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 39615
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 2737 . . . . 5 ((LTrnβ€˜πΎ)β€˜π‘Š) = ((LTrnβ€˜πΎ)β€˜π‘Š)
5 eqid 2737 . . . . 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 39610 . . . 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 2824 . . 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 1470 . 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 7954 . 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  {csn 4587   class class class wbr 5106   ↦ cmpt 5189   I cid 5531   Γ— cxp 5632   β†Ύ cres 5636  β€˜cfv 6497  1st c1st 7920  Basecbs 17084  lecple 17141  LHypclh 38450  LTrncltrn 38567  DIsoAcdia 39494  DIsoBcdib 39604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-1st 7922  df-disoa 39495  df-dib 39605
This theorem is referenced by:  dibelval1st1  39616  dibelval1st2N  39617  diblss  39636
  Copyright terms: Public domain W3C validator