Theorem cdlemk 41173

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 2734	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2734	. . 3 ⊢ (join‘𝐾) = (join‘𝐾)
3		eqid 2734	. . 3 ⊢ (meet‘𝐾) = (meet‘𝐾)
4		eqid 2734	. . 3 ⊢ (oc‘𝐾) = (oc‘𝐾)
5		eqid 2734	. . 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 2734	. . 3 ⊢ ((oc‘𝐾)‘𝑊) = ((oc‘𝐾)‘𝑊)
10		eqid 2734	. . 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 2734	. . 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 2734	. . 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 2734	. . 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 41172	. 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 6831	. . . 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 2736	. . 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 3574	. 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 1086   = wceq 1541   ∈ wcel 2113   ≠ wne 2930  ∀wral 3049  ∃wrex 3058  ifcif 4477   ↦ cmpt 5177   I cid 5516  ^◡ccnv 5621   ↾ cres 5624   ∘ ccom 5626  ‘cfv 6490  ℩crio 7312  (class class class)co 7356  Basecbs 17134  occoc 17183  joincjn 18232  meetcmee 18233  Atomscatm 39462  HLchlt 39549  LHypclh 40183  LTrncltrn 40300  trLctrl 40357  TEndoctendo 40951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-riotaBAD 39152

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-iin 4947  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-undef 8213  df-map 8763  df-proset 18215  df-poset 18234  df-plt 18249  df-lub 18265  df-glb 18266  df-join 18267  df-meet 18268  df-p0 18344  df-p1 18345  df-lat 18353  df-clat 18420  df-oposet 39375  df-ol 39377  df-oml 39378  df-covers 39465  df-ats 39466  df-atl 39497  df-cvlat 39521  df-hlat 39550  df-llines 39697  df-lplanes 39698  df-lvols 39699  df-lines 39700  df-psubsp 39702  df-pmap 39703  df-padd 39995  df-lhyp 40187  df-laut 40188  df-ldil 40303  df-ltrn 40304  df-trl 40358  df-tendo 40954

This theorem is referenced by:  tendoex  41174  cdleml2N  41176