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Theorem spanss 31106
Description: Ordering relationship for the spans of subsets of Hilbert space. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
spanss ((𝐡 βŠ† β„‹ ∧ 𝐴 βŠ† 𝐡) β†’ (spanβ€˜π΄) βŠ† (spanβ€˜π΅))
Proof of Theorem spanss
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 sstr2 3984 . . . . . 6 (𝐴 βŠ† 𝐡 β†’ (𝐡 βŠ† π‘₯ β†’ 𝐴 βŠ† π‘₯))
21adantr 480 . . . . 5 ((𝐴 βŠ† 𝐡 ∧ π‘₯ ∈ Sβ„‹ ) β†’ (𝐡 βŠ† π‘₯ β†’ 𝐴 βŠ† π‘₯))
32ss2rabdv 4068 . . . 4 (𝐴 βŠ† 𝐡 β†’ {π‘₯ ∈ Sβ„‹ ∣ 𝐡 βŠ† π‘₯} βŠ† {π‘₯ ∈ Sβ„‹ ∣ 𝐴 βŠ† π‘₯})
4 intss 4966 . . . 4 ({π‘₯ ∈ Sβ„‹ ∣ 𝐡 βŠ† π‘₯} βŠ† {π‘₯ ∈ Sβ„‹ ∣ 𝐴 βŠ† π‘₯} β†’ ∩ {π‘₯ ∈ Sβ„‹ ∣ 𝐴 βŠ† π‘₯} βŠ† ∩ {π‘₯ ∈ Sβ„‹ ∣ 𝐡 βŠ† π‘₯})
53, 4syl 17 . . 3 (𝐴 βŠ† 𝐡 β†’ ∩ {π‘₯ ∈ Sβ„‹ ∣ 𝐴 βŠ† π‘₯} βŠ† ∩ {π‘₯ ∈ Sβ„‹ ∣ 𝐡 βŠ† π‘₯})
65adantl 481 . 2 ((𝐡 βŠ† β„‹ ∧ 𝐴 βŠ† 𝐡) β†’ ∩ {π‘₯ ∈ Sβ„‹ ∣ 𝐴 βŠ† π‘₯} βŠ† ∩ {π‘₯ ∈ Sβ„‹ ∣ 𝐡 βŠ† π‘₯})
7 sstr 3985 . . . 4 ((𝐴 βŠ† 𝐡 ∧ 𝐡 βŠ† β„‹) β†’ 𝐴 βŠ† β„‹)
87ancoms 458 . . 3 ((𝐡 βŠ† β„‹ ∧ 𝐴 βŠ† 𝐡) β†’ 𝐴 βŠ† β„‹)
9 spanval 31091 . . 3 (𝐴 βŠ† β„‹ β†’ (spanβ€˜π΄) = ∩ {π‘₯ ∈ Sβ„‹ ∣ 𝐴 βŠ† π‘₯})
108, 9syl 17 . 2 ((𝐡 βŠ† β„‹ ∧ 𝐴 βŠ† 𝐡) β†’ (spanβ€˜π΄) = ∩ {π‘₯ ∈ Sβ„‹ ∣ 𝐴 βŠ† π‘₯})
11 spanval 31091 . . 3 (𝐡 βŠ† β„‹ β†’ (spanβ€˜π΅) = ∩ {π‘₯ ∈ Sβ„‹ ∣ 𝐡 βŠ† π‘₯})
1211adantr 480 . 2 ((𝐡 βŠ† β„‹ ∧ 𝐴 βŠ† 𝐡) β†’ (spanβ€˜π΅) = ∩ {π‘₯ ∈ Sβ„‹ ∣ 𝐡 βŠ† π‘₯})
136, 10, 123sstr4d 4024 1 ((𝐡 βŠ† β„‹ ∧ 𝐴 βŠ† 𝐡) β†’ (spanβ€˜π΄) βŠ† (spanβ€˜π΅))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  {crab 3426   βŠ† wss 3943  βˆ© cint 4943  β€˜cfv 6536   β„‹chba 30677   Sβ„‹ csh 30686  spancspn 30690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-cnex 11165  ax-1cn 11167  ax-addcl 11169  ax-hilex 30757  ax-hfvadd 30758  ax-hv0cl 30761  ax-hfvmul 30763
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7852  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408  df-map 8821  df-nn 12214  df-hlim 30730  df-sh 30965  df-ch 30979  df-span 31067
This theorem is referenced by:  spanssoc  31107  span0  31300  spanuni  31302  spansnpji  31336  shatomistici  32119
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