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Theorem cdleme43aN 37617
Description: Part of proof of Lemma E in [Crawley] p. 113. TODO: FIX COMMENT p. 115 penultimate line: g(f(r)) = (p v q) ^ (g(s) v v1). (Contributed by NM, 20-Mar-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdleme43.b 𝐵 = (Base‘𝐾)
cdleme43.l = (le‘𝐾)
cdleme43.j = (join‘𝐾)
cdleme43.m = (meet‘𝐾)
cdleme43.a 𝐴 = (Atoms‘𝐾)
cdleme43.h 𝐻 = (LHyp‘𝐾)
cdleme43.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdleme43.x 𝑋 = ((𝑄 𝑃) 𝑊)
cdleme43.c 𝐶 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))
cdleme43.f 𝑍 = ((𝑃 𝑄) (𝐶 ((𝑅 𝑆) 𝑊)))
cdleme43.d 𝐷 = ((𝑆 𝑋) (𝑃 ((𝑄 𝑆) 𝑊)))
cdleme43.g 𝐺 = ((𝑄 𝑃) (𝐷 ((𝑍 𝑆) 𝑊)))
cdleme43.e 𝐸 = ((𝐷 𝑈) (𝑄 ((𝑃 𝐷) 𝑊)))
cdleme43.v 𝑉 = ((𝑍 𝑆) 𝑊)
cdleme43.y 𝑌 = ((𝑅 𝐷) 𝑊)
Assertion
Ref Expression
cdleme43aN ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → 𝐺 = ((𝑃 𝑄) (𝐷 𝑉)))

Proof of Theorem cdleme43aN
StepHypRef Expression
1 cdleme43.j . . . 4 = (join‘𝐾)
2 cdleme43.a . . . 4 𝐴 = (Atoms‘𝐾)
31, 2hlatjcom 36496 . . 3 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) = (𝑄 𝑃))
4 cdleme43.v . . . . 5 𝑉 = ((𝑍 𝑆) 𝑊)
54oveq2i 7159 . . . 4 (𝐷 𝑉) = (𝐷 ((𝑍 𝑆) 𝑊))
65a1i 11 . . 3 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝐷 𝑉) = (𝐷 ((𝑍 𝑆) 𝑊)))
73, 6oveq12d 7166 . 2 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → ((𝑃 𝑄) (𝐷 𝑉)) = ((𝑄 𝑃) (𝐷 ((𝑍 𝑆) 𝑊))))
8 cdleme43.g . 2 𝐺 = ((𝑄 𝑃) (𝐷 ((𝑍 𝑆) 𝑊)))
97, 8syl6reqr 2873 1 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → 𝐺 = ((𝑃 𝑄) (𝐷 𝑉)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1082   = wceq 1531  wcel 2108  cfv 6348  (class class class)co 7148  Basecbs 16475  lecple 16564  joincjn 17546  meetcmee 17547  Atomscatm 36391  HLchlt 36478  LHypclh 37112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-lub 17576  df-join 17578  df-lat 17648  df-ats 36395  df-atl 36426  df-cvlat 36450  df-hlat 36479
This theorem is referenced by: (None)
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