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Theorem islvol 34378
Description: The predicate "is a 3-dim lattice volume". (Contributed by NM, 1-Jul-2012.)
Hypotheses
Ref Expression
lvolset.b 𝐵 = (Base‘𝐾)
lvolset.c 𝐶 = ( ⋖ ‘𝐾)
lvolset.p 𝑃 = (LPlanes‘𝐾)
lvolset.v 𝑉 = (LVols‘𝐾)
Assertion
Ref Expression
islvol (𝐾𝐴 → (𝑋𝑉 ↔ (𝑋𝐵 ∧ ∃𝑦𝑃 𝑦𝐶𝑋)))
Distinct variable groups:   𝑦,𝑃   𝑦,𝐾   𝑦,𝑋
Allowed substitution hints:   𝐴(𝑦)   𝐵(𝑦)   𝐶(𝑦)   𝑉(𝑦)

Proof of Theorem islvol
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 lvolset.b . . . 4 𝐵 = (Base‘𝐾)
2 lvolset.c . . . 4 𝐶 = ( ⋖ ‘𝐾)
3 lvolset.p . . . 4 𝑃 = (LPlanes‘𝐾)
4 lvolset.v . . . 4 𝑉 = (LVols‘𝐾)
51, 2, 3, 4lvolset 34377 . . 3 (𝐾𝐴𝑉 = {𝑥𝐵 ∣ ∃𝑦𝑃 𝑦𝐶𝑥})
65eleq2d 2684 . 2 (𝐾𝐴 → (𝑋𝑉𝑋 ∈ {𝑥𝐵 ∣ ∃𝑦𝑃 𝑦𝐶𝑥}))
7 breq2 4627 . . . 4 (𝑥 = 𝑋 → (𝑦𝐶𝑥𝑦𝐶𝑋))
87rexbidv 3047 . . 3 (𝑥 = 𝑋 → (∃𝑦𝑃 𝑦𝐶𝑥 ↔ ∃𝑦𝑃 𝑦𝐶𝑋))
98elrab 3351 . 2 (𝑋 ∈ {𝑥𝐵 ∣ ∃𝑦𝑃 𝑦𝐶𝑥} ↔ (𝑋𝐵 ∧ ∃𝑦𝑃 𝑦𝐶𝑋))
106, 9syl6bb 276 1 (𝐾𝐴 → (𝑋𝑉 ↔ (𝑋𝐵 ∧ ∃𝑦𝑃 𝑦𝐶𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wrex 2909  {crab 2912   class class class wbr 4623  cfv 5857  Basecbs 15800  ccvr 34068  LPlanesclpl 34297  LVolsclvol 34298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-iota 5820  df-fun 5859  df-fv 5865  df-lvols 34305
This theorem is referenced by:  islvol4  34379  lvoli  34380  lvolbase  34383  lvolnle3at  34387
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