cdlemg2fvlem - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemg2fvlem		Structured version Visualization version GIF version

Theorem cdlemg2fvlem 39453

Description: Lemma for cdlemg2fv 39458. (Contributed by NM, 23-Apr-2013.)

**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
cdlemg2fvlem	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (𝐹‘𝑋) = ((𝐹‘𝑃) ∨ (𝑋 ∧ 𝑊)))

Distinct variable groups: 𝑡,𝑠,𝑥,𝑦,𝑧,𝐴 𝐵,𝑠,𝑡,𝑥,𝑦,𝑧 𝐷,𝑠,𝑥,𝑦,𝑧 𝑥,𝐸,𝑦,𝑧 𝐻,𝑠,𝑡,𝑥,𝑦,𝑧 ∨ ,𝑠,𝑡,𝑥,𝑦,𝑧 𝐾,𝑠,𝑡,𝑥,𝑦,𝑧 ≤ ,𝑠,𝑡,𝑥,𝑦,𝑧 ∧ ,𝑠,𝑡,𝑥,𝑦,𝑧 𝑃,𝑠,𝑡,𝑥,𝑦,𝑧 𝑈,𝑠,𝑡,𝑥,𝑦,𝑧 𝑊,𝑠,𝑡,𝑥,𝑦,𝑧 𝑋,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 𝐵,𝑝,𝑞 ∧ ,𝑝,𝑞 𝑋,𝑝,𝑞 Allowed substitution hints: 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)

**Proof of Theorem cdlemg2fvlem**
Step	Hyp	Ref	Expression
1		simp1 1136	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simp3l 1201	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → 𝐹 ∈ 𝑇)
3		simp2r 1200	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊))
4		simp2l 1199	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
5		simp3r 1202	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋)
6	4, 5	jca 512	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋))
7		cdlemg2.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
8		cdlemg2.l	. . 3 ⊢ ≤ = (le‘𝐾)
9		cdlemg2.j	. . 3 ⊢ ∨ = (join‘𝐾)
10		cdlemg2.m	. . 3 ⊢ ∧ = (meet‘𝐾)
11		cdlemg2.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
12		cdlemg2.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
13		cdlemg2.t	. . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
14		cdlemg2ex.u	. . 3 ⊢ 𝑈 = ((𝑝 ∨ 𝑞) ∧ 𝑊)
15		cdlemg2ex.d	. . 3 ⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑞 ∨ ((𝑝 ∨ 𝑡) ∧ 𝑊)))
16		cdlemg2ex.e	. . 3 ⊢ 𝐸 = ((𝑝 ∨ 𝑞) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊)))
17		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))
18		fveq1 6887	. . . 4 ⊢ (𝐹 = 𝐺 → (𝐹‘𝑋) = (𝐺‘𝑋))
19		fveq1 6887	. . . . 5 ⊢ (𝐹 = 𝐺 → (𝐹‘𝑃) = (𝐺‘𝑃))
20	19	oveq1d 7420	. . . 4 ⊢ (𝐹 = 𝐺 → ((𝐹‘𝑃) ∨ (𝑋 ∧ 𝑊)) = ((𝐺‘𝑃) ∨ (𝑋 ∧ 𝑊)))
21	18, 20	eqeq12d 2748	. . 3 ⊢ (𝐹 = 𝐺 → ((𝐹‘𝑋) = ((𝐹‘𝑃) ∨ (𝑋 ∧ 𝑊)) ↔ (𝐺‘𝑋) = ((𝐺‘𝑃) ∨ (𝑋 ∧ 𝑊))))
22	7, 8, 9, 10, 11, 12, 14, 15, 16, 17	cdleme48fvg 39359	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (𝐺‘𝑋) = ((𝐺‘𝑃) ∨ (𝑋 ∧ 𝑊)))
23	22	3expb 1120	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊)) ∧ ((𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋))) → (𝐺‘𝑋) = ((𝐺‘𝑃) ∨ (𝑋 ∧ 𝑊)))
24	7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 21, 23	cdlemg2ce 39451	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋))) → (𝐹‘𝑋) = ((𝐹‘𝑃) ∨ (𝑋 ∧ 𝑊)))
25	1, 2, 3, 6, 24	syl112anc 1374	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (𝐹‘𝑋) = ((𝐹‘𝑃) ∨ (𝑋 ∧ 𝑊)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 ⦋csb 3892 ifcif 4527 class class class wbr 5147 ↦ cmpt 5230 ‘cfv 6540 ℩crio 7360 (class class class)co 7405 Basecbs 17140 lecple 17200 joincjn 18260 meetcmee 18261 Atomscatm 38121 HLchlt 38208 LHypclh 38843 LTrncltrn 38960

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-riotaBAD 37811

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7971 df-2nd 7972 df-undef 8254 df-map 8818 df-proset 18244 df-poset 18262 df-plt 18279 df-lub 18295 df-glb 18296 df-join 18297 df-meet 18298 df-p0 18374 df-p1 18375 df-lat 18381 df-clat 18448 df-oposet 38034 df-ol 38036 df-oml 38037 df-covers 38124 df-ats 38125 df-atl 38156 df-cvlat 38180 df-hlat 38209 df-llines 38357 df-lplanes 38358 df-lvols 38359 df-lines 38360 df-psubsp 38362 df-pmap 38363 df-padd 38655 df-lhyp 38847 df-laut 38848 df-ldil 38963 df-ltrn 38964 df-trl 39018

This theorem is referenced by: cdlemg2fv 39458

W3C validator