cdlemefs32fva1 - Mathbox for Norm Megill

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Theorem cdlemefs32fva1 40793

Description: Part of proof of Lemma E in [Crawley] p. 113. TODO: FIX COMMENT. (Contributed by NM, 29-Mar-2013.)

**Hypotheses**
Ref	Expression
cdlemefs32.b	⊢ 𝐵 = (Base‘𝐾)
cdlemefs32.l	⊢ ≤ = (le‘𝐾)
cdlemefs32.j	⊢ ∨ = (join‘𝐾)
cdlemefs32.m	⊢ ∧ = (meet‘𝐾)
cdlemefs32.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemefs32.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemefs32.u	⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
cdlemefs32.d	⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
cdlemefs32.e	⊢ 𝐸 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊)))
cdlemefs32.i	⊢ 𝐼 = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐸))
cdlemefs32.n	⊢ 𝑁 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐶)
cdleme29fs.o	⊢ 𝑂 = (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (𝑁 ∨ (𝑥 ∧ 𝑊))))
cdleme29fs.f	⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ if((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊), 𝑂, 𝑥))

**Assertion**
Ref	Expression
cdlemefs32fva1	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → (𝐹‘𝑅) = ⦋𝑅 / 𝑠⦌𝑁)

Distinct variable groups: 𝑡,𝑠,𝑥,𝑦,𝑧,𝐴 𝐵,𝑠,𝑡,𝑥,𝑦,𝑧 𝑦,𝐷 𝑦,𝐸 𝐻,𝑠,𝑡,𝑦 ∨ ,𝑠,𝑡,𝑥,𝑦,𝑧 𝐾,𝑠,𝑡,𝑦 ≤ ,𝑠,𝑡,𝑥,𝑦,𝑧 ∧ ,𝑠,𝑡,𝑥,𝑦,𝑧 𝑥,𝑁,𝑧 𝑃,𝑠,𝑡,𝑦,𝑧 𝑄,𝑠,𝑡,𝑦,𝑧 𝑅,𝑠,𝑡,𝑦 𝑡,𝑈,𝑦 𝑊,𝑠,𝑡,𝑥,𝑦,𝑧 𝐷,𝑠 𝑧,𝐻 𝑧,𝐾 𝑧,𝑅,𝑥 𝑥,𝑃 𝑥,𝑄 Allowed substitution hints: 𝐶(𝑥,𝑦,𝑧,𝑡,𝑠) 𝐷(𝑥,𝑧,𝑡) 𝑈(𝑥,𝑧,𝑠) 𝐸(𝑥,𝑧,𝑡,𝑠) 𝐹(𝑥,𝑦,𝑧,𝑡,𝑠) 𝐻(𝑥) 𝐼(𝑥,𝑦,𝑧,𝑡,𝑠) 𝐾(𝑥) 𝑁(𝑦,𝑡,𝑠) 𝑂(𝑥,𝑦,𝑧,𝑡,𝑠)

**Proof of Theorem cdlemefs32fva1**
Step	Hyp	Ref	Expression
1		cdlemefs32.b	. 2 ⊢ 𝐵 = (Base‘𝐾)
2		cdlemefs32.l	. 2 ⊢ ≤ = (le‘𝐾)
3		cdlemefs32.j	. 2 ⊢ ∨ = (join‘𝐾)
4		cdlemefs32.m	. 2 ⊢ ∧ = (meet‘𝐾)
5		cdlemefs32.a	. 2 ⊢ 𝐴 = (Atoms‘𝐾)
6		cdlemefs32.h	. 2 ⊢ 𝐻 = (LHyp‘𝐾)
7		breq1 5103	. 2 ⊢ (𝑠 = 𝑅 → (𝑠 ≤ (𝑃 ∨ 𝑄) ↔ 𝑅 ≤ (𝑃 ∨ 𝑄)))
8		simp1 1137	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ 𝑠 ≤ (𝑃 ∨ 𝑄)))) → ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)))
9		simp3l 1203	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ 𝑠 ≤ (𝑃 ∨ 𝑄)))) → 𝑠 ∈ 𝐴)
10		simp3rl 1248	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ 𝑠 ≤ (𝑃 ∨ 𝑄)))) → ¬ 𝑠 ≤ 𝑊)
11	9, 10	jca 511	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ 𝑠 ≤ (𝑃 ∨ 𝑄)))) → (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊))
12		simp3rr 1249	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ 𝑠 ≤ (𝑃 ∨ 𝑄)))) → 𝑠 ≤ (𝑃 ∨ 𝑄))
13		simp2 1138	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ 𝑠 ≤ (𝑃 ∨ 𝑄)))) → 𝑃 ≠ 𝑄)
14		cdlemefs32.u	. . . 4 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
15		cdlemefs32.d	. . . 4 ⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
16		cdlemefs32.e	. . . 4 ⊢ 𝐸 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊)))
17		cdlemefs32.i	. . . 4 ⊢ 𝐼 = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐸))
18		cdlemefs32.n	. . . 4 ⊢ 𝑁 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐶)
19	1, 2, 3, 4, 5, 6, 14, 15, 16, 17, 18	cdlemefs27cl 40783	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊) ∧ 𝑠 ≤ (𝑃 ∨ 𝑄) ∧ 𝑃 ≠ 𝑄)) → 𝑁 ∈ 𝐵)
20	8, 11, 12, 13, 19	syl13anc 1375	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ 𝑠 ≤ (𝑃 ∨ 𝑄)))) → 𝑁 ∈ 𝐵)
21	1, 2, 3, 4, 5, 6, 14, 15, 16, 17, 18	cdlemefs32snb 40785	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ⦋𝑅 / 𝑠⦌𝑁 ∈ 𝐵)
22		cdleme29fs.o	. 2 ⊢ 𝑂 = (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (𝑁 ∨ (𝑥 ∧ 𝑊))))
23		cdleme29fs.f	. 2 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ if((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊), 𝑂, 𝑥))
24	1, 2, 3, 4, 5, 6, 7, 20, 21, 22, 23	cdlemefrs32fva1 40771	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → (𝐹‘𝑅) = ⦋𝑅 / 𝑠⦌𝑁)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 ⦋csb 3851 ifcif 4481 class class class wbr 5100 ↦ cmpt 5181 ‘cfv 6500 ℩crio 7324 (class class class)co 7368 Basecbs 17148 lecple 17196 joincjn 18246 meetcmee 18247 Atomscatm 39633 HLchlt 39720 LHypclh 40354

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-riotaBAD 39323

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5527 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-1st 7943 df-2nd 7944 df-undef 8225 df-proset 18229 df-poset 18248 df-plt 18263 df-lub 18279 df-glb 18280 df-join 18281 df-meet 18282 df-p0 18358 df-p1 18359 df-lat 18367 df-clat 18434 df-oposet 39546 df-ol 39548 df-oml 39549 df-covers 39636 df-ats 39637 df-atl 39668 df-cvlat 39692 df-hlat 39721 df-llines 39868 df-lplanes 39869 df-lvols 39870 df-lines 39871 df-psubsp 39873 df-pmap 39874 df-padd 40166 df-lhyp 40358

This theorem is referenced by: cdlemefs31fv1 40794 cdleme46fsvlpq 40875

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