cdlemeg46v1v2 - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemeg46v1v2		Structured version Visualization version GIF version

Theorem cdlemeg46v1v2 38540

Description: TODO FIX COMMENT v₁ = v₂ p. 116 3rd line. (Contributed by NM, 2-Apr-2013.)

**Hypotheses**
Ref	Expression
cdlemef46g.b	⊢ 𝐵 = (Base‘𝐾)
cdlemef46g.l	⊢ ≤ = (le‘𝐾)
cdlemef46g.j	⊢ ∨ = (join‘𝐾)
cdlemef46g.m	⊢ ∧ = (meet‘𝐾)
cdlemef46g.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemef46g.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemef46g.u	⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
cdlemef46g.d	⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
cdlemefs46g.e	⊢ 𝐸 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊)))
cdlemef46g.f	⊢ 𝐹 = (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))
cdlemef46.v	⊢ 𝑉 = ((𝑄 ∨ 𝑃) ∧ 𝑊)
cdlemef46.n	⊢ 𝑁 = ((𝑣 ∨ 𝑉) ∧ (𝑃 ∨ ((𝑄 ∨ 𝑣) ∧ 𝑊)))
cdlemefs46.o	⊢ 𝑂 = ((𝑄 ∨ 𝑃) ∧ (𝑁 ∨ ((𝑢 ∨ 𝑣) ∧ 𝑊)))
cdlemef46.g	⊢ 𝐺 = (𝑎 ∈ 𝐵 ↦ if((𝑄 ≠ 𝑃 ∧ ¬ 𝑎 ≤ 𝑊), (℩𝑐 ∈ 𝐵 ∀𝑢 ∈ 𝐴 ((¬ 𝑢 ≤ 𝑊 ∧ (𝑢 ∨ (𝑎 ∧ 𝑊)) = 𝑎) → 𝑐 = (if(𝑢 ≤ (𝑄 ∨ 𝑃), (℩𝑏 ∈ 𝐵 ∀𝑣 ∈ 𝐴 ((¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑃)) → 𝑏 = 𝑂)), ⦋𝑢 / 𝑣⦌𝑁) ∨ (𝑎 ∧ 𝑊)))), 𝑎))
cdlemeg46.y	⊢ 𝑌 = ((𝑅 ∨ (𝐺‘𝑆)) ∧ 𝑊)
cdlemeg46.x	⊢ 𝑋 = (((𝐹‘𝑅) ∨ 𝑆) ∧ 𝑊)

**Assertion**
Ref	Expression
cdlemeg46v1v2	⊢ ((((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))) → 𝑋 = 𝑌)

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 𝑄,𝑠,𝑡,𝑥,𝑦,𝑧 𝑅,𝑠,𝑡,𝑥,𝑦,𝑧 𝑈,𝑠,𝑡,𝑥,𝑦,𝑧 𝑊,𝑠,𝑡,𝑥,𝑦,𝑧 𝑆,𝑠,𝑡,𝑥,𝑦,𝑧 𝑎,𝑏,𝑐,𝑢,𝑣,𝐴 𝐵,𝑎,𝑏,𝑐,𝑢,𝑣 𝑣,𝐷 𝐺,𝑠,𝑡,𝑥,𝑦,𝑧 𝐻,𝑎,𝑏,𝑐,𝑢,𝑣 ∨ ,𝑎,𝑏,𝑐,𝑢,𝑣 𝐾,𝑎,𝑏,𝑐,𝑢,𝑣 ≤ ,𝑎,𝑏,𝑐,𝑢,𝑣 ∧ ,𝑎,𝑏,𝑐,𝑢,𝑣 𝑁,𝑎,𝑏,𝑐 𝑂,𝑎,𝑏,𝑐 𝑃,𝑎,𝑏,𝑐,𝑢,𝑣 𝑄,𝑎,𝑏,𝑐,𝑢,𝑣 𝑅,𝑎,𝑏,𝑐,𝑢,𝑣 𝑆,𝑎,𝑏,𝑐,𝑢,𝑣 𝑉,𝑎,𝑏,𝑐 𝑊,𝑎,𝑏,𝑐,𝑢,𝑣 𝑥,𝑢,𝑦,𝑧,𝑁 𝑥,𝑂,𝑦,𝑧 𝑣,𝑡 𝑢,𝑉 𝑥,𝑣,𝑦,𝑧,𝑉 𝐷,𝑎,𝑏,𝑐 𝐸,𝑎,𝑏,𝑐 𝐹,𝑎,𝑏,𝑐,𝑢,𝑣 𝑡,𝑁 𝑈,𝑎,𝑏,𝑐,𝑣 𝑡,𝑉 𝑠,𝑎,𝑡,𝑏,𝑐 𝑌,𝑠,𝑡,𝑥,𝑧 Allowed substitution hints: 𝐷(𝑢,𝑡) 𝑈(𝑢) 𝐸(𝑣,𝑢,𝑡,𝑠) 𝐹(𝑥,𝑦,𝑧,𝑡,𝑠) 𝐺(𝑣,𝑢,𝑎,𝑏,𝑐) 𝑁(𝑣,𝑠) 𝑂(𝑣,𝑢,𝑡,𝑠) 𝑉(𝑠) 𝑋(𝑥,𝑦,𝑧,𝑣,𝑢,𝑡,𝑠,𝑎,𝑏,𝑐) 𝑌(𝑦,𝑣,𝑢,𝑎,𝑏,𝑐)

**Proof of Theorem cdlemeg46v1v2**
Step	Hyp	Ref	Expression
1		cdlemeg46.x	. 2 ⊢ 𝑋 = (((𝐹‘𝑅) ∨ 𝑆) ∧ 𝑊)
2		cdlemef46g.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
3		cdlemef46g.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
4		cdlemef46g.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
5		cdlemef46g.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
6		cdlemef46g.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
7		cdlemef46g.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
8		cdlemef46g.u	. . . . 5 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
9		cdlemef46g.d	. . . . 5 ⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
10		cdlemefs46g.e	. . . . 5 ⊢ 𝐸 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊)))
11		cdlemef46g.f	. . . . 5 ⊢ 𝐹 = (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))
12		cdlemef46.v	. . . . 5 ⊢ 𝑉 = ((𝑄 ∨ 𝑃) ∧ 𝑊)
13		cdlemef46.n	. . . . 5 ⊢ 𝑁 = ((𝑣 ∨ 𝑉) ∧ (𝑃 ∨ ((𝑄 ∨ 𝑣) ∧ 𝑊)))
14		cdlemefs46.o	. . . . 5 ⊢ 𝑂 = ((𝑄 ∨ 𝑃) ∧ (𝑁 ∨ ((𝑢 ∨ 𝑣) ∧ 𝑊)))
15		cdlemef46.g	. . . . 5 ⊢ 𝐺 = (𝑎 ∈ 𝐵 ↦ if((𝑄 ≠ 𝑃 ∧ ¬ 𝑎 ≤ 𝑊), (℩𝑐 ∈ 𝐵 ∀𝑢 ∈ 𝐴 ((¬ 𝑢 ≤ 𝑊 ∧ (𝑢 ∨ (𝑎 ∧ 𝑊)) = 𝑎) → 𝑐 = (if(𝑢 ≤ (𝑄 ∨ 𝑃), (℩𝑏 ∈ 𝐵 ∀𝑣 ∈ 𝐴 ((¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑃)) → 𝑏 = 𝑂)), ⦋𝑢 / 𝑣⦌𝑁) ∨ (𝑎 ∧ 𝑊)))), 𝑎))
16		cdlemeg46.y	. . . . 5 ⊢ 𝑌 = ((𝑅 ∨ (𝐺‘𝑆)) ∧ 𝑊)
17	2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16	cdlemeg46fsfv 38538	. . . 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))) → ((𝐹‘𝑅) ∨ 𝑆) = ((𝐹‘𝑅) ∨ 𝑌))
18	17	oveq1d 7290	. . 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))) → (((𝐹‘𝑅) ∨ 𝑆) ∧ 𝑊) = (((𝐹‘𝑅) ∨ 𝑌) ∧ 𝑊))
19	2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16	cdlemeg46frv 38539	. . 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))) → (((𝐹‘𝑅) ∨ 𝑌) ∧ 𝑊) = 𝑌)
20	18, 19	eqtrd 2778	. 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) ∨ 𝑆) ∧ 𝑊) = 𝑌)
21	1, 20	eqtrid 2790	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))) → 𝑋 = 𝑌)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ⦋csb 3832 ifcif 4459 class class class wbr 5074 ↦ cmpt 5157 ‘cfv 6433 ℩crio 7231 (class class class)co 7275 Basecbs 16912 lecple 16969 joincjn 18029 meetcmee 18030 Atomscatm 37277 HLchlt 37364 LHypclh 37998

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-riotaBAD 36967

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-1st 7831 df-2nd 7832 df-undef 8089 df-proset 18013 df-poset 18031 df-plt 18048 df-lub 18064 df-glb 18065 df-join 18066 df-meet 18067 df-p0 18143 df-p1 18144 df-lat 18150 df-clat 18217 df-oposet 37190 df-ol 37192 df-oml 37193 df-covers 37280 df-ats 37281 df-atl 37312 df-cvlat 37336 df-hlat 37365 df-llines 37512 df-lplanes 37513 df-lvols 37514 df-lines 37515 df-psubsp 37517 df-pmap 37518 df-padd 37810 df-lhyp 38002

This theorem is referenced by: cdlemeg46vrg 38541

W3C validator