Theorem cdleme50lebi 34649

Description: Part of proof of Lemma D in [Crawley] p. 113. TODO: fix comment. (Contributed by NM, 9-Apr-2013.)

Hypotheses

Ref	Expression
cdlemef50.b	⊢ 𝐵 = (Base‘𝐾)
cdlemef50.l	⊢ ≤ = (le‘𝐾)
cdlemef50.j	⊢ ∨ = (join‘𝐾)
cdlemef50.m	⊢ ∧ = (meet‘𝐾)
cdlemef50.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemef50.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemef50.u	⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
cdlemef50.d	⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
cdlemefs50.e	⊢ 𝐸 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊)))
cdlemef50.f	⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ if((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊), (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 ≤ (𝑃 ∨ 𝑄), (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐸)), ⦋𝑠 / 𝑡⦌𝐷) ∨ (𝑥 ∧ 𝑊)))), 𝑥))

Assertion

Ref	Expression
cdleme50lebi	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋 ≤ 𝑌 ↔ (𝐹‘𝑋) ≤ (𝐹‘𝑌)))

Distinct variable groups:   𝑡,𝑠,𝑥,𝑦,𝑧, ∧   ∨ ,𝑠,𝑡,𝑥,𝑦,𝑧   ≤ ,𝑠,𝑡,𝑥,𝑦,𝑧   𝐴,𝑠,𝑡,𝑥,𝑦,𝑧   𝐵,𝑠,𝑡,𝑥,𝑦,𝑧   𝐷,𝑠,𝑥,𝑦,𝑧   𝑥,𝐸,𝑦,𝑧   𝐻,𝑠,𝑡,𝑥,𝑦,𝑧   𝐾,𝑠,𝑡,𝑥,𝑦,𝑧   𝑃,𝑠,𝑡,𝑥,𝑦,𝑧   𝑄,𝑠,𝑡,𝑥,𝑦,𝑧   𝑈,𝑠,𝑡,𝑥,𝑦,𝑧   𝑊,𝑠,𝑡,𝑥,𝑦,𝑧   𝑋,𝑠,𝑡,𝑥,𝑦,𝑧   𝑌,𝑠,𝑡,𝑥,𝑦,𝑧

Allowed substitution hints:   𝐷(𝑡)   𝐸(𝑡,𝑠)   𝐹(𝑥,𝑦,𝑧,𝑡,𝑠)

Proof of Theorem cdleme50lebi

Dummy variables 𝑎 𝑏 𝑐 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cdlemef50.b	. 2 ⊢ 𝐵 = (Base‘𝐾)
2		cdlemef50.l	. 2 ⊢ ≤ = (le‘𝐾)
3		cdlemef50.j	. 2 ⊢ ∨ = (join‘𝐾)
4		cdlemef50.m	. 2 ⊢ ∧ = (meet‘𝐾)
5		cdlemef50.a	. 2 ⊢ 𝐴 = (Atoms‘𝐾)
6		cdlemef50.h	. 2 ⊢ 𝐻 = (LHyp‘𝐾)
7		cdlemef50.u	. 2 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
8		cdlemef50.d	. 2 ⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
9		cdlemefs50.e	. 2 ⊢ 𝐸 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊)))
10		cdlemef50.f	. 2 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ if((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊), (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 ≤ (𝑃 ∨ 𝑄), (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐸)), ⦋𝑠 / 𝑡⦌𝐷) ∨ (𝑥 ∧ 𝑊)))), 𝑥))
11		eqid 2609	. 2 ⊢ ((𝑄 ∨ 𝑃) ∧ 𝑊) = ((𝑄 ∨ 𝑃) ∧ 𝑊)
12		eqid 2609	. 2 ⊢ ((𝑣 ∨ ((𝑄 ∨ 𝑃) ∧ 𝑊)) ∧ (𝑃 ∨ ((𝑄 ∨ 𝑣) ∧ 𝑊))) = ((𝑣 ∨ ((𝑄 ∨ 𝑃) ∧ 𝑊)) ∧ (𝑃 ∨ ((𝑄 ∨ 𝑣) ∧ 𝑊)))
13		eqid 2609	. 2 ⊢ ((𝑄 ∨ 𝑃) ∧ (((𝑣 ∨ ((𝑄 ∨ 𝑃) ∧ 𝑊)) ∧ (𝑃 ∨ ((𝑄 ∨ 𝑣) ∧ 𝑊))) ∨ ((𝑢 ∨ 𝑣) ∧ 𝑊))) = ((𝑄 ∨ 𝑃) ∧ (((𝑣 ∨ ((𝑄 ∨ 𝑃) ∧ 𝑊)) ∧ (𝑃 ∨ ((𝑄 ∨ 𝑣) ∧ 𝑊))) ∨ ((𝑢 ∨ 𝑣) ∧ 𝑊)))
14		eqid 2609	. 2 ⊢ (𝑎 ∈ 𝐵 ↦ if((𝑄 ≠ 𝑃 ∧ ¬ 𝑎 ≤ 𝑊), (℩𝑐 ∈ 𝐵 ∀𝑢 ∈ 𝐴 ((¬ 𝑢 ≤ 𝑊 ∧ (𝑢 ∨ (𝑎 ∧ 𝑊)) = 𝑎) → 𝑐 = (if(𝑢 ≤ (𝑄 ∨ 𝑃), (℩𝑏 ∈ 𝐵 ∀𝑣 ∈ 𝐴 ((¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑃)) → 𝑏 = ((𝑄 ∨ 𝑃) ∧ (((𝑣 ∨ ((𝑄 ∨ 𝑃) ∧ 𝑊)) ∧ (𝑃 ∨ ((𝑄 ∨ 𝑣) ∧ 𝑊))) ∨ ((𝑢 ∨ 𝑣) ∧ 𝑊))))), ⦋𝑢 / 𝑣⦌((𝑣 ∨ ((𝑄 ∨ 𝑃) ∧ 𝑊)) ∧ (𝑃 ∨ ((𝑄 ∨ 𝑣) ∧ 𝑊)))) ∨ (𝑎 ∧ 𝑊)))), 𝑎)) = (𝑎 ∈ 𝐵 ↦ if((𝑄 ≠ 𝑃 ∧ ¬ 𝑎 ≤ 𝑊), (℩𝑐 ∈ 𝐵 ∀𝑢 ∈ 𝐴 ((¬ 𝑢 ≤ 𝑊 ∧ (𝑢 ∨ (𝑎 ∧ 𝑊)) = 𝑎) → 𝑐 = (if(𝑢 ≤ (𝑄 ∨ 𝑃), (℩𝑏 ∈ 𝐵 ∀𝑣 ∈ 𝐴 ((¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑃)) → 𝑏 = ((𝑄 ∨ 𝑃) ∧ (((𝑣 ∨ ((𝑄 ∨ 𝑃) ∧ 𝑊)) ∧ (𝑃 ∨ ((𝑄 ∨ 𝑣) ∧ 𝑊))) ∨ ((𝑢 ∨ 𝑣) ∧ 𝑊))))), ⦋𝑢 / 𝑣⦌((𝑣 ∨ ((𝑄 ∨ 𝑃) ∧ 𝑊)) ∧ (𝑃 ∨ ((𝑄 ∨ 𝑣) ∧ 𝑊)))) ∨ (𝑎 ∧ 𝑊)))), 𝑎))
15	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14	cdlemeg49lebilem 34648	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋 ≤ 𝑌 ↔ (𝐹‘𝑋) ≤ (𝐹‘𝑌)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 194   ∧ wa 382   ∧ w3a 1030   = wceq 1474   ∈ wcel 1976   ≠ wne 2779  ∀wral 2895  ⦋csb 3498  ifcif 4035   class class class wbr 4577   ↦ cmpt 4637  ‘cfv 5790  ℩crio 6488  (class class class)co 6527  Basecbs 15641  lecple 15721  joincjn 16713  meetcmee 16714  Atomscatm 33371  HLchlt 33458  LHypclh 34091

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-riotaBAD 33060

This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-iin 4452  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-1st 7036  df-2nd 7037  df-undef 7263  df-preset 16697  df-poset 16715  df-plt 16727  df-lub 16743  df-glb 16744  df-join 16745  df-meet 16746  df-p0 16808  df-p1 16809  df-lat 16815  df-clat 16877  df-oposet 33284  df-ol 33286  df-oml 33287  df-covers 33374  df-ats 33375  df-atl 33406  df-cvlat 33430  df-hlat 33459  df-llines 33605  df-lplanes 33606  df-lvols 33607  df-lines 33608  df-psubsp 33610  df-pmap 33611  df-padd 33903  df-lhyp 34095

This theorem is referenced by:  cdleme50eq  34650  cdleme50laut  34656