Theorem cdlemeg46fvcl 36655

Description: TODO: fix comment. (Contributed by NM, 9-Apr-2013.)

Hypotheses

Ref	Expression
cdlemef47.b	⊢ 𝐵 = (Base‘𝐾)
cdlemef47.l	⊢ ≤ = (le‘𝐾)
cdlemef47.j	⊢ ∨ = (join‘𝐾)
cdlemef47.m	⊢ ∧ = (meet‘𝐾)
cdlemef47.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemef47.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemef47.v	⊢ 𝑉 = ((𝑄 ∨ 𝑃) ∧ 𝑊)
cdlemef47.n	⊢ 𝑁 = ((𝑣 ∨ 𝑉) ∧ (𝑃 ∨ ((𝑄 ∨ 𝑣) ∧ 𝑊)))
cdlemefs47.o	⊢ 𝑂 = ((𝑄 ∨ 𝑃) ∧ (𝑁 ∨ ((𝑢 ∨ 𝑣) ∧ 𝑊)))
cdlemef47.g	⊢ 𝐺 = (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 = 𝑂)), ⦋𝑢 / 𝑣⦌𝑁) ∨ (𝑎 ∧ 𝑊)))), 𝑎))

Assertion

Ref	Expression
cdlemeg46fvcl	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑋 ∈ 𝐵) → (𝐺‘𝑋) ∈ 𝐵)

Distinct variable groups:   𝑎,𝑏,𝑐,𝑢,𝑣,𝐴   𝐵,𝑎,𝑏,𝑐,𝑢,𝑣   𝐻,𝑎,𝑏,𝑐,𝑢,𝑣   ∨ ,𝑎,𝑏,𝑐,𝑢,𝑣   𝐾,𝑎,𝑏,𝑐,𝑢,𝑣   ≤ ,𝑎,𝑏,𝑐,𝑢,𝑣   ∧ ,𝑎,𝑏,𝑐,𝑢,𝑣   𝑁,𝑎,𝑏,𝑐,𝑢   𝑂,𝑎,𝑏,𝑐   𝑃,𝑎,𝑏,𝑐,𝑢,𝑣   𝑄,𝑎,𝑏,𝑐,𝑢,𝑣   𝑉,𝑎,𝑏,𝑐,𝑢,𝑣   𝑊,𝑎,𝑏,𝑐,𝑢,𝑣   𝑋,𝑎,𝑐,𝑢,𝑣

Allowed substitution hints:   𝐺(𝑣,𝑢,𝑎,𝑏,𝑐)   𝑁(𝑣)   𝑂(𝑣,𝑢)   𝑋(𝑏)

Proof of Theorem cdlemeg46fvcl

Step	Hyp	Ref	Expression
1		simpl1 1199	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑋 ∈ 𝐵) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simpl3 1203	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑋 ∈ 𝐵) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
3		simpl2 1201	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑋 ∈ 𝐵) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
4		simpr 479	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵)
5		cdlemef47.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
6		cdlemef47.l	. . 3 ⊢ ≤ = (le‘𝐾)
7		cdlemef47.j	. . 3 ⊢ ∨ = (join‘𝐾)
8		cdlemef47.m	. . 3 ⊢ ∧ = (meet‘𝐾)
9		cdlemef47.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
10		cdlemef47.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
11		cdlemef47.v	. . 3 ⊢ 𝑉 = ((𝑄 ∨ 𝑃) ∧ 𝑊)
12		vex 3400	. . . 4 ⊢ 𝑢 ∈ V
13		cdlemef47.n	. . . . 5 ⊢ 𝑁 = ((𝑣 ∨ 𝑉) ∧ (𝑃 ∨ ((𝑄 ∨ 𝑣) ∧ 𝑊)))
14		eqid 2777	. . . . 5 ⊢ ((𝑢 ∨ 𝑉) ∧ (𝑃 ∨ ((𝑄 ∨ 𝑢) ∧ 𝑊))) = ((𝑢 ∨ 𝑉) ∧ (𝑃 ∨ ((𝑄 ∨ 𝑢) ∧ 𝑊)))
15	13, 14	cdleme31sc 36533	. . . 4 ⊢ (𝑢 ∈ V → ⦋𝑢 / 𝑣⦌𝑁 = ((𝑢 ∨ 𝑉) ∧ (𝑃 ∨ ((𝑄 ∨ 𝑢) ∧ 𝑊))))
16	12, 15	ax-mp 5	. . 3 ⊢ ⦋𝑢 / 𝑣⦌𝑁 = ((𝑢 ∨ 𝑉) ∧ (𝑃 ∨ ((𝑄 ∨ 𝑢) ∧ 𝑊)))
17		cdlemefs47.o	. . 3 ⊢ 𝑂 = ((𝑄 ∨ 𝑃) ∧ (𝑁 ∨ ((𝑢 ∨ 𝑣) ∧ 𝑊)))
18		eqid 2777	. . 3 ⊢ (℩𝑏 ∈ 𝐵 ∀𝑣 ∈ 𝐴 ((¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑃)) → 𝑏 = 𝑂)) = (℩𝑏 ∈ 𝐵 ∀𝑣 ∈ 𝐴 ((¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑃)) → 𝑏 = 𝑂))
19		eqid 2777	. . 3 ⊢ if(𝑢 ≤ (𝑄 ∨ 𝑃), (℩𝑏 ∈ 𝐵 ∀𝑣 ∈ 𝐴 ((¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑃)) → 𝑏 = 𝑂)), ⦋𝑢 / 𝑣⦌𝑁) = if(𝑢 ≤ (𝑄 ∨ 𝑃), (℩𝑏 ∈ 𝐵 ∀𝑣 ∈ 𝐴 ((¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑃)) → 𝑏 = 𝑂)), ⦋𝑢 / 𝑣⦌𝑁)
20		eqid 2777	. . 3 ⊢ (℩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 = 𝑂)), ⦋𝑢 / 𝑣⦌𝑁) ∨ (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 = 𝑂)), ⦋𝑢 / 𝑣⦌𝑁) ∨ (𝑎 ∧ 𝑊))))
21		cdlemef47.g	. . 3 ⊢ 𝐺 = (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 = 𝑂)), ⦋𝑢 / 𝑣⦌𝑁) ∨ (𝑎 ∧ 𝑊)))), 𝑎))
22	5, 6, 7, 8, 9, 10, 11, 16, 13, 17, 18, 19, 20, 21	cdleme32fvcl 36589	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝑋 ∈ 𝐵) → (𝐺‘𝑋) ∈ 𝐵)
23	1, 2, 3, 4, 22	syl31anc 1441	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑋 ∈ 𝐵) → (𝐺‘𝑋) ∈ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 386   ∧ w3a 1071   = wceq 1601   ∈ wcel 2106   ≠ wne 2968  ∀wral 3089  Vcvv 3397  ⦋csb 3750  ifcif 4306   class class class wbr 4886   ↦ cmpt 4965  ‘cfv 6135  ℩crio 6882  (class class class)co 6922  Basecbs 16255  lecple 16345  joincjn 17330  meetcmee 17331  Atomscatm 35412  HLchlt 35499  LHypclh 36133

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-riotaBAD 35102

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-iun 4755  df-iin 4756  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-1st 7445  df-2nd 7446  df-undef 7681  df-proset 17314  df-poset 17332  df-plt 17344  df-lub 17360  df-glb 17361  df-join 17362  df-meet 17363  df-p0 17425  df-p1 17426  df-lat 17432  df-clat 17494  df-oposet 35325  df-ol 35327  df-oml 35328  df-covers 35415  df-ats 35416  df-atl 35447  df-cvlat 35471  df-hlat 35500  df-llines 35647  df-lplanes 35648  df-lvols 35649  df-lines 35650  df-psubsp 35652  df-pmap 35653  df-padd 35945  df-lhyp 36137

This theorem is referenced by:  cdleme50rnlem  36693  cdleme51finvfvN  36704