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Theorem spanss 29137
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 ((𝐵 ⊆ ℋ ∧ 𝐴𝐵) → (span‘𝐴) ⊆ (span‘𝐵))

Proof of Theorem spanss
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 sstr2 3961 . . . . . 6 (𝐴𝐵 → (𝐵𝑥𝐴𝑥))
21ralrimivw 3178 . . . . 5 (𝐴𝐵 → ∀𝑥S (𝐵𝑥𝐴𝑥))
3 ss2rab 4034 . . . . 5 ({𝑥S𝐵𝑥} ⊆ {𝑥S𝐴𝑥} ↔ ∀𝑥S (𝐵𝑥𝐴𝑥))
42, 3sylibr 237 . . . 4 (𝐴𝐵 → {𝑥S𝐵𝑥} ⊆ {𝑥S𝐴𝑥})
5 intss 4884 . . . 4 ({𝑥S𝐵𝑥} ⊆ {𝑥S𝐴𝑥} → {𝑥S𝐴𝑥} ⊆ {𝑥S𝐵𝑥})
64, 5syl 17 . . 3 (𝐴𝐵 {𝑥S𝐴𝑥} ⊆ {𝑥S𝐵𝑥})
76adantl 485 . 2 ((𝐵 ⊆ ℋ ∧ 𝐴𝐵) → {𝑥S𝐴𝑥} ⊆ {𝑥S𝐵𝑥})
8 sstr 3962 . . . 4 ((𝐴𝐵𝐵 ⊆ ℋ) → 𝐴 ⊆ ℋ)
98ancoms 462 . . 3 ((𝐵 ⊆ ℋ ∧ 𝐴𝐵) → 𝐴 ⊆ ℋ)
10 spanval 29122 . . 3 (𝐴 ⊆ ℋ → (span‘𝐴) = {𝑥S𝐴𝑥})
119, 10syl 17 . 2 ((𝐵 ⊆ ℋ ∧ 𝐴𝐵) → (span‘𝐴) = {𝑥S𝐴𝑥})
12 spanval 29122 . . 3 (𝐵 ⊆ ℋ → (span‘𝐵) = {𝑥S𝐵𝑥})
1312adantr 484 . 2 ((𝐵 ⊆ ℋ ∧ 𝐴𝐵) → (span‘𝐵) = {𝑥S𝐵𝑥})
147, 11, 133sstr4d 4001 1 ((𝐵 ⊆ ℋ ∧ 𝐴𝐵) → (span‘𝐴) ⊆ (span‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wral 3133  {crab 3137  wss 3920   cint 4863  cfv 6344  chba 28708   S csh 28717  spancspn 28721
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7456  ax-cnex 10592  ax-1cn 10594  ax-addcl 10596  ax-hilex 28788  ax-hfvadd 28789  ax-hv0cl 28792  ax-hfvmul 28794
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3483  df-sbc 3760  df-csb 3868  df-dif 3923  df-un 3925  df-in 3927  df-ss 3937  df-pss 3939  df-nul 4278  df-if 4452  df-pw 4525  df-sn 4552  df-pr 4554  df-tp 4556  df-op 4558  df-uni 4826  df-int 4864  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-tr 5160  df-id 5448  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-we 5504  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-pred 6136  df-ord 6182  df-on 6183  df-lim 6184  df-suc 6185  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7576  df-wrecs 7944  df-recs 8005  df-rdg 8043  df-map 8405  df-nn 11638  df-hlim 28761  df-sh 28996  df-ch 29010  df-span 29098
This theorem is referenced by:  spanssoc  29138  span0  29331  spanuni  29333  spansnpji  29367  shatomistici  30150
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