cdleme41sn4aw - Mathbox for Norm Megill

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Theorem cdleme41sn4aw 39948

Description: Part of proof of Lemma E in [Crawley] p. 113. Show that f(r) is for on and off 𝑃 ∨ 𝑄 line. TODO: FIX COMMENT. (Contributed by NM, 19-Mar-2013.)

**Hypotheses**
Ref	Expression
cdleme41.b	⊢ 𝐵 = (Base‘𝐾)
cdleme41.l	⊢ ≤ = (le‘𝐾)
cdleme41.j	⊢ ∨ = (join‘𝐾)
cdleme41.m	⊢ ∧ = (meet‘𝐾)
cdleme41.a	⊢ 𝐴 = (Atoms‘𝐾)
cdleme41.h	⊢ 𝐻 = (LHyp‘𝐾)
cdleme41.u	⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
cdleme41.d	⊢ 𝐷 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊)))
cdleme41.e	⊢ 𝐸 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
cdleme41.g	⊢ 𝐺 = ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊)))
cdleme41.i	⊢ 𝐼 = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐺))
cdleme41.n	⊢ 𝑁 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐷)

**Assertion**
Ref	Expression
cdleme41sn4aw	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆)) → ⦋𝑅 / 𝑠⦌𝑁 ≠ ⦋𝑆 / 𝑠⦌𝑁)

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

**Proof of Theorem cdleme41sn4aw**
Step	Hyp	Ref	Expression
1		simp1 1134	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆)) → ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)))
2		simp21 1204	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆)) → 𝑃 ≠ 𝑄)
3		simp23 1206	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆)) → (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊))
4		simp22 1205	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆)) → (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))
5		simp32 1208	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆)) → 𝑆 ≤ (𝑃 ∨ 𝑄))
6		simp31 1207	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆)) → ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))
7		simp33 1209	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆)) → 𝑅 ≠ 𝑆)
8	7	necomd 2993	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆)) → 𝑆 ≠ 𝑅)
9		cdleme41.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
10		cdleme41.l	. . . 4 ⊢ ≤ = (le‘𝐾)
11		cdleme41.j	. . . 4 ⊢ ∨ = (join‘𝐾)
12		cdleme41.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
13		cdleme41.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
14		cdleme41.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
15		cdleme41.u	. . . 4 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
16		cdleme41.d	. . . 4 ⊢ 𝐷 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊)))
17		cdleme41.e	. . . 4 ⊢ 𝐸 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
18		cdleme41.g	. . . 4 ⊢ 𝐺 = ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊)))
19		cdleme41.i	. . . 4 ⊢ 𝐼 = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐺))
20		cdleme41.n	. . . 4 ⊢ 𝑁 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐷)
21	9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20	cdleme41sn3aw 39947	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≠ 𝑅)) → ⦋𝑆 / 𝑠⦌𝑁 ≠ ⦋𝑅 / 𝑠⦌𝑁)
22	1, 2, 3, 4, 5, 6, 8, 21	syl133anc 1391	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆)) → ⦋𝑆 / 𝑠⦌𝑁 ≠ ⦋𝑅 / 𝑠⦌𝑁)
23	22	necomd 2993	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆)) → ⦋𝑅 / 𝑠⦌𝑁 ≠ ⦋𝑆 / 𝑠⦌𝑁)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 ∀wral 3058 ⦋csb 3892 ifcif 4529 class class class wbr 5148 ‘cfv 6548 ℩crio 7375 (class class class)co 7420 Basecbs 17180 lecple 17240 joincjn 18303 meetcmee 18304 Atomscatm 38735 HLchlt 38822 LHypclh 39457

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-riotaBAD 38425

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 7993 df-2nd 7994 df-undef 8279 df-proset 18287 df-poset 18305 df-plt 18322 df-lub 18338 df-glb 18339 df-join 18340 df-meet 18341 df-p0 18417 df-p1 18418 df-lat 18424 df-clat 18491 df-oposet 38648 df-ol 38650 df-oml 38651 df-covers 38738 df-ats 38739 df-atl 38770 df-cvlat 38794 df-hlat 38823 df-llines 38971 df-lplanes 38972 df-lvols 38973 df-lines 38974 df-psubsp 38976 df-pmap 38977 df-padd 39269 df-lhyp 39461

This theorem is referenced by: cdleme41snaw 39949

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