΄) = β© {π₯ β Sβ β£ π΄ β π₯}) | |
10 | 8, 9 | syl 17 | . 2 β’ ((π΅ β β β§ π΄ β π΅) β (spanβπ΄) = β© {π₯ β Sβ β£ π΄ β π₯}) |
11 | spanval 30317 | . . 3 β’ (π΅ β β β (spanβπ΅) = β© {π₯ β Sβ β£ π΅ β π₯}) | |
12 | 11 | adantr 482 | . 2 β’ ((π΅ β β β§ π΄ β π΅) β (spanβπ΅) = β© {π₯ β Sβ β£ π΅ β π₯}) |
13 | 6, 10, 12 | 3sstr4d 3992 | 1 β’ ((π΅ β β β§ π΄ β π΅) β (spanβπ΄) β (spanβπ΅)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 {crab 3406 β wss 3911 β© cint 4908 βcfv 6497 βchba 29903 Sβ csh 29912 spancspn 29916 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-1cn 11114 ax-addcl 11116 ax-hilex 29983 ax-hfvadd 29984 ax-hv0cl 29987 ax-hfvmul 29989 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-map 8770 df-nn 12159 df-hlim 29956 df-sh 30191 df-ch 30205 df-span 30293 |
This theorem is referenced by: spanssoc 30333 span0 30526 spanuni 30528 spansnpji 30562 shatomistici 31345 |
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