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Theorem cdlemk56 40306
Description: Part of Lemma K of [Crawley] p. 118. Line 11, p. 120, "tau is in Delta" i.e. 𝑈 is a trace-preserving endormorphism. (Contributed by NM, 31-Jul-2013.)
Hypotheses
Ref Expression
cdlemk5.b 𝐵 = (Base‘𝐾)
cdlemk5.l = (le‘𝐾)
cdlemk5.j = (join‘𝐾)
cdlemk5.m = (meet‘𝐾)
cdlemk5.a 𝐴 = (Atoms‘𝐾)
cdlemk5.h 𝐻 = (LHyp‘𝐾)
cdlemk5.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemk5.r 𝑅 = ((trL‘𝐾)‘𝑊)
cdlemk5.z 𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))
cdlemk5.y 𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))
cdlemk5.x 𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑔)) → (𝑧𝑃) = 𝑌))
cdlemk5.u 𝑈 = (𝑔𝑇 ↦ if(𝐹 = 𝑁, 𝑔, 𝑋))
cdlemk5.e 𝐸 = ((TEndo‘𝐾)‘𝑊)
Assertion
Ref Expression
cdlemk56 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑁𝑇) ∧ (𝑅𝐹) = (𝑅𝑁) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝑈𝐸)
Distinct variable groups:   ,𝑔   ,𝑔   𝐵,𝑔   𝑃,𝑔   𝑅,𝑔   𝑇,𝑔   𝑔,𝑍   𝑔,𝑏,𝑧,   ,𝑏   𝑧,𝑔,   ,𝑏,𝑧   𝐴,𝑏,𝑔,𝑧   𝐵,𝑏,𝑧   𝐹,𝑏,𝑔,𝑧   𝐻,𝑏,𝑔,𝑧   𝐾,𝑏,𝑔,𝑧   𝑁,𝑏,𝑔,𝑧   𝑃,𝑏,𝑧   𝑅,𝑏,𝑧   𝑇,𝑏,𝑧   𝑊,𝑏,𝑔,𝑧   𝑧,𝑌
Allowed substitution hints:   𝑈(𝑧,𝑔,𝑏)   𝐸(𝑧,𝑔,𝑏)   𝑋(𝑧,𝑔,𝑏)   𝑌(𝑔,𝑏)   𝑍(𝑧,𝑏)

Proof of Theorem cdlemk56
Dummy variables 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cdlemk5.l . 2 = (le‘𝐾)
2 cdlemk5.h . 2 𝐻 = (LHyp‘𝐾)
3 cdlemk5.t . 2 𝑇 = ((LTrn‘𝐾)‘𝑊)
4 cdlemk5.r . 2 𝑅 = ((trL‘𝐾)‘𝑊)
5 cdlemk5.e . 2 𝐸 = ((TEndo‘𝐾)‘𝑊)
6 simp11 1202 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑁𝑇) ∧ (𝑅𝐹) = (𝑅𝑁) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
7 vex 3477 . . . . . 6 𝑔 ∈ V
8 cdlemk5.x . . . . . . 7 𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑔)) → (𝑧𝑃) = 𝑌))
9 riotaex 7372 . . . . . . 7 (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑔)) → (𝑧𝑃) = 𝑌)) ∈ V
108, 9eqeltri 2828 . . . . . 6 𝑋 ∈ V
117, 10ifex 4578 . . . . 5 if(𝐹 = 𝑁, 𝑔, 𝑋) ∈ V
1211rgenw 3064 . . . 4 𝑔𝑇 if(𝐹 = 𝑁, 𝑔, 𝑋) ∈ V
13 cdlemk5.u . . . . 5 𝑈 = (𝑔𝑇 ↦ if(𝐹 = 𝑁, 𝑔, 𝑋))
1413fnmpt 6690 . . . 4 (∀𝑔𝑇 if(𝐹 = 𝑁, 𝑔, 𝑋) ∈ V → 𝑈 Fn 𝑇)
1512, 14mp1i 13 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑁𝑇) ∧ (𝑅𝐹) = (𝑅𝑁) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝑈 Fn 𝑇)
16 simpl11 1247 . . . . 5 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑁𝑇) ∧ (𝑅𝐹) = (𝑅𝑁) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ 𝑓𝑇) → (𝐾 ∈ HL ∧ 𝑊𝐻))
17 simpl2 1191 . . . . 5 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑁𝑇) ∧ (𝑅𝐹) = (𝑅𝑁) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ 𝑓𝑇) → (𝑅𝐹) = (𝑅𝑁))
18 simpl12 1248 . . . . 5 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑁𝑇) ∧ (𝑅𝐹) = (𝑅𝑁) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ 𝑓𝑇) → 𝐹𝑇)
19 simpl13 1249 . . . . 5 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑁𝑇) ∧ (𝑅𝐹) = (𝑅𝑁) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ 𝑓𝑇) → 𝑁𝑇)
20 simpr 484 . . . . 5 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑁𝑇) ∧ (𝑅𝐹) = (𝑅𝑁) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ 𝑓𝑇) → 𝑓𝑇)
21 simpl3 1192 . . . . 5 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑁𝑇) ∧ (𝑅𝐹) = (𝑅𝑁) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ 𝑓𝑇) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
22 cdlemk5.b . . . . . 6 𝐵 = (Base‘𝐾)
23 cdlemk5.j . . . . . 6 = (join‘𝐾)
24 cdlemk5.m . . . . . 6 = (meet‘𝐾)
25 cdlemk5.a . . . . . 6 𝐴 = (Atoms‘𝐾)
26 cdlemk5.z . . . . . 6 𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))
27 cdlemk5.y . . . . . 6 𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))
2822, 1, 23, 24, 25, 2, 3, 4, 26, 27, 8, 13cdlemk35u 40299 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐹) = (𝑅𝑁)) ∧ (𝐹𝑇𝑁𝑇𝑓𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑈𝑓) ∈ 𝑇)
2916, 17, 18, 19, 20, 21, 28syl231anc 1389 . . . 4 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑁𝑇) ∧ (𝑅𝐹) = (𝑅𝑁) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ 𝑓𝑇) → (𝑈𝑓) ∈ 𝑇)
3029ralrimiva 3145 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑁𝑇) ∧ (𝑅𝐹) = (𝑅𝑁) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ∀𝑓𝑇 (𝑈𝑓) ∈ 𝑇)
31 ffnfv 7120 . . 3 (𝑈:𝑇𝑇 ↔ (𝑈 Fn 𝑇 ∧ ∀𝑓𝑇 (𝑈𝑓) ∈ 𝑇))
3215, 30, 31sylanbrc 582 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑁𝑇) ∧ (𝑅𝐹) = (𝑅𝑁) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝑈:𝑇𝑇)
33 simp11 1202 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑁𝑇) ∧ (𝑅𝐹) = (𝑅𝑁) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ 𝑓𝑇𝑇) → ((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑁𝑇))
34 simp12 1203 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑁𝑇) ∧ (𝑅𝐹) = (𝑅𝑁) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ 𝑓𝑇𝑇) → (𝑅𝐹) = (𝑅𝑁))
35 simp2 1136 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑁𝑇) ∧ (𝑅𝐹) = (𝑅𝑁) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ 𝑓𝑇𝑇) → 𝑓𝑇)
36 simp3 1137 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑁𝑇) ∧ (𝑅𝐹) = (𝑅𝑁) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ 𝑓𝑇𝑇) → 𝑇)
37 simp13 1204 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑁𝑇) ∧ (𝑅𝐹) = (𝑅𝑁) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ 𝑓𝑇𝑇) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
3822, 1, 23, 24, 25, 2, 3, 4, 26, 27, 8, 13cdlemk55u 40301 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑁𝑇) ∧ ((𝑅𝐹) = (𝑅𝑁) ∧ 𝑓𝑇𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑈‘(𝑓)) = ((𝑈𝑓) ∘ (𝑈)))
3933, 34, 35, 36, 37, 38syl131anc 1382 . 2 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑁𝑇) ∧ (𝑅𝐹) = (𝑅𝑁) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ 𝑓𝑇𝑇) → (𝑈‘(𝑓)) = ((𝑈𝑓) ∘ (𝑈)))
40 simpl1 1190 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑁𝑇) ∧ (𝑅𝐹) = (𝑅𝑁) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ 𝑓𝑇) → ((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑁𝑇))
4122, 1, 23, 24, 25, 2, 3, 4, 26, 27, 8, 13cdlemk39u 40303 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑁𝑇) ∧ ((𝑅𝐹) = (𝑅𝑁) ∧ 𝑓𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑅‘(𝑈𝑓)) (𝑅𝑓))
4240, 17, 20, 21, 41syl121anc 1374 . 2 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑁𝑇) ∧ (𝑅𝐹) = (𝑅𝑁) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ 𝑓𝑇) → (𝑅‘(𝑈𝑓)) (𝑅𝑓))
431, 2, 3, 4, 5, 6, 32, 39, 42istendod 40097 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑁𝑇) ∧ (𝑅𝐹) = (𝑅𝑁) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝑈𝐸)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1086   = wceq 1540  wcel 2105  wne 2939  wral 3060  Vcvv 3473  ifcif 4528   class class class wbr 5148  cmpt 5231   I cid 5573  ccnv 5675  cres 5678  ccom 5680   Fn wfn 6538  wf 6539  cfv 6543  crio 7367  (class class class)co 7412  Basecbs 17151  lecple 17211  joincjn 18274  meetcmee 18275  Atomscatm 38597  HLchlt 38684  LHypclh 39319  LTrncltrn 39436  trLctrl 39493  TEndoctendo 40087
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-riotaBAD 38287
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7979  df-2nd 7980  df-undef 8264  df-map 8828  df-proset 18258  df-poset 18276  df-plt 18293  df-lub 18309  df-glb 18310  df-join 18311  df-meet 18312  df-p0 18388  df-p1 18389  df-lat 18395  df-clat 18462  df-oposet 38510  df-ol 38512  df-oml 38513  df-covers 38600  df-ats 38601  df-atl 38632  df-cvlat 38656  df-hlat 38685  df-llines 38833  df-lplanes 38834  df-lvols 38835  df-lines 38836  df-psubsp 38838  df-pmap 38839  df-padd 39131  df-lhyp 39323  df-laut 39324  df-ldil 39439  df-ltrn 39440  df-trl 39494  df-tendo 40090
This theorem is referenced by:  cdlemk56w  40308
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