cdlemk56w - Mathbox for Norm Megill

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Theorem cdlemk56w 41243

Description: Use a fixed element to eliminate 𝑃 in cdlemk56 41241. (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 1136	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simp2l 1200	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) → 𝐹 ∈ 𝑇)
3		simp2r 1201	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) → 𝑁 ∈ 𝑇)
4		simp3 1138	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) → (𝑅‘𝐹) = (𝑅‘𝑁))
5		eqid 2736	. . . . 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 6835	. . . . . 6 ⊢ ( ⊥ ‘𝑊) = ((oc‘𝐾)‘𝑊)
11	8, 10	eqtri 2759	. . . . 5 ⊢ 𝑃 = ((oc‘𝐾)‘𝑊)
12	5, 6, 7, 11	lhpocnel2 40289	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃(le‘𝐾)𝑊))
13	12	3ad2ant1 1133	. . 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 41241	. . 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 1386	. 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 41242	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) → (𝑈‘𝐹) = 𝑁)
27	25, 26	jca 511	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 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ≠ wne 2932 ∀wral 3051 ifcif 4479 class class class wbr 5098 ↦ cmpt 5179 I cid 5518 ^◡ccnv 5623 ↾ cres 5626 ∘ ccom 5628 ‘cfv 6492 ℩crio 7314 (class class class)co 7358 Basecbs 17136 lecple 17184 occoc 17185 joincjn 18234 meetcmee 18235 Atomscatm 39533 HLchlt 39620 LHypclh 40254 LTrncltrn 40371 trLctrl 40428 TEndoctendo 41022

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-riotaBAD 39223

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-id 5519 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7933 df-2nd 7934 df-undef 8215 df-map 8765 df-proset 18217 df-poset 18236 df-plt 18251 df-lub 18267 df-glb 18268 df-join 18269 df-meet 18270 df-p0 18346 df-p1 18347 df-lat 18355 df-clat 18422 df-oposet 39446 df-ol 39448 df-oml 39449 df-covers 39536 df-ats 39537 df-atl 39568 df-cvlat 39592 df-hlat 39621 df-llines 39768 df-lplanes 39769 df-lvols 39770 df-lines 39771 df-psubsp 39773 df-pmap 39774 df-padd 40066 df-lhyp 40258 df-laut 40259 df-ldil 40374 df-ltrn 40375 df-trl 40429 df-tendo 41025

This theorem is referenced by: cdlemk 41244 cdleml6 41251

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