Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dibelval1st Structured version   Visualization version   GIF version

Theorem dibelval1st 40533
Description: Membership in value of the partial isomorphism B for a lattice 𝐾. (Contributed by NM, 13-Feb-2014.)
Hypotheses
Ref Expression
dibelval1.b 𝐡 = (Baseβ€˜πΎ)
dibelval1.l ≀ = (leβ€˜πΎ)
dibelval1.h 𝐻 = (LHypβ€˜πΎ)
dibelval1.j 𝐽 = ((DIsoAβ€˜πΎ)β€˜π‘Š)
dibelval1.i 𝐼 = ((DIsoBβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
dibelval1st (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š) ∧ π‘Œ ∈ (πΌβ€˜π‘‹)) β†’ (1st β€˜π‘Œ) ∈ (π½β€˜π‘‹))

Proof of Theorem dibelval1st
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 dibelval1.b . . . . 5 𝐡 = (Baseβ€˜πΎ)
2 dibelval1.l . . . . 5 ≀ = (leβ€˜πΎ)
3 dibelval1.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
4 eqid 2726 . . . . 5 ((LTrnβ€˜πΎ)β€˜π‘Š) = ((LTrnβ€˜πΎ)β€˜π‘Š)
5 eqid 2726 . . . . 5 (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ ( I β†Ύ 𝐡)) = (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ ( I β†Ύ 𝐡))
6 dibelval1.j . . . . 5 𝐽 = ((DIsoAβ€˜πΎ)β€˜π‘Š)
7 dibelval1.i . . . . 5 𝐼 = ((DIsoBβ€˜πΎ)β€˜π‘Š)
81, 2, 3, 4, 5, 6, 7dibval2 40528 . . . 4 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ (πΌβ€˜π‘‹) = ((π½β€˜π‘‹) Γ— {(𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ ( I β†Ύ 𝐡))}))
98eleq2d 2813 . . 3 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ (π‘Œ ∈ (πΌβ€˜π‘‹) ↔ π‘Œ ∈ ((π½β€˜π‘‹) Γ— {(𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ ( I β†Ύ 𝐡))})))
109biimp3a 1465 . 2 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š) ∧ π‘Œ ∈ (πΌβ€˜π‘‹)) β†’ π‘Œ ∈ ((π½β€˜π‘‹) Γ— {(𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ ( I β†Ύ 𝐡))}))
11 xp1st 8006 . 2 (π‘Œ ∈ ((π½β€˜π‘‹) Γ— {(𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ ( I β†Ύ 𝐡))}) β†’ (1st β€˜π‘Œ) ∈ (π½β€˜π‘‹))
1210, 11syl 17 1 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š) ∧ π‘Œ ∈ (πΌβ€˜π‘‹)) β†’ (1st β€˜π‘Œ) ∈ (π½β€˜π‘‹))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  {csn 4623   class class class wbr 5141   ↦ cmpt 5224   I cid 5566   Γ— cxp 5667   β†Ύ cres 5671  β€˜cfv 6537  1st c1st 7972  Basecbs 17153  lecple 17213  LHypclh 39368  LTrncltrn 39485  DIsoAcdia 40412  DIsoBcdib 40522
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 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  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-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-1st 7974  df-disoa 40413  df-dib 40523
This theorem is referenced by:  dibelval1st1  40534  dibelval1st2N  40535  diblss  40554
  Copyright terms: Public domain W3C validator