Theorem cdlemk 38915

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 2738	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2738	. . 3 ⊢ (join‘𝐾) = (join‘𝐾)
3		eqid 2738	. . 3 ⊢ (meet‘𝐾) = (meet‘𝐾)
4		eqid 2738	. . 3 ⊢ (oc‘𝐾) = (oc‘𝐾)
5		eqid 2738	. . 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 2738	. . 3 ⊢ ((oc‘𝐾)‘𝑊) = ((oc‘𝐾)‘𝑊)
10		eqid 2738	. . 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 2738	. . 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 2738	. . 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 2738	. . 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 38914	. 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 6755	. . . 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 2740	. . 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 3552	. 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 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  ∃wrex 3064  ifcif 4456   ↦ cmpt 5153   I cid 5479  ^◡ccnv 5579   ↾ cres 5582   ∘ ccom 5584  ‘cfv 6418  ℩crio 7211  (class class class)co 7255  Basecbs 16840  occoc 16896  joincjn 17944  meetcmee 17945  Atomscatm 37204  HLchlt 37291  LHypclh 37925  LTrncltrn 38042  trLctrl 38099  TEndoctendo 38693

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-riotaBAD 36894

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-undef 8060  df-map 8575  df-proset 17928  df-poset 17946  df-plt 17963  df-lub 17979  df-glb 17980  df-join 17981  df-meet 17982  df-p0 18058  df-p1 18059  df-lat 18065  df-clat 18132  df-oposet 37117  df-ol 37119  df-oml 37120  df-covers 37207  df-ats 37208  df-atl 37239  df-cvlat 37263  df-hlat 37292  df-llines 37439  df-lplanes 37440  df-lvols 37441  df-lines 37442  df-psubsp 37444  df-pmap 37445  df-padd 37737  df-lhyp 37929  df-laut 37930  df-ldil 38045  df-ltrn 38046  df-trl 38100  df-tendo 38696

This theorem is referenced by:  tendoex  38916  cdleml2N  38918