Theorem cdleme42mN 39662

Description: Part of proof of Lemma E in [Crawley] p. 113. TODO: FIX COMMENT . f preserves join: f(r ∨ s) = f(r) ∨ s, p. 115 10th line from bottom. (Contributed by NM, 20-Mar-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cdleme41.b	⊢ 𝐵 = (Base‘𝐾)
cdleme41.l	⊢ ≤ = (le‘𝐾)
cdleme41.j	⊢ ∨ = (join‘𝐾)
cdleme41.m	⊢ ∧ = (meet‘𝐾)
cdleme41.a	⊢ 𝐴 = (Atoms‘𝐾)
cdleme41.h	⊢ 𝐻 = (LHyp‘𝐾)
cdleme41.u	⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
cdleme41.d	⊢ 𝐷 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊)))
cdleme41.e	⊢ 𝐸 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
cdleme41.g	⊢ 𝐺 = ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊)))
cdleme41.i	⊢ 𝐼 = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐺))
cdleme41.n	⊢ 𝑁 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐷)
cdleme41.o	⊢ 𝑂 = (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (𝑁 ∨ (𝑥 ∧ 𝑊))))
cdleme41.f	⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ if((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊), 𝑂, 𝑥))

Assertion

Ref	Expression
cdleme42mN	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊))) → (𝐹‘(𝑅 ∨ 𝑆)) = ((𝐹‘𝑅) ∨ (𝐹‘𝑆)))

Distinct variable groups:   𝐴,𝑠   ∨ ,𝑠   ≤ ,𝑠   ∧ ,𝑠   𝑃,𝑠   𝑄,𝑠   𝑅,𝑠   𝑆,𝑠   𝑈,𝑠   𝑊,𝑠   𝑦,𝑡,𝐴,𝑠   𝐵,𝑠,𝑡,𝑦   𝑦,𝐷   𝑦,𝐺   𝐸,𝑠,𝑦   𝐻,𝑠,𝑡,𝑦   𝑡, ∨ ,𝑦   𝐾,𝑠,𝑡,𝑦   𝑡, ≤ ,𝑦   𝑡, ∧ ,𝑦   𝑡,𝑃,𝑦   𝑡,𝑄,𝑦   𝑡,𝑅,𝑦   𝑡,𝑆,𝑦   𝑡,𝑈,𝑦   𝑡,𝑊,𝑦   𝑥,𝑧,𝐴   𝑥,𝐵,𝑧   𝑧,𝐸,𝑠   𝑧,𝐻   𝑥, ∨ ,𝑧   𝑧,𝐾   𝑥, ≤ ,𝑧   𝑥, ∧ ,𝑧   𝑥,𝑁,𝑧   𝑥,𝑃,𝑧   𝑥,𝑄,𝑧   𝑥,𝑅,𝑧   𝑥,𝑆,𝑧   𝑥,𝑈,𝑧   𝑥,𝑊,𝑧,𝑠,𝑡,𝑦

Allowed substitution hints:   𝐷(𝑥,𝑧,𝑡,𝑠)   𝐸(𝑥,𝑡)   𝐹(𝑥,𝑦,𝑧,𝑡,𝑠)   𝐺(𝑥,𝑧,𝑡,𝑠)   𝐻(𝑥)   𝐼(𝑥,𝑦,𝑧,𝑡,𝑠)   𝐾(𝑥)   𝑁(𝑦,𝑡,𝑠)   𝑂(𝑥,𝑦,𝑧,𝑡,𝑠)

Proof of Theorem cdleme42mN

Step	Hyp	Ref	Expression
1		simpl 482	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊))) → ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)))
2		simpr2 1194	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊))) → (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))
3		simpr3 1195	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊))) → (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊))
4		simpr1 1193	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊))) → 𝑃 ≠ 𝑄)
5		cdleme41.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
6		cdleme41.l	. . . 4 ⊢ ≤ = (le‘𝐾)
7		cdleme41.j	. . . 4 ⊢ ∨ = (join‘𝐾)
8		cdleme41.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
9		cdleme41.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
10		cdleme41.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
11		cdleme41.u	. . . 4 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
12		cdleme41.d	. . . 4 ⊢ 𝐷 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊)))
13		cdleme41.e	. . . 4 ⊢ 𝐸 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
14		cdleme41.g	. . . 4 ⊢ 𝐺 = ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊)))
15		cdleme41.i	. . . 4 ⊢ 𝐼 = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐺))
16		cdleme41.n	. . . 4 ⊢ 𝑁 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐷)
17		cdleme41.o	. . . 4 ⊢ 𝑂 = (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (𝑁 ∨ (𝑥 ∧ 𝑊))))
18		cdleme41.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ if((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊), 𝑂, 𝑥))
19		eqid 2731	. . . 4 ⊢ ((𝑅 ∨ 𝑆) ∧ 𝑊) = ((𝑅 ∨ 𝑆) ∧ 𝑊)
20	5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19	cdleme42g 39656	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) → (𝐹‘(𝑅 ∨ 𝑆)) = ((𝐹‘𝑅) ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊)))
21	1, 2, 3, 4, 20	syl121anc 1374	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊))) → (𝐹‘(𝑅 ∨ 𝑆)) = ((𝐹‘𝑅) ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊)))
22	5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19	cdleme42ke 39660	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊))) → ((𝐹‘𝑅) ∨ (𝐹‘𝑆)) = ((𝐹‘𝑅) ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊)))
23	21, 22	eqtr4d 2774	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊))) → (𝐹‘(𝑅 ∨ 𝑆)) = ((𝐹‘𝑅) ∨ (𝐹‘𝑆)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   ≠ wne 2939  ∀wral 3060  ifcif 4529   class class class wbr 5149   ↦ cmpt 5232  ‘cfv 6544  ℩crio 7367  (class class class)co 7412  Basecbs 17149  lecple 17209  joincjn 18269  meetcmee 18270  Atomscatm 38437  HLchlt 38524  LHypclh 39159

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728  ax-riotaBAD 38127

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7978  df-2nd 7979  df-undef 8261  df-proset 18253  df-poset 18271  df-plt 18288  df-lub 18304  df-glb 18305  df-join 18306  df-meet 18307  df-p0 18383  df-p1 18384  df-lat 18390  df-clat 18457  df-oposet 38350  df-ol 38352  df-oml 38353  df-covers 38440  df-ats 38441  df-atl 38472  df-cvlat 38496  df-hlat 38525  df-llines 38673  df-lplanes 38674  df-lvols 38675  df-lines 38676  df-psubsp 38678  df-pmap 38679  df-padd 38971  df-lhyp 39163

This theorem is referenced by:  cdleme42mgN  39663  cdlemeg46fjgN  39696