cdlemefs31fv1 - Mathbox for Norm Megill

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Theorem cdlemefs31fv1 41449

Description: Value of (𝐹‘𝑅) when 𝑅 ≤ (𝑃 ∨ 𝑄). TODO This may be useful for shortening others that now use riotasv 39984 3d . TODO: FIX COMMENT. ***END OF VALUE AT ATOM STUFF TO REPLACE ONES BELOW***

       "cdleme3xsn1aw" decreased using "cdlemefs32sn1aw"
       "cdleme32sn1aw" decreased from 3302 to 36 using "cdlemefs32sn1aw".
       "cdleme32sn2aw" decreased from 1687 to 26 using "cdlemefr32sn2aw".
       "cdleme32snaw" decreased from 376 to 375 using "cdlemefs32sn1aw".
       "cdleme32snaw" decreased from 375 to 368 using "cdlemefr32sn2aw".
       "cdleme35sn3a" decreased from 547 to 523 using "cdleme43frv1sn".

(Contributed by NM, 27-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))
cdleme43fsv.y	⊢ 𝑌 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊)))
cdleme43fsv.z	⊢ 𝑍 = ((𝑃 ∨ 𝑄) ∧ (𝑌 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊)))

**Assertion**
Ref	Expression
cdlemefs31fv1	⊢ ((((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))) → (𝐹‘𝑅) = 𝑍)

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 𝑦,𝑌 𝐷,𝑠 𝑧,𝐻 𝑧,𝐾 𝑧,𝑅 𝑡,𝑆,𝑦 𝑡,𝑍 𝑥,𝑅 𝑥,𝑃 𝑥,𝑄 Allowed substitution hints: 𝐶(𝑥, 𝑦, 𝑧, 𝑡, 𝑠) 𝐷(𝑥, 𝑧, 𝑡) 𝑆(𝑥, 𝑧, 𝑠) 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) 𝑌(𝑥, 𝑧, 𝑡, 𝑠) 𝑍(𝑥, 𝑦, 𝑧, 𝑠)

**Proof of Theorem cdlemefs31fv1**
Step	Hyp	Ref	Expression
1		simp1 1154	. . 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))) → ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)))
2		simp21 1225	. . 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))) → 𝑃 ≠ 𝑄)
3		simp22 1226	. . 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 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))
4		simp3l 1220	. . 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 ≤ (𝑃 ∨ 𝑄))
5		cdlemefs32.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
6		cdlemefs32.l	. . . 4 ⊢ ≤ = (le‘𝐾)
7		cdlemefs32.j	. . . 4 ⊢ ∨ = (join‘𝐾)
8		cdlemefs32.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
9		cdlemefs32.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
10		cdlemefs32.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
11		cdlemefs32.u	. . . 4 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
12		cdlemefs32.d	. . . 4 ⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
13		cdlemefs32.e	. . . 4 ⊢ 𝐸 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊)))
14		cdlemefs32.i	. . . 4 ⊢ 𝐼 = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐸))
15		cdlemefs32.n	. . . 4 ⊢ 𝑁 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐶)
16		cdleme29fs.o	. . . 4 ⊢ 𝑂 = (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (𝑁 ∨ (𝑥 ∧ 𝑊))))
17		cdleme29fs.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ if((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊), 𝑂, 𝑥))
18	5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17	cdlemefs32fva1 41448	. . 3 ⊢ ((((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)
19	1, 2, 3, 4, 18	syl121anc 1402	. 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))) → (𝐹‘𝑅) = ⦋𝑅 / 𝑠⦌𝑁)
20		cdleme43fsv.y	. . 3 ⊢ 𝑌 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊)))
21		cdleme43fsv.z	. . 3 ⊢ 𝑍 = ((𝑃 ∨ 𝑄) ∧ (𝑌 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊)))
22	5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 20, 21	cdleme43fsv1sn 41446	. 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⦌𝑁 = 𝑍)
23	19, 22	eqtrd 2796	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))) → (𝐹‘𝑅) = 𝑍)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2956 ∀wral 3077 ⦋csb 3847 ifcif 4482 class class class wbr 5103 ↦ cmpt 5186 ‘cfv 6531 ℩crio 7368 (class class class)co 7412 Basecbs 17367 lecple 17415 joincjn 18465 meetcmee 18466 Atomscatm 40288 HLchlt 40375 LHypclh 41009

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 7740 ax-riotaBAD 39978

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 6487 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7990 df-2nd 7991 df-undef 8274 df-proset 18448 df-poset 18467 df-plt 18482 df-lub 18498 df-glb 18499 df-join 18500 df-meet 18501 df-p0 18577 df-p1 18578 df-lat 18586 df-clat 18653 df-oposet 40201 df-ol 40203 df-oml 40204 df-covers 40291 df-ats 40292 df-atl 40323 df-cvlat 40347 df-hlat 40376 df-llines 40523 df-lplanes 40524 df-lvols 40525 df-lines 40526 df-psubsp 40528 df-pmap 40529 df-padd 40821 df-lhyp 41013

This theorem is used by: cdlemefs44 41451

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