cdlemkyuu - Mathbox for Norm Megill

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

Theorem cdlemkyuu 38951

Description: cdlemkyu 38950 with some hypotheses eliminated. TODO: Clean all this up. (Contributed by NM, 21-Jul-2013.)

**Hypotheses**
Ref	Expression
cdlemk5.b	⊢ 𝐵 = (Base‘𝐾)
cdlemk5.l	⊢ ≤ = (le‘𝐾)
cdlemk5.j	⊢ ∨ = (join‘𝐾)
cdlemk5.m	⊢ ∧ = (meet‘𝐾)
cdlemk5.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemk5.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemk5.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemk5.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
cdlemk5.z	⊢ 𝑍 = ((𝑃 ∨ (𝑅‘𝑏)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝑏 ∘ ^◡𝐹))))
cdlemk5.y	⊢ 𝑌 = ((𝑃 ∨ (𝑅‘𝑔)) ∧ (𝑍 ∨ (𝑅‘(𝑔 ∘ ^◡𝑏))))
cdlemk5c.s	⊢ 𝑆 = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (𝑖‘𝑃) = ((𝑃 ∨ (𝑅‘𝑓)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝑓 ∘ ^◡𝐹))))))
cdlemk5a.u2	⊢ 𝐶 = (𝑒 ∈ 𝑇 ↦ (℩𝑗 ∈ 𝑇 (𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝑒)) ∧ (((𝑆‘𝑏)‘𝑃) ∨ (𝑅‘(𝑒 ∘ ^◡𝑏))))))

**Assertion**
Ref	Expression
cdlemkyuu	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) ∧ (𝐺 ∈ 𝑇 ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝑏 ∈ 𝑇 ∧ (𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝐺)))) → ⦋𝐺 / 𝑔⦌𝑌 = ((𝐶‘𝐺)‘𝑃))

Distinct variable groups: ∧ ,𝑔 ∨ ,𝑔 𝐵,𝑔 𝑃,𝑔 𝑅,𝑔 𝑇,𝑔 𝑔,𝑍 𝑔,𝑏 𝑔,𝐺,𝑒 𝑓,𝑔,𝑖,𝑗,𝑒, ∧ ≤ ,𝑖,𝑗 ∨ ,𝑒,𝑓,𝑖,𝑗 𝐴,𝑖,𝑗 𝑓,𝐹,𝑖,𝑗 𝑒,𝐺,𝑗 𝑖,𝐻,𝑗 𝑖,𝐾,𝑗 𝑓,𝑁,𝑖,𝑗 𝑃,𝑒,𝑓,𝑖,𝑗 𝑅,𝑒,𝑓,𝑖,𝑗 𝑒,𝑏,𝑗,𝑆 𝑇,𝑒,𝑓,𝑖,𝑗 𝑒,𝑊,𝑓,𝑖,𝑗 𝑓,𝑏,𝑖 Allowed substitution hints: 𝐴(𝑒,𝑓,𝑔,𝑏) 𝐵(𝑒,𝑓,𝑖,𝑗,𝑏) 𝐶(𝑒,𝑓,𝑔,𝑖,𝑗,𝑏) 𝑃(𝑏) 𝑅(𝑏) 𝑆(𝑓,𝑔,𝑖) 𝑇(𝑏) 𝐹(𝑒,𝑔,𝑏) 𝐺(𝑓,𝑖,𝑏) 𝐻(𝑒,𝑓,𝑔,𝑏) ∨ (𝑏) 𝐾(𝑒,𝑓,𝑔,𝑏) ≤ (𝑒,𝑓,𝑔,𝑏) ∧ (𝑏) 𝑁(𝑒,𝑔,𝑏) 𝑊(𝑔,𝑏) 𝑌(𝑒,𝑓,𝑔,𝑖,𝑗,𝑏) 𝑍(𝑒,𝑓,𝑖,𝑗,𝑏)

Proof of Theorem cdlemkyuu Dummy variable 𝑑 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cdlemk5.b	. 2 ⊢ 𝐵 = (Base‘𝐾)
2		cdlemk5.l	. 2 ⊢ ≤ = (le‘𝐾)
3		cdlemk5.j	. 2 ⊢ ∨ = (join‘𝐾)
4		cdlemk5.m	. 2 ⊢ ∧ = (meet‘𝐾)
5		cdlemk5.a	. 2 ⊢ 𝐴 = (Atoms‘𝐾)
6		cdlemk5.h	. 2 ⊢ 𝐻 = (LHyp‘𝐾)
7		cdlemk5.t	. 2 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
8		cdlemk5.r	. 2 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
9		cdlemk5.z	. 2 ⊢ 𝑍 = ((𝑃 ∨ (𝑅‘𝑏)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝑏 ∘ ^◡𝐹))))
10		cdlemk5.y	. 2 ⊢ 𝑌 = ((𝑃 ∨ (𝑅‘𝑔)) ∧ (𝑍 ∨ (𝑅‘(𝑔 ∘ ^◡𝑏))))
11		cdlemk5c.s	. 2 ⊢ 𝑆 = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (𝑖‘𝑃) = ((𝑃 ∨ (𝑅‘𝑓)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝑓 ∘ ^◡𝐹))))))
12		eqid 2740	. 2 ⊢ (𝑑 ∈ 𝑇, 𝑒 ∈ 𝑇 ↦ (℩𝑗 ∈ 𝑇 (𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝑒)) ∧ (((𝑆‘𝑑)‘𝑃) ∨ (𝑅‘(𝑒 ∘ ^◡𝑑)))))) = (𝑑 ∈ 𝑇, 𝑒 ∈ 𝑇 ↦ (℩𝑗 ∈ 𝑇 (𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝑒)) ∧ (((𝑆‘𝑑)‘𝑃) ∨ (𝑅‘(𝑒 ∘ ^◡𝑑))))))
13		eqid 2740	. 2 ⊢ (𝑆‘𝑏) = (𝑆‘𝑏)
14		cdlemk5a.u2	. 2 ⊢ 𝐶 = (𝑒 ∈ 𝑇 ↦ (℩𝑗 ∈ 𝑇 (𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝑒)) ∧ (((𝑆‘𝑏)‘𝑃) ∨ (𝑅‘(𝑒 ∘ ^◡𝑏))))))
15	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14	cdlemkyu 38950	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) ∧ (𝐺 ∈ 𝑇 ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝑏 ∈ 𝑇 ∧ (𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝐺)))) → ⦋𝐺 / 𝑔⦌𝑌 = ((𝐶‘𝐺)‘𝑃))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1542 ∈ wcel 2110 ≠ wne 2945 ⦋csb 3837 class class class wbr 5079 ↦ cmpt 5162 I cid 5489 ^◡ccnv 5589 ↾ cres 5592 ∘ ccom 5594 ‘cfv 6432 ℩crio 7228 (class class class)co 7272 ∈ cmpo 7274 Basecbs 16923 lecple 16980 joincjn 18040 meetcmee 18041 Atomscatm 37286 HLchlt 37373 LHypclh 38007 LTrncltrn 38124 trLctrl 38181

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7583 ax-riotaBAD 36976

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-id 5490 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7229 df-ov 7275 df-oprab 7276 df-mpo 7277 df-1st 7825 df-2nd 7826 df-undef 8081 df-map 8609 df-proset 18024 df-poset 18042 df-plt 18059 df-lub 18075 df-glb 18076 df-join 18077 df-meet 18078 df-p0 18154 df-p1 18155 df-lat 18161 df-clat 18228 df-oposet 37199 df-ol 37201 df-oml 37202 df-covers 37289 df-ats 37290 df-atl 37321 df-cvlat 37345 df-hlat 37374 df-llines 37521 df-lplanes 37522 df-lvols 37523 df-lines 37524 df-psubsp 37526 df-pmap 37527 df-padd 37819 df-lhyp 38011 df-laut 38012 df-ldil 38127 df-ltrn 38128 df-trl 38182

This theorem is referenced by: cdlemk11ta 38952 cdlemk19ylem 38953

W3C validator