Theorem cdlemk 39650

Description: Lemma K of [Crawley] p. 118. Final result, lines 11 and 12 on p. 120: given two translations f and k with the same trace, there exists a trace-preserving endomorphism tau whose value at f is k. We use 𝐹, 𝑁, and 𝑢 to represent f, k, and tau. (Contributed by NM, 1-Aug-2013.)

Hypotheses

Ref	Expression
cdlemk7.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemk7.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemk7.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
cdlemk7.e	⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)

Assertion

Ref	Expression
cdlemk	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) → ∃𝑢 ∈ 𝐸 (𝑢‘𝐹) = 𝑁)

Distinct variable groups:   𝑢,𝐸   𝑢,𝐹   𝑢,𝐾   𝑢,𝑁   𝑢,𝑅   𝑢,𝑇   𝑢,𝑊

Allowed substitution hint:   𝐻(𝑢)

Proof of Theorem cdlemk

Dummy variables 𝑓 𝑏 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2731	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2731	. . 3 ⊢ (join‘𝐾) = (join‘𝐾)
3		eqid 2731	. . 3 ⊢ (meet‘𝐾) = (meet‘𝐾)
4		eqid 2731	. . 3 ⊢ (oc‘𝐾) = (oc‘𝐾)
5		eqid 2731	. . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
6		cdlemk7.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
7		cdlemk7.t	. . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
8		cdlemk7.r	. . 3 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
9		eqid 2731	. . 3 ⊢ ((oc‘𝐾)‘𝑊) = ((oc‘𝐾)‘𝑊)
10		eqid 2731	. . 3 ⊢ ((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅‘𝑏))(meet‘𝐾)((𝑁‘((oc‘𝐾)‘𝑊))(join‘𝐾)(𝑅‘(𝑏 ∘ ◡𝐹)))) = ((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅‘𝑏))(meet‘𝐾)((𝑁‘((oc‘𝐾)‘𝑊))(join‘𝐾)(𝑅‘(𝑏 ∘ ◡𝐹))))
11		eqid 2731	. . 3 ⊢ ((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅‘𝑓))(meet‘𝐾)(((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅‘𝑏))(meet‘𝐾)((𝑁‘((oc‘𝐾)‘𝑊))(join‘𝐾)(𝑅‘(𝑏 ∘ ◡𝐹))))(join‘𝐾)(𝑅‘(𝑓 ∘ ◡𝑏)))) = ((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅‘𝑓))(meet‘𝐾)(((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅‘𝑏))(meet‘𝐾)((𝑁‘((oc‘𝐾)‘𝑊))(join‘𝐾)(𝑅‘(𝑏 ∘ ◡𝐹))))(join‘𝐾)(𝑅‘(𝑓 ∘ ◡𝑏))))
12		eqid 2731	. . 3 ⊢ (℩𝑧 ∈ 𝑇 ∀𝑏 ∈ 𝑇 ((𝑏 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝑓)) → (𝑧‘((oc‘𝐾)‘𝑊)) = ((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅‘𝑓))(meet‘𝐾)(((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅‘𝑏))(meet‘𝐾)((𝑁‘((oc‘𝐾)‘𝑊))(join‘𝐾)(𝑅‘(𝑏 ∘ ◡𝐹))))(join‘𝐾)(𝑅‘(𝑓 ∘ ◡𝑏)))))) = (℩𝑧 ∈ 𝑇 ∀𝑏 ∈ 𝑇 ((𝑏 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝑓)) → (𝑧‘((oc‘𝐾)‘𝑊)) = ((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅‘𝑓))(meet‘𝐾)(((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅‘𝑏))(meet‘𝐾)((𝑁‘((oc‘𝐾)‘𝑊))(join‘𝐾)(𝑅‘(𝑏 ∘ ◡𝐹))))(join‘𝐾)(𝑅‘(𝑓 ∘ ◡𝑏))))))
13		eqid 2731	. . 3 ⊢ (𝑓 ∈ 𝑇 ↦ if(𝐹 = 𝑁, 𝑓, (℩𝑧 ∈ 𝑇 ∀𝑏 ∈ 𝑇 ((𝑏 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝑓)) → (𝑧‘((oc‘𝐾)‘𝑊)) = ((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅‘𝑓))(meet‘𝐾)(((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅‘𝑏))(meet‘𝐾)((𝑁‘((oc‘𝐾)‘𝑊))(join‘𝐾)(𝑅‘(𝑏 ∘ ◡𝐹))))(join‘𝐾)(𝑅‘(𝑓 ∘ ◡𝑏)))))))) = (𝑓 ∈ 𝑇 ↦ if(𝐹 = 𝑁, 𝑓, (℩𝑧 ∈ 𝑇 ∀𝑏 ∈ 𝑇 ((𝑏 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝑓)) → (𝑧‘((oc‘𝐾)‘𝑊)) = ((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅‘𝑓))(meet‘𝐾)(((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅‘𝑏))(meet‘𝐾)((𝑁‘((oc‘𝐾)‘𝑊))(join‘𝐾)(𝑅‘(𝑏 ∘ ◡𝐹))))(join‘𝐾)(𝑅‘(𝑓 ∘ ◡𝑏))))))))
14		cdlemk7.e	. . 3 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
15	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14	cdlemk56w 39649	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) → ((𝑓 ∈ 𝑇 ↦ if(𝐹 = 𝑁, 𝑓, (℩𝑧 ∈ 𝑇 ∀𝑏 ∈ 𝑇 ((𝑏 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝑓)) → (𝑧‘((oc‘𝐾)‘𝑊)) = ((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅‘𝑓))(meet‘𝐾)(((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅‘𝑏))(meet‘𝐾)((𝑁‘((oc‘𝐾)‘𝑊))(join‘𝐾)(𝑅‘(𝑏 ∘ ◡𝐹))))(join‘𝐾)(𝑅‘(𝑓 ∘ ◡𝑏)))))))) ∈ 𝐸 ∧ ((𝑓 ∈ 𝑇 ↦ if(𝐹 = 𝑁, 𝑓, (℩𝑧 ∈ 𝑇 ∀𝑏 ∈ 𝑇 ((𝑏 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝑓)) → (𝑧‘((oc‘𝐾)‘𝑊)) = ((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅‘𝑓))(meet‘𝐾)(((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅‘𝑏))(meet‘𝐾)((𝑁‘((oc‘𝐾)‘𝑊))(join‘𝐾)(𝑅‘(𝑏 ∘ ◡𝐹))))(join‘𝐾)(𝑅‘(𝑓 ∘ ◡𝑏))))))))‘𝐹) = 𝑁))
16		fveq1 6877	. . . 4 ⊢ (𝑢 = (𝑓 ∈ 𝑇 ↦ if(𝐹 = 𝑁, 𝑓, (℩𝑧 ∈ 𝑇 ∀𝑏 ∈ 𝑇 ((𝑏 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝑓)) → (𝑧‘((oc‘𝐾)‘𝑊)) = ((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅‘𝑓))(meet‘𝐾)(((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅‘𝑏))(meet‘𝐾)((𝑁‘((oc‘𝐾)‘𝑊))(join‘𝐾)(𝑅‘(𝑏 ∘ ◡𝐹))))(join‘𝐾)(𝑅‘(𝑓 ∘ ◡𝑏)))))))) → (𝑢‘𝐹) = ((𝑓 ∈ 𝑇 ↦ if(𝐹 = 𝑁, 𝑓, (℩𝑧 ∈ 𝑇 ∀𝑏 ∈ 𝑇 ((𝑏 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝑓)) → (𝑧‘((oc‘𝐾)‘𝑊)) = ((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅‘𝑓))(meet‘𝐾)(((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅‘𝑏))(meet‘𝐾)((𝑁‘((oc‘𝐾)‘𝑊))(join‘𝐾)(𝑅‘(𝑏 ∘ ◡𝐹))))(join‘𝐾)(𝑅‘(𝑓 ∘ ◡𝑏))))))))‘𝐹))
17	16	eqeq1d 2733	. . 3 ⊢ (𝑢 = (𝑓 ∈ 𝑇 ↦ if(𝐹 = 𝑁, 𝑓, (℩𝑧 ∈ 𝑇 ∀𝑏 ∈ 𝑇 ((𝑏 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝑓)) → (𝑧‘((oc‘𝐾)‘𝑊)) = ((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅‘𝑓))(meet‘𝐾)(((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅‘𝑏))(meet‘𝐾)((𝑁‘((oc‘𝐾)‘𝑊))(join‘𝐾)(𝑅‘(𝑏 ∘ ◡𝐹))))(join‘𝐾)(𝑅‘(𝑓 ∘ ◡𝑏)))))))) → ((𝑢‘𝐹) = 𝑁 ↔ ((𝑓 ∈ 𝑇 ↦ if(𝐹 = 𝑁, 𝑓, (℩𝑧 ∈ 𝑇 ∀𝑏 ∈ 𝑇 ((𝑏 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝑓)) → (𝑧‘((oc‘𝐾)‘𝑊)) = ((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅‘𝑓))(meet‘𝐾)(((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅‘𝑏))(meet‘𝐾)((𝑁‘((oc‘𝐾)‘𝑊))(join‘𝐾)(𝑅‘(𝑏 ∘ ◡𝐹))))(join‘𝐾)(𝑅‘(𝑓 ∘ ◡𝑏))))))))‘𝐹) = 𝑁))
18	17	rspcev 3609	. 2 ⊢ (((𝑓 ∈ 𝑇 ↦ if(𝐹 = 𝑁, 𝑓, (℩𝑧 ∈ 𝑇 ∀𝑏 ∈ 𝑇 ((𝑏 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝑓)) → (𝑧‘((oc‘𝐾)‘𝑊)) = ((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅‘𝑓))(meet‘𝐾)(((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅‘𝑏))(meet‘𝐾)((𝑁‘((oc‘𝐾)‘𝑊))(join‘𝐾)(𝑅‘(𝑏 ∘ ◡𝐹))))(join‘𝐾)(𝑅‘(𝑓 ∘ ◡𝑏)))))))) ∈ 𝐸 ∧ ((𝑓 ∈ 𝑇 ↦ if(𝐹 = 𝑁, 𝑓, (℩𝑧 ∈ 𝑇 ∀𝑏 ∈ 𝑇 ((𝑏 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝑓)) → (𝑧‘((oc‘𝐾)‘𝑊)) = ((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅‘𝑓))(meet‘𝐾)(((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅‘𝑏))(meet‘𝐾)((𝑁‘((oc‘𝐾)‘𝑊))(join‘𝐾)(𝑅‘(𝑏 ∘ ◡𝐹))))(join‘𝐾)(𝑅‘(𝑓 ∘ ◡𝑏))))))))‘𝐹) = 𝑁) → ∃𝑢 ∈ 𝐸 (𝑢‘𝐹) = 𝑁)
19	15, 18	syl 17	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) → ∃𝑢 ∈ 𝐸 (𝑢‘𝐹) = 𝑁)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2939  ∀wral 3060  ∃wrex 3069  ifcif 4522   ↦ cmpt 5224   I cid 5566  ^◡ccnv 5668   ↾ cres 5671   ∘ ccom 5673  ‘cfv 6532  ℩crio 7348  (class class class)co 7393  Basecbs 17126  occoc 17187  joincjn 18246  meetcmee 18247  Atomscatm 37938  HLchlt 38025  LHypclh 38660  LTrncltrn 38777  trLctrl 38834  TEndoctendo 39428

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708  ax-riotaBAD 37628

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-iun 4992  df-iin 4993  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-riota 7349  df-ov 7396  df-oprab 7397  df-mpo 7398  df-1st 7957  df-2nd 7958  df-undef 8240  df-map 8805  df-proset 18230  df-poset 18248  df-plt 18265  df-lub 18281  df-glb 18282  df-join 18283  df-meet 18284  df-p0 18360  df-p1 18361  df-lat 18367  df-clat 18434  df-oposet 37851  df-ol 37853  df-oml 37854  df-covers 37941  df-ats 37942  df-atl 37973  df-cvlat 37997  df-hlat 38026  df-llines 38174  df-lplanes 38175  df-lvols 38176  df-lines 38177  df-psubsp 38179  df-pmap 38180  df-padd 38472  df-lhyp 38664  df-laut 38665  df-ldil 38780  df-ltrn 38781  df-trl 38835  df-tendo 39431

This theorem is referenced by:  tendoex  39651  cdleml2N  39653