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Theorem spanss 31171
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 3987 . . . . . 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 4071 . . . 4 (𝐴 βŠ† 𝐡 β†’ {π‘₯ ∈ Sβ„‹ ∣ 𝐡 βŠ† π‘₯} βŠ† {π‘₯ ∈ Sβ„‹ ∣ 𝐴 βŠ† π‘₯})
4 intss 4972 . . . 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 3988 . . . 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 31156 . . 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 31156 . . 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 4027 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 1534   ∈ wcel 2099  {crab 3429   βŠ† wss 3947  βˆ© cint 4949  β€˜cfv 6548   β„‹chba 30742   Sβ„‹ csh 30751  spancspn 30755
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-cnex 11195  ax-1cn 11197  ax-addcl 11199  ax-hilex 30822  ax-hfvadd 30823  ax-hv0cl 30826  ax-hfvmul 30828
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-2nd 7994  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-map 8847  df-nn 12244  df-hlim 30795  df-sh 31030  df-ch 31044  df-span 31132
This theorem is referenced by:  spanssoc  31172  span0  31365  spanuni  31367  spansnpji  31401  shatomistici  32184
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