Theorem cdlemg2jlemOLDN 41650

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. TODO: Combine with cdlemg2jOLDN 41655? (Contributed by NM, 22-Apr-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cdlemg2.b	⊢ 𝐵 = (Base‘𝐾)
cdlemg2.l	⊢ ≤ = (le‘𝐾)
cdlemg2.j	⊢ ∨ = (join‘𝐾)
cdlemg2.m	⊢ ∧ = (meet‘𝐾)
cdlemg2.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemg2.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemg2.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemg2ex.u	⊢ 𝑈 = ((𝑝 ∨ 𝑞) ∧ 𝑊)
cdlemg2ex.d	⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑞 ∨ ((𝑝 ∨ 𝑡) ∧ 𝑊)))
cdlemg2ex.e	⊢ 𝐸 = ((𝑝 ∨ 𝑞) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊)))
cdlemg2ex.g	⊢ 𝐺 = (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
cdlemg2jlemOLDN	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → (𝐹‘(𝑃 ∨ 𝑄)) = ((𝐹‘𝑃) ∨ (𝐹‘𝑄)))

Distinct variable groups:   ∧ ,𝑠,𝑡,𝑥,𝑦,𝑧   𝐾,𝑠,𝑡,𝑥,𝑦,𝑧   𝑈,𝑠,𝑡,𝑥,𝑦,𝑧   𝑊,𝑠,𝑡,𝑥,𝑦,𝑧   ∨ ,𝑠,𝑡,𝑥,𝑦,𝑧   𝐵,𝑠,𝑡,𝑥,𝑦,𝑧   𝐴,𝑠,𝑡,𝑥,𝑦,𝑧   ≤ ,𝑠,𝑡,𝑥,𝑦,𝑧   𝐻,𝑠,𝑡,𝑥,𝑦,𝑧   𝑥,𝐸,𝑦,𝑧   𝑃,𝑠,𝑡,𝑥,𝑦,𝑧   𝐷,𝑠,𝑥,𝑦,𝑧   𝑄,𝑠,𝑡,𝑥,𝑦,𝑧   𝑞,𝑝,𝐴   𝐹,𝑝,𝑞   𝐻,𝑝,𝑞   𝐾,𝑝,𝑞   ≤ ,𝑝,𝑞   𝑇,𝑝,𝑞   𝑊,𝑝,𝑞,𝑠,𝑡,𝑥,𝑦,𝑧   ∨ ,𝑝,𝑞   𝑃,𝑝,𝑞   𝑄,𝑝,𝑞

Allowed substitution hints:   𝐵(𝑞, 𝑝)   𝐷(𝑡, 𝑞, 𝑝)   𝑇(𝑥, 𝑦, 𝑧, 𝑡, 𝑠)   𝑈(𝑞, 𝑝)   𝐸(𝑡, 𝑠, 𝑞, 𝑝)   𝐹(𝑥, 𝑦, 𝑧, 𝑡, 𝑠)   𝐺(𝑥, 𝑦, 𝑧, 𝑡, 𝑠, 𝑞, 𝑝)   ∧ (𝑞, 𝑝)

Proof of Theorem cdlemg2jlemOLDN

Step	Hyp	Ref	Expression
1		cdlemg2.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
2		cdlemg2.l	. . 3 ⊢ ≤ = (le‘𝐾)
3		cdlemg2.j	. . 3 ⊢ ∨ = (join‘𝐾)
4		cdlemg2.m	. . 3 ⊢ ∧ = (meet‘𝐾)
5		cdlemg2.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
6		cdlemg2.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
7		cdlemg2.t	. . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
8		cdlemg2ex.u	. . 3 ⊢ 𝑈 = ((𝑝 ∨ 𝑞) ∧ 𝑊)
9		cdlemg2ex.d	. . 3 ⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑞 ∨ ((𝑝 ∨ 𝑡) ∧ 𝑊)))
10		cdlemg2ex.e	. . 3 ⊢ 𝐸 = ((𝑝 ∨ 𝑞) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊)))
11		cdlemg2ex.g	. . 3 ⊢ 𝐺 = (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))
12		fveq1 6884	. . . 4 ⊢ (𝐹 = 𝐺 → (𝐹‘(𝑃 ∨ 𝑄)) = (𝐺‘(𝑃 ∨ 𝑄)))
13		fveq1 6884	. . . . 5 ⊢ (𝐹 = 𝐺 → (𝐹‘𝑃) = (𝐺‘𝑃))
14		fveq1 6884	. . . . 5 ⊢ (𝐹 = 𝐺 → (𝐹‘𝑄) = (𝐺‘𝑄))
15	13, 14	oveq12d 7438	. . . 4 ⊢ (𝐹 = 𝐺 → ((𝐹‘𝑃) ∨ (𝐹‘𝑄)) = ((𝐺‘𝑃) ∨ (𝐺‘𝑄)))
16	12, 15	eqeq12d 2777	. . 3 ⊢ (𝐹 = 𝐺 → ((𝐹‘(𝑃 ∨ 𝑄)) = ((𝐹‘𝑃) ∨ (𝐹‘𝑄)) ↔ (𝐺‘(𝑃 ∨ 𝑄)) = ((𝐺‘𝑃) ∨ (𝐺‘𝑄))))
17		vex 3455	. . . . 5 ⊢ 𝑠 ∈ V
18		eqid 2761	. . . . . 6 ⊢ ((𝑠 ∨ 𝑈) ∧ (𝑞 ∨ ((𝑝 ∨ 𝑠) ∧ 𝑊))) = ((𝑠 ∨ 𝑈) ∧ (𝑞 ∨ ((𝑝 ∨ 𝑠) ∧ 𝑊)))
19	9, 18	cdleme31sc 41441	. . . . 5 ⊢ (𝑠 ∈ V → ⦋𝑠 / 𝑡⦌𝐷 = ((𝑠 ∨ 𝑈) ∧ (𝑞 ∨ ((𝑝 ∨ 𝑠) ∧ 𝑊))))
20	17, 19	ax-mp 5	. . . 4 ⊢ ⦋𝑠 / 𝑡⦌𝐷 = ((𝑠 ∨ 𝑈) ∧ (𝑞 ∨ ((𝑝 ∨ 𝑠) ∧ 𝑊)))
21		eqid 2761	. . . 4 ⊢ (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑝 ∨ 𝑞)) → 𝑦 = 𝐸)) = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑝 ∨ 𝑞)) → 𝑦 = 𝐸))
22		eqid 2761	. . . 4 ⊢ if(𝑠 ≤ (𝑝 ∨ 𝑞), (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑝 ∨ 𝑞)) → 𝑦 = 𝐸)), ⦋𝑠 / 𝑡⦌𝐷) = if(𝑠 ≤ (𝑝 ∨ 𝑞), (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑝 ∨ 𝑞)) → 𝑦 = 𝐸)), ⦋𝑠 / 𝑡⦌𝐷)
23		eqid 2761	. . . 4 ⊢ (℩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)))) = (℩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))))
24	1, 2, 3, 4, 5, 6, 8, 20, 9, 10, 21, 22, 23, 11	cdleme42mgN 41545	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → (𝐺‘(𝑃 ∨ 𝑄)) = ((𝐺‘𝑃) ∨ (𝐺‘𝑄)))
25	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 24	cdlemg2ce 41649	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → (𝐹‘(𝑃 ∨ 𝑄)) = ((𝐹‘𝑃) ∨ (𝐹‘𝑄)))
26	25	3com23 1144	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → (𝐹‘(𝑃 ∨ 𝑄)) = ((𝐹‘𝑃) ∨ (𝐹‘𝑄)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077  Vcvv 3451  ⦋csb 3847  ifcif 4482   class class class wbr 5103   ↦ cmpt 5186  ‘cfv 6538  ℩crio 7376  (class class class)co 7420  Basecbs 17387  lecple 17435  joincjn 18485  meetcmee 18486  Atomscatm 40320  HLchlt 40407  LHypclh 41041  LTrncltrn 41158

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-riotaBAD 40010

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-undef 8290  df-map 8849  df-proset 18468  df-poset 18487  df-plt 18502  df-lub 18518  df-glb 18519  df-join 18520  df-meet 18521  df-p0 18597  df-p1 18598  df-lat 18606  df-clat 18673  df-oposet 40233  df-ol 40235  df-oml 40236  df-covers 40323  df-ats 40324  df-atl 40355  df-cvlat 40379  df-hlat 40408  df-llines 40555  df-lplanes 40556  df-lvols 40557  df-lines 40558  df-psubsp 40560  df-pmap 40561  df-padd 40853  df-lhyp 41045  df-laut 41046  df-ldil 41161  df-ltrn 41162  df-trl 41216

This theorem is used by:  cdlemg2jOLDN  41655