Theorem cdlemk 41638

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 2769	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2769	. . 3 ⊢ (join‘𝐾) = (join‘𝐾)
3		eqid 2769	. . 3 ⊢ (meet‘𝐾) = (meet‘𝐾)
4		eqid 2769	. . 3 ⊢ (oc‘𝐾) = (oc‘𝐾)
5		eqid 2769	. . 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 2769	. . 3 ⊢ ((oc‘𝐾)‘𝑊) = ((oc‘𝐾)‘𝑊)
10		eqid 2769	. . 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 2769	. . 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 2769	. . 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 2769	. . 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 41637	. 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 6881	. . . 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 2771	. . 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 3590	. 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 18	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) → ∃𝑢 ∈ 𝐸 (𝑢‘𝐹) = 𝑁)

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085  ∃wrex 3095  ifcif 4492   ↦ cmpt 5196   I cid 5556  ^◡ccnv 5661   ↾ cres 5664   ∘ ccom 5666  ‘cfv 6537  ℩crio 7367  (class class class)co 7411  Basecbs 17269  occoc 17318  joincjn 18367  meetcmee 18368  Atomscatm 39927  HLchlt 40014  LHypclh 40648  LTrncltrn 40765  trLctrl 40822  TEndoctendo 41416

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-riotaBAD 39617

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-iin 4963  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7986  df-2nd 7987  df-undef 8269  df-map 8826  df-proset 18350  df-poset 18369  df-plt 18384  df-lub 18400  df-glb 18401  df-join 18402  df-meet 18403  df-p0 18479  df-p1 18480  df-lat 18488  df-clat 18555  df-oposet 39840  df-ol 39842  df-oml 39843  df-covers 39930  df-ats 39931  df-atl 39962  df-cvlat 39986  df-hlat 40015  df-llines 40162  df-lplanes 40163  df-lvols 40164  df-lines 40165  df-psubsp 40167  df-pmap 40168  df-padd 40460  df-lhyp 40652  df-laut 40653  df-ldil 40768  df-ltrn 40769  df-trl 40823  df-tendo 41419

This theorem is referenced by:  tendoex  41639  cdleml2N  41641