Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dibelval1st2N Structured version   Visualization version   GIF version

Theorem dibelval1st2N 40676
Description: Membership in value of the partial isomorphism B for a lattice 𝐾. (Contributed by NM, 13-Feb-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
dibelval1st2.b 𝐡 = (Baseβ€˜πΎ)
dibelval1st2.l ≀ = (leβ€˜πΎ)
dibelval1st2.h 𝐻 = (LHypβ€˜πΎ)
dibelval1st2.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
dibelval1st2.r 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
dibelval1st2.i 𝐼 = ((DIsoBβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
dibelval1st2N (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š) ∧ π‘Œ ∈ (πΌβ€˜π‘‹)) β†’ (π‘…β€˜(1st β€˜π‘Œ)) ≀ 𝑋)

Proof of Theorem dibelval1st2N
StepHypRef Expression
1 dibelval1st2.b . . 3 𝐡 = (Baseβ€˜πΎ)
2 dibelval1st2.l . . 3 ≀ = (leβ€˜πΎ)
3 dibelval1st2.h . . 3 𝐻 = (LHypβ€˜πΎ)
4 eqid 2725 . . 3 ((DIsoAβ€˜πΎ)β€˜π‘Š) = ((DIsoAβ€˜πΎ)β€˜π‘Š)
5 dibelval1st2.i . . 3 𝐼 = ((DIsoBβ€˜πΎ)β€˜π‘Š)
61, 2, 3, 4, 5dibelval1st 40674 . 2 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š) ∧ π‘Œ ∈ (πΌβ€˜π‘‹)) β†’ (1st β€˜π‘Œ) ∈ (((DIsoAβ€˜πΎ)β€˜π‘Š)β€˜π‘‹))
7 dibelval1st2.t . . 3 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
8 dibelval1st2.r . . 3 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
91, 2, 3, 7, 8, 4diatrl 40569 . 2 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š) ∧ (1st β€˜π‘Œ) ∈ (((DIsoAβ€˜πΎ)β€˜π‘Š)β€˜π‘‹)) β†’ (π‘…β€˜(1st β€˜π‘Œ)) ≀ 𝑋)
106, 9syld3an3 1406 1 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š) ∧ π‘Œ ∈ (πΌβ€˜π‘‹)) β†’ (π‘…β€˜(1st β€˜π‘Œ)) ≀ 𝑋)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   class class class wbr 5144  β€˜cfv 6543  1st c1st 7985  Basecbs 17174  lecple 17234  LHypclh 39509  LTrncltrn 39626  trLctrl 39683  DIsoAcdia 40553  DIsoBcdib 40663
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 2166  ax-ext 2696  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  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-1st 7987  df-disoa 40554  df-dib 40664
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator