Theorem cdlemk 37048

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 2825	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2825	. . 3 ⊢ (join‘𝐾) = (join‘𝐾)
3		eqid 2825	. . 3 ⊢ (meet‘𝐾) = (meet‘𝐾)
4		eqid 2825	. . 3 ⊢ (oc‘𝐾) = (oc‘𝐾)
5		eqid 2825	. . 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 2825	. . 3 ⊢ ((oc‘𝐾)‘𝑊) = ((oc‘𝐾)‘𝑊)
10		eqid 2825	. . 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 2825	. . 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 2825	. . 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 2825	. . 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 37047	. 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 6436	. . . 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 2827	. . 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 3526	. 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 386   ∧ w3a 1111   = wceq 1656   ∈ wcel 2164   ≠ wne 2999  ∀wral 3117  ∃wrex 3118  ifcif 4308   ↦ cmpt 4954   I cid 5251  ^◡ccnv 5345   ↾ cres 5348   ∘ ccom 5350  ‘cfv 6127  ℩crio 6870  (class class class)co 6910  Basecbs 16229  occoc 16320  joincjn 17304  meetcmee 17305  Atomscatm 35337  HLchlt 35424  LHypclh 36058  LTrncltrn 36175  trLctrl 36232  TEndoctendo 36826

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-riotaBAD 35027

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-fal 1670  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-iin 4745  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-1st 7433  df-2nd 7434  df-undef 7669  df-map 8129  df-proset 17288  df-poset 17306  df-plt 17318  df-lub 17334  df-glb 17335  df-join 17336  df-meet 17337  df-p0 17399  df-p1 17400  df-lat 17406  df-clat 17468  df-oposet 35250  df-ol 35252  df-oml 35253  df-covers 35340  df-ats 35341  df-atl 35372  df-cvlat 35396  df-hlat 35425  df-llines 35572  df-lplanes 35573  df-lvols 35574  df-lines 35575  df-psubsp 35577  df-pmap 35578  df-padd 35870  df-lhyp 36062  df-laut 36063  df-ldil 36178  df-ltrn 36179  df-trl 36233  df-tendo 36829

This theorem is referenced by:  tendoex  37049  cdleml2N  37051