cdlemg2fvlem - Mathbox for Norm Megill

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Theorem cdlemg2fvlem 40632

Description: Lemma for cdlemg2fv 40637. (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: ∧ ,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 𝑋,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,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 1202	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → 𝐹 ∈ 𝑇)
3		simp2r 1201	. 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 1200	. . 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 1203	. . 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 511	. 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 6821	. . . 4 ⊢ (𝐹 = 𝐺 → (𝐹‘𝑋) = (𝐺‘𝑋))
19		fveq1 6821	. . . . 5 ⊢ (𝐹 = 𝐺 → (𝐹‘𝑃) = (𝐺‘𝑃))
20	19	oveq1d 7361	. . . 4 ⊢ (𝐹 = 𝐺 → ((𝐹‘𝑃) ∨ (𝑋 ∧ 𝑊)) = ((𝐺‘𝑃) ∨ (𝑋 ∧ 𝑊)))
21	18, 20	eqeq12d 2747	. . 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 40538	. . . 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 40630	. 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 1376	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 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 ∀wral 3047 ⦋csb 3850 ifcif 4475 class class class wbr 5091 ↦ cmpt 5172 ‘cfv 6481 ℩crio 7302 (class class class)co 7346 Basecbs 17117 lecple 17165 joincjn 18214 meetcmee 18215 Atomscatm 39301 HLchlt 39388 LHypclh 40022 LTrncltrn 40139

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 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5217 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-riotaBAD 38991

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-op 4583 df-uni 4860 df-iun 4943 df-iin 4944 df-br 5092 df-opab 5154 df-mpt 5173 df-id 5511 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-1st 7921 df-2nd 7922 df-undef 8203 df-map 8752 df-proset 18197 df-poset 18216 df-plt 18231 df-lub 18247 df-glb 18248 df-join 18249 df-meet 18250 df-p0 18326 df-p1 18327 df-lat 18335 df-clat 18402 df-oposet 39214 df-ol 39216 df-oml 39217 df-covers 39304 df-ats 39305 df-atl 39336 df-cvlat 39360 df-hlat 39389 df-llines 39536 df-lplanes 39537 df-lvols 39538 df-lines 39539 df-psubsp 39541 df-pmap 39542 df-padd 39834 df-lhyp 40026 df-laut 40027 df-ldil 40142 df-ltrn 40143 df-trl 40197

This theorem is referenced by: cdlemg2fv 40637

W3C validator