cdlemk56w - Mathbox for Norm Megill

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Theorem cdlemk56w 40672

Description: Use a fixed element to eliminate 𝑃 in cdlemk56 40670. (Contributed by NM, 1-Aug-2013.)

**Hypotheses**
Ref	Expression
cdlemk6.b	⊢ 𝐵 = (Base‘𝐾)
cdlemk6.j	⊢ ∨ = (join‘𝐾)
cdlemk6.m	⊢ ∧ = (meet‘𝐾)
cdlemk6.o	⊢ ⊥ = (oc‘𝐾)
cdlemk6.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemk6.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemk6.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemk6.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
cdlemk6.p	⊢ 𝑃 = ( ⊥ ‘𝑊)
cdlemk6.z	⊢ 𝑍 = ((𝑃 ∨ (𝑅‘𝑏)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝑏 ∘ ^◡𝐹))))
cdlemk6.y	⊢ 𝑌 = ((𝑃 ∨ (𝑅‘𝑔)) ∧ (𝑍 ∨ (𝑅‘(𝑔 ∘ ^◡𝑏))))
cdlemk6.x	⊢ 𝑋 = (℩𝑧 ∈ 𝑇 ∀𝑏 ∈ 𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝑔)) → (𝑧‘𝑃) = 𝑌))
cdlemk6.u	⊢ 𝑈 = (𝑔 ∈ 𝑇 ↦ if(𝐹 = 𝑁, 𝑔, 𝑋))
cdlemk6.e	⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)

**Assertion**
Ref	Expression
cdlemk56w	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) → (𝑈 ∈ 𝐸 ∧ (𝑈‘𝐹) = 𝑁))

Distinct variable groups: 𝑔,𝑏,𝑧, ∧ ∨ ,𝑏,𝑔,𝑧 𝐴,𝑏,𝑔,𝑧 𝐵,𝑏,𝑔,𝑧 𝐹,𝑏,𝑔,𝑧 𝐻,𝑏,𝑔,𝑧 𝐾,𝑏,𝑔,𝑧 𝑁,𝑏,𝑔,𝑧 𝑃,𝑏,𝑔,𝑧 𝑅,𝑏,𝑔,𝑧 𝑇,𝑏,𝑔,𝑧 𝑊,𝑏,𝑔,𝑧 𝑧,𝑌 𝑔,𝑍 Allowed substitution hints: 𝑈(𝑧,𝑔,𝑏) 𝐸(𝑧,𝑔,𝑏) ⊥ (𝑧,𝑔,𝑏) 𝑋(𝑧,𝑔,𝑏) 𝑌(𝑔,𝑏) 𝑍(𝑧,𝑏)

**Proof of Theorem cdlemk56w**
Step	Hyp	Ref	Expression
1		simp1 1133	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simp2l 1196	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) → 𝐹 ∈ 𝑇)
3		simp2r 1197	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) → 𝑁 ∈ 𝑇)
4		simp3 1135	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) → (𝑅‘𝐹) = (𝑅‘𝑁))
5		eqid 2726	. . . . 5 ⊢ (le‘𝐾) = (le‘𝐾)
6		cdlemk6.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
7		cdlemk6.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
8		cdlemk6.p	. . . . . 6 ⊢ 𝑃 = ( ⊥ ‘𝑊)
9		cdlemk6.o	. . . . . . 7 ⊢ ⊥ = (oc‘𝐾)
10	9	fveq1i 6902	. . . . . 6 ⊢ ( ⊥ ‘𝑊) = ((oc‘𝐾)‘𝑊)
11	8, 10	eqtri 2754	. . . . 5 ⊢ 𝑃 = ((oc‘𝐾)‘𝑊)
12	5, 6, 7, 11	lhpocnel2 39718	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃(le‘𝐾)𝑊))
13	12	3ad2ant1 1130	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃(le‘𝐾)𝑊))
14		cdlemk6.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
15		cdlemk6.j	. . . 4 ⊢ ∨ = (join‘𝐾)
16		cdlemk6.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
17		cdlemk6.t	. . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
18		cdlemk6.r	. . . 4 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
19		cdlemk6.z	. . . 4 ⊢ 𝑍 = ((𝑃 ∨ (𝑅‘𝑏)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝑏 ∘ ^◡𝐹))))
20		cdlemk6.y	. . . 4 ⊢ 𝑌 = ((𝑃 ∨ (𝑅‘𝑔)) ∧ (𝑍 ∨ (𝑅‘(𝑔 ∘ ^◡𝑏))))
21		cdlemk6.x	. . . 4 ⊢ 𝑋 = (℩𝑧 ∈ 𝑇 ∀𝑏 ∈ 𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝑔)) → (𝑧‘𝑃) = 𝑌))
22		cdlemk6.u	. . . 4 ⊢ 𝑈 = (𝑔 ∈ 𝑇 ↦ if(𝐹 = 𝑁, 𝑔, 𝑋))
23		cdlemk6.e	. . . 4 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
24	14, 5, 15, 16, 6, 7, 17, 18, 19, 20, 21, 22, 23	cdlemk56 40670	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝐹) = (𝑅‘𝑁) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃(le‘𝐾)𝑊)) → 𝑈 ∈ 𝐸)
25	1, 2, 3, 4, 13, 24	syl311anc 1381	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) → 𝑈 ∈ 𝐸)
26	14, 15, 16, 9, 6, 7, 17, 18, 8, 19, 20, 21, 22	cdlemk19w 40671	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) → (𝑈‘𝐹) = 𝑁)
27	25, 26	jca 510	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) → (𝑈 ∈ 𝐸 ∧ (𝑈‘𝐹) = 𝑁))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∀wral 3051 ifcif 4533 class class class wbr 5153 ↦ cmpt 5236 I cid 5579 ^◡ccnv 5681 ↾ cres 5684 ∘ ccom 5686 ‘cfv 6554 ℩crio 7379 (class class class)co 7424 Basecbs 17213 lecple 17273 occoc 17274 joincjn 18336 meetcmee 18337 Atomscatm 38961 HLchlt 39048 LHypclh 39683 LTrncltrn 39800 trLctrl 39857 TEndoctendo 40451

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 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-riotaBAD 38651

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-iin 5004 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-1st 8003 df-2nd 8004 df-undef 8288 df-map 8857 df-proset 18320 df-poset 18338 df-plt 18355 df-lub 18371 df-glb 18372 df-join 18373 df-meet 18374 df-p0 18450 df-p1 18451 df-lat 18457 df-clat 18524 df-oposet 38874 df-ol 38876 df-oml 38877 df-covers 38964 df-ats 38965 df-atl 38996 df-cvlat 39020 df-hlat 39049 df-llines 39197 df-lplanes 39198 df-lvols 39199 df-lines 39200 df-psubsp 39202 df-pmap 39203 df-padd 39495 df-lhyp 39687 df-laut 39688 df-ldil 39803 df-ltrn 39804 df-trl 39858 df-tendo 40454

This theorem is referenced by: cdlemk 40673 cdleml6 40680

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