Theorem cdlemeg46fvcl 40465

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 1191	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑋 ∈ 𝐵) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simpl3 1193	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑋 ∈ 𝐵) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
3		simpl2 1192	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑋 ∈ 𝐵) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
4		simpr 484	. 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 3492	. . . 4 ⊢ 𝑢 ∈ V
13		cdlemef47.n	. . . . 5 ⊢ 𝑁 = ((𝑣 ∨ 𝑉) ∧ (𝑃 ∨ ((𝑄 ∨ 𝑣) ∧ 𝑊)))
14		eqid 2740	. . . . 5 ⊢ ((𝑢 ∨ 𝑉) ∧ (𝑃 ∨ ((𝑄 ∨ 𝑢) ∧ 𝑊))) = ((𝑢 ∨ 𝑉) ∧ (𝑃 ∨ ((𝑄 ∨ 𝑢) ∧ 𝑊)))
15	13, 14	cdleme31sc 40343	. . . 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 2740	. . 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 2740	. . 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 2740	. . 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 40399	. 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 1373	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 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  Vcvv 3488  ⦋csb 3921  ifcif 4548   class class class wbr 5166   ↦ cmpt 5249  ‘cfv 6575  ℩crio 7405  (class class class)co 7450  Basecbs 17260  lecple 17320  joincjn 18383  meetcmee 18384  Atomscatm 39221  HLchlt 39308  LHypclh 39943

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7772  ax-riotaBAD 38911

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6527  df-fun 6577  df-fn 6578  df-f 6579  df-f1 6580  df-fo 6581  df-f1o 6582  df-fv 6583  df-riota 7406  df-ov 7453  df-oprab 7454  df-mpo 7455  df-1st 8032  df-2nd 8033  df-undef 8316  df-proset 18367  df-poset 18385  df-plt 18402  df-lub 18418  df-glb 18419  df-join 18420  df-meet 18421  df-p0 18497  df-p1 18498  df-lat 18504  df-clat 18571  df-oposet 39134  df-ol 39136  df-oml 39137  df-covers 39224  df-ats 39225  df-atl 39256  df-cvlat 39280  df-hlat 39309  df-llines 39457  df-lplanes 39458  df-lvols 39459  df-lines 39460  df-psubsp 39462  df-pmap 39463  df-padd 39755  df-lhyp 39947

This theorem is referenced by:  cdleme50rnlem  40503  cdleme51finvfvN  40514