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Theorem cdlemk 40954
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 2736 . . 3 (Base‘𝐾) = (Base‘𝐾)
2 eqid 2736 . . 3 (join‘𝐾) = (join‘𝐾)
3 eqid 2736 . . 3 (meet‘𝐾) = (meet‘𝐾)
4 eqid 2736 . . 3 (oc‘𝐾) = (oc‘𝐾)
5 eqid 2736 . . 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 2736 . . 3 ((oc‘𝐾)‘𝑊) = ((oc‘𝐾)‘𝑊)
10 eqid 2736 . . 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 2736 . . 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 2736 . . 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 2736 . . 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 40953 . 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 6903 . . . 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 2738 . . 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 3621 . 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 395  w3a 1087   = wceq 1540  wcel 2108  wne 2939  wral 3060  wrex 3069  ifcif 4524  cmpt 5223   I cid 5575  ccnv 5682  cres 5685  ccom 5687  cfv 6559  crio 7385  (class class class)co 7429  Basecbs 17243  occoc 17301  joincjn 18353  meetcmee 18354  Atomscatm 39242  HLchlt 39329  LHypclh 39964  LTrncltrn 40081  trLctrl 40138  TEndoctendo 40732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5277  ax-sep 5294  ax-nul 5304  ax-pow 5363  ax-pr 5430  ax-un 7751  ax-riotaBAD 38932
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-iun 4991  df-iin 4992  df-br 5142  df-opab 5204  df-mpt 5224  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-iota 6512  df-fun 6561  df-fn 6562  df-f 6563  df-f1 6564  df-fo 6565  df-f1o 6566  df-fv 6567  df-riota 7386  df-ov 7432  df-oprab 7433  df-mpo 7434  df-1st 8010  df-2nd 8011  df-undef 8294  df-map 8864  df-proset 18336  df-poset 18355  df-plt 18371  df-lub 18387  df-glb 18388  df-join 18389  df-meet 18390  df-p0 18466  df-p1 18467  df-lat 18473  df-clat 18540  df-oposet 39155  df-ol 39157  df-oml 39158  df-covers 39245  df-ats 39246  df-atl 39277  df-cvlat 39301  df-hlat 39330  df-llines 39478  df-lplanes 39479  df-lvols 39480  df-lines 39481  df-psubsp 39483  df-pmap 39484  df-padd 39776  df-lhyp 39968  df-laut 39969  df-ldil 40084  df-ltrn 40085  df-trl 40139  df-tendo 40735
This theorem is referenced by:  tendoex  40955  cdleml2N  40957
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