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Theorem cdlemk 35083
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 ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝐹) = (𝑅𝑁)) → ∃𝑢𝐸 (𝑢𝐹) = 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.
StepHypRef Expression
1 eqid 2609 . . 3 (Base‘𝐾) = (Base‘𝐾)
2 eqid 2609 . . 3 (join‘𝐾) = (join‘𝐾)
3 eqid 2609 . . 3 (meet‘𝐾) = (meet‘𝐾)
4 eqid 2609 . . 3 (oc‘𝐾) = (oc‘𝐾)
5 eqid 2609 . . 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 2609 . . 3 ((oc‘𝐾)‘𝑊) = ((oc‘𝐾)‘𝑊)
10 eqid 2609 . . 3 ((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅𝑏))(meet‘𝐾)((𝑁‘((oc‘𝐾)‘𝑊))(join‘𝐾)(𝑅‘(𝑏𝐹)))) = ((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅𝑏))(meet‘𝐾)((𝑁‘((oc‘𝐾)‘𝑊))(join‘𝐾)(𝑅‘(𝑏𝐹))))
11 eqid 2609 . . 3 ((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅𝑓))(meet‘𝐾)(((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅𝑏))(meet‘𝐾)((𝑁‘((oc‘𝐾)‘𝑊))(join‘𝐾)(𝑅‘(𝑏𝐹))))(join‘𝐾)(𝑅‘(𝑓𝑏)))) = ((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅𝑓))(meet‘𝐾)(((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅𝑏))(meet‘𝐾)((𝑁‘((oc‘𝐾)‘𝑊))(join‘𝐾)(𝑅‘(𝑏𝐹))))(join‘𝐾)(𝑅‘(𝑓𝑏))))
12 eqid 2609 . . 3 (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑓)) → (𝑧‘((oc‘𝐾)‘𝑊)) = ((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅𝑓))(meet‘𝐾)(((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅𝑏))(meet‘𝐾)((𝑁‘((oc‘𝐾)‘𝑊))(join‘𝐾)(𝑅‘(𝑏𝐹))))(join‘𝐾)(𝑅‘(𝑓𝑏)))))) = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑓)) → (𝑧‘((oc‘𝐾)‘𝑊)) = ((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅𝑓))(meet‘𝐾)(((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅𝑏))(meet‘𝐾)((𝑁‘((oc‘𝐾)‘𝑊))(join‘𝐾)(𝑅‘(𝑏𝐹))))(join‘𝐾)(𝑅‘(𝑓𝑏))))))
13 eqid 2609 . . 3 (𝑓𝑇 ↦ if(𝐹 = 𝑁, 𝑓, (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑓)) → (𝑧‘((oc‘𝐾)‘𝑊)) = ((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅𝑓))(meet‘𝐾)(((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅𝑏))(meet‘𝐾)((𝑁‘((oc‘𝐾)‘𝑊))(join‘𝐾)(𝑅‘(𝑏𝐹))))(join‘𝐾)(𝑅‘(𝑓𝑏)))))))) = (𝑓𝑇 ↦ if(𝐹 = 𝑁, 𝑓, (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑓)) → (𝑧‘((oc‘𝐾)‘𝑊)) = ((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅𝑓))(meet‘𝐾)(((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅𝑏))(meet‘𝐾)((𝑁‘((oc‘𝐾)‘𝑊))(join‘𝐾)(𝑅‘(𝑏𝐹))))(join‘𝐾)(𝑅‘(𝑓𝑏))))))))
14 cdlemk7.e . . 3 𝐸 = ((TEndo‘𝐾)‘𝑊)
151, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14cdlemk56w 35082 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝐹) = (𝑅𝑁)) → ((𝑓𝑇 ↦ if(𝐹 = 𝑁, 𝑓, (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑓)) → (𝑧‘((oc‘𝐾)‘𝑊)) = ((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅𝑓))(meet‘𝐾)(((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅𝑏))(meet‘𝐾)((𝑁‘((oc‘𝐾)‘𝑊))(join‘𝐾)(𝑅‘(𝑏𝐹))))(join‘𝐾)(𝑅‘(𝑓𝑏)))))))) ∈ 𝐸 ∧ ((𝑓𝑇 ↦ if(𝐹 = 𝑁, 𝑓, (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑓)) → (𝑧‘((oc‘𝐾)‘𝑊)) = ((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅𝑓))(meet‘𝐾)(((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅𝑏))(meet‘𝐾)((𝑁‘((oc‘𝐾)‘𝑊))(join‘𝐾)(𝑅‘(𝑏𝐹))))(join‘𝐾)(𝑅‘(𝑓𝑏))))))))‘𝐹) = 𝑁))
16 fveq1 6087 . . . 4 (𝑢 = (𝑓𝑇 ↦ if(𝐹 = 𝑁, 𝑓, (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑓)) → (𝑧‘((oc‘𝐾)‘𝑊)) = ((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅𝑓))(meet‘𝐾)(((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅𝑏))(meet‘𝐾)((𝑁‘((oc‘𝐾)‘𝑊))(join‘𝐾)(𝑅‘(𝑏𝐹))))(join‘𝐾)(𝑅‘(𝑓𝑏)))))))) → (𝑢𝐹) = ((𝑓𝑇 ↦ if(𝐹 = 𝑁, 𝑓, (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑓)) → (𝑧‘((oc‘𝐾)‘𝑊)) = ((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅𝑓))(meet‘𝐾)(((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅𝑏))(meet‘𝐾)((𝑁‘((oc‘𝐾)‘𝑊))(join‘𝐾)(𝑅‘(𝑏𝐹))))(join‘𝐾)(𝑅‘(𝑓𝑏))))))))‘𝐹))
1716eqeq1d 2611 . . 3 (𝑢 = (𝑓𝑇 ↦ if(𝐹 = 𝑁, 𝑓, (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑓)) → (𝑧‘((oc‘𝐾)‘𝑊)) = ((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅𝑓))(meet‘𝐾)(((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅𝑏))(meet‘𝐾)((𝑁‘((oc‘𝐾)‘𝑊))(join‘𝐾)(𝑅‘(𝑏𝐹))))(join‘𝐾)(𝑅‘(𝑓𝑏)))))))) → ((𝑢𝐹) = 𝑁 ↔ ((𝑓𝑇 ↦ if(𝐹 = 𝑁, 𝑓, (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑓)) → (𝑧‘((oc‘𝐾)‘𝑊)) = ((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅𝑓))(meet‘𝐾)(((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅𝑏))(meet‘𝐾)((𝑁‘((oc‘𝐾)‘𝑊))(join‘𝐾)(𝑅‘(𝑏𝐹))))(join‘𝐾)(𝑅‘(𝑓𝑏))))))))‘𝐹) = 𝑁))
1817rspcev 3281 . 2 (((𝑓𝑇 ↦ if(𝐹 = 𝑁, 𝑓, (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑓)) → (𝑧‘((oc‘𝐾)‘𝑊)) = ((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅𝑓))(meet‘𝐾)(((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅𝑏))(meet‘𝐾)((𝑁‘((oc‘𝐾)‘𝑊))(join‘𝐾)(𝑅‘(𝑏𝐹))))(join‘𝐾)(𝑅‘(𝑓𝑏)))))))) ∈ 𝐸 ∧ ((𝑓𝑇 ↦ if(𝐹 = 𝑁, 𝑓, (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑓)) → (𝑧‘((oc‘𝐾)‘𝑊)) = ((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅𝑓))(meet‘𝐾)(((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝑅𝑏))(meet‘𝐾)((𝑁‘((oc‘𝐾)‘𝑊))(join‘𝐾)(𝑅‘(𝑏𝐹))))(join‘𝐾)(𝑅‘(𝑓𝑏))))))))‘𝐹) = 𝑁) → ∃𝑢𝐸 (𝑢𝐹) = 𝑁)
1915, 18syl 17 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝐹) = (𝑅𝑁)) → ∃𝑢𝐸 (𝑢𝐹) = 𝑁)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030   = wceq 1474  wcel 1976  wne 2779  wral 2895  wrex 2896  ifcif 4035  cmpt 4637   I cid 4938  ccnv 5027  cres 5030  ccom 5032  cfv 5790  crio 6488  (class class class)co 6527  Basecbs 15641  occoc 15722  joincjn 16713  meetcmee 16714  Atomscatm 33371  HLchlt 33458  LHypclh 34091  LTrncltrn 34208  trLctrl 34266  TEndoctendo 34861
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-riotaBAD 33060
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-iin 4452  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-1st 7036  df-2nd 7037  df-undef 7263  df-map 7723  df-preset 16697  df-poset 16715  df-plt 16727  df-lub 16743  df-glb 16744  df-join 16745  df-meet 16746  df-p0 16808  df-p1 16809  df-lat 16815  df-clat 16877  df-oposet 33284  df-ol 33286  df-oml 33287  df-covers 33374  df-ats 33375  df-atl 33406  df-cvlat 33430  df-hlat 33459  df-llines 33605  df-lplanes 33606  df-lvols 33607  df-lines 33608  df-psubsp 33610  df-pmap 33611  df-padd 33903  df-lhyp 34095  df-laut 34096  df-ldil 34211  df-ltrn 34212  df-trl 34267  df-tendo 34864
This theorem is referenced by:  tendoex  35084  cdleml2N  35086
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