Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dicssdvh Structured version   Visualization version   GIF version
Theorem dicssdvh 40361
Description: The partial isomorphism C maps to a set of vectors in full vector space H. (Contributed by NM, 19-Jan-2014.)
Hypotheses
Ref Expression
dicssdvh.l ≀ = (leβ€˜πΎ)
dicssdvh.a 𝐴 = (Atomsβ€˜πΎ)
dicssdvh.h 𝐻 = (LHypβ€˜πΎ)
dicssdvh.i 𝐼 = ((DIsoCβ€˜πΎ)β€˜π‘Š)
dicssdvh.u π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
dicssdvh.v 𝑉 = (Baseβ€˜π‘ˆ)
Assertion
Ref Expression
dicssdvh (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) β†’ (πΌβ€˜π‘„) βŠ† 𝑉)
Proof of Theorem dicssdvh
Dummy variables 𝑓 𝑔 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simprl 768 . . . . . . 7 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑓 = (π‘ β€˜(℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑄)) ∧ 𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š))) β†’ 𝑓 = (π‘ β€˜(℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑄)))
2 simpll 764 . . . . . . . 8 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑓 = (π‘ β€˜(℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑄)) ∧ 𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š))) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
3 simprr 770 . . . . . . . 8 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑓 = (π‘ β€˜(℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑄)) ∧ 𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š))) β†’ 𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š))
4 dicssdvh.l . . . . . . . . . . 11 ≀ = (leβ€˜πΎ)
5 eqid 2731 . . . . . . . . . . 11 (ocβ€˜πΎ) = (ocβ€˜πΎ)
6 dicssdvh.a . . . . . . . . . . 11 𝐴 = (Atomsβ€˜πΎ)
7 dicssdvh.h . . . . . . . . . . 11 𝐻 = (LHypβ€˜πΎ)
84, 5, 6, 7lhpocnel 39193 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (((ocβ€˜πΎ)β€˜π‘Š) ∈ 𝐴 ∧ Β¬ ((ocβ€˜πΎ)β€˜π‘Š) ≀ π‘Š))
98ad2antrr 723 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑓 = (π‘ β€˜(℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑄)) ∧ 𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š))) β†’ (((ocβ€˜πΎ)β€˜π‘Š) ∈ 𝐴 ∧ Β¬ ((ocβ€˜πΎ)β€˜π‘Š) ≀ π‘Š))
10 simplr 766 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑓 = (π‘ β€˜(℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑄)) ∧ 𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š))) β†’ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š))
11 eqid 2731 . . . . . . . . . 10 ((LTrnβ€˜πΎ)β€˜π‘Š) = ((LTrnβ€˜πΎ)β€˜π‘Š)
12 eqid 2731 . . . . . . . . . 10 (℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑄) = (℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑄)
134, 6, 7, 11, 12ltrniotacl 39754 . . . . . . . . 9 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (((ocβ€˜πΎ)β€˜π‘Š) ∈ 𝐴 ∧ Β¬ ((ocβ€˜πΎ)β€˜π‘Š) ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) β†’ (℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑄) ∈ ((LTrnβ€˜πΎ)β€˜π‘Š))
142, 9, 10, 13syl3anc 1370 . . . . . . . 8 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑓 = (π‘ β€˜(℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑄)) ∧ 𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š))) β†’ (℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑄) ∈ ((LTrnβ€˜πΎ)β€˜π‘Š))
15 eqid 2731 . . . . . . . . 9 ((TEndoβ€˜πΎ)β€˜π‘Š) = ((TEndoβ€˜πΎ)β€˜π‘Š)
167, 11, 15tendocl 39942 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š) ∧ (℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑄) ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)) β†’ (π‘ β€˜(℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑄)) ∈ ((LTrnβ€˜πΎ)β€˜π‘Š))
172, 3, 14, 16syl3anc 1370 . . . . . . 7 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑓 = (π‘ β€˜(℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑄)) ∧ 𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š))) β†’ (π‘ β€˜(℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑄)) ∈ ((LTrnβ€˜πΎ)β€˜π‘Š))
181, 17eqeltrd 2832 . . . . . 6 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑓 = (π‘ β€˜(℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑄)) ∧ 𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š))) β†’ 𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š))
1918, 3, 3jca31 514 . . . . 5 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑓 = (π‘ β€˜(℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑄)) ∧ 𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š))) β†’ ((𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ∧ 𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š)) ∧ 𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š)))
2019ex 412 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) β†’ ((𝑓 = (π‘ β€˜(℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑄)) ∧ 𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š)) β†’ ((𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ∧ 𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š)) ∧ 𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š))))
2120ssopab2dv 5551 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) β†’ {βŸ¨π‘“, π‘ βŸ© ∣ (𝑓 = (π‘ β€˜(℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑄)) ∧ 𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š))} βŠ† {βŸ¨π‘“, π‘ βŸ© ∣ ((𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ∧ 𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š)) ∧ 𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š))})
22 opabssxp 5768 . . 3 {βŸ¨π‘“, π‘ βŸ© ∣ ((𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ∧ 𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š)) ∧ 𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š))} βŠ† (((LTrnβ€˜πΎ)β€˜π‘Š) Γ— ((TEndoβ€˜πΎ)β€˜π‘Š))
2321, 22sstrdi 3994 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) β†’ {βŸ¨π‘“, π‘ βŸ© ∣ (𝑓 = (π‘ β€˜(℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑄)) ∧ 𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š))} βŠ† (((LTrnβ€˜πΎ)β€˜π‘Š) Γ— ((TEndoβ€˜πΎ)β€˜π‘Š)))
24 eqid 2731 . . 3 ((ocβ€˜πΎ)β€˜π‘Š) = ((ocβ€˜πΎ)β€˜π‘Š)
25 dicssdvh.i . . 3 𝐼 = ((DIsoCβ€˜πΎ)β€˜π‘Š)
264, 6, 7, 24, 11, 15, 25dicval 40351 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) β†’ (πΌβ€˜π‘„) = {βŸ¨π‘“, π‘ βŸ© ∣ (𝑓 = (π‘ β€˜(℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑄)) ∧ 𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š))})
27 dicssdvh.u . . . 4 π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
28 dicssdvh.v . . . 4 𝑉 = (Baseβ€˜π‘ˆ)
297, 11, 15, 27, 28dvhvbase 40262 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝑉 = (((LTrnβ€˜πΎ)β€˜π‘Š) Γ— ((TEndoβ€˜πΎ)β€˜π‘Š)))
3029adantr 480 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) β†’ 𝑉 = (((LTrnβ€˜πΎ)β€˜π‘Š) Γ— ((TEndoβ€˜πΎ)β€˜π‘Š)))
3123, 26, 303sstr4d 4029 1 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) β†’ (πΌβ€˜π‘„) βŠ† 𝑉)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105   βŠ† wss 3948   class class class wbr 5148  {copab 5210   Γ— cxp 5674  β€˜cfv 6543  β„©crio 7367  Basecbs 17149  lecple 17209  occoc 17210  Atomscatm 38437  HLchlt 38524  LHypclh 39159  LTrncltrn 39276  TEndoctendo 39927  DVecHcdvh 40253  DIsoCcdic 40347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11170  ax-resscn 11171  ax-1cn 11172  ax-icn 11173  ax-addcl 11174  ax-addrcl 11175  ax-mulcl 11176  ax-mulrcl 11177  ax-mulcom 11178  ax-addass 11179  ax-mulass 11180  ax-distr 11181  ax-i2m1 11182  ax-1ne0 11183  ax-1rid 11184  ax-rnegex 11185  ax-rrecex 11186  ax-cnre 11187  ax-pre-lttri 11188  ax-pre-lttrn 11189  ax-pre-ltadd 11190  ax-pre-mulgt0 11191  ax-riotaBAD 38127
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-undef 8262  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-1o 8470  df-er 8707  df-map 8826  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-nn 12218  df-2 12280  df-3 12281  df-4 12282  df-5 12283  df-6 12284  df-n0 12478  df-z 12564  df-uz 12828  df-fz 13490  df-struct 17085  df-slot 17120  df-ndx 17132  df-base 17150  df-plusg 17215  df-sca 17218  df-vsca 17219  df-proset 18253  df-poset 18271  df-plt 18288  df-lub 18304  df-glb 18305  df-join 18306  df-meet 18307  df-p0 18383  df-p1 18384  df-lat 18390  df-clat 18457  df-oposet 38350  df-ol 38352  df-oml 38353  df-covers 38440  df-ats 38441  df-atl 38472  df-cvlat 38496  df-hlat 38525  df-llines 38673  df-lplanes 38674  df-lvols 38675  df-lines 38676  df-psubsp 38678  df-pmap 38679  df-padd 38971  df-lhyp 39163  df-laut 39164  df-ldil 39279  df-ltrn 39280  df-trl 39334  df-tendo 39930  df-dvech 40254  df-dic 40348
This theorem is referenced by:  dicelval1stN  40363  dicelval2nd  40364  dicvaddcl  40365  dicvscacl  40366  diclss  40368
  Copyright terms: Public domain W3C validator