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Theorem dihwN 40155
Description: Value of isomorphism H at the fiducial hyperplane π‘Š. (Contributed by NM, 25-Aug-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
dihw.b 𝐡 = (Baseβ€˜πΎ)
dihw.h 𝐻 = (LHypβ€˜πΎ)
dihw.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
dihw.o 0 = (𝑓 ∈ 𝑇 ↦ ( I β†Ύ 𝐡))
dihw.i 𝐼 = ((DIsoHβ€˜πΎ)β€˜π‘Š)
dihw.k (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
Assertion
Ref Expression
dihwN (πœ‘ β†’ (πΌβ€˜π‘Š) = (𝑇 Γ— { 0 }))
Distinct variable groups:   𝑓,𝐾   𝑓,π‘Š
Allowed substitution hints:   πœ‘(𝑓)   𝐡(𝑓)   𝑇(𝑓)   𝐻(𝑓)   𝐼(𝑓)   0 (𝑓)

Proof of Theorem dihwN
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 dihw.k . . 3 (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
21simprd 496 . . . . 5 (πœ‘ β†’ π‘Š ∈ 𝐻)
3 dihw.b . . . . . 6 𝐡 = (Baseβ€˜πΎ)
4 dihw.h . . . . . 6 𝐻 = (LHypβ€˜πΎ)
53, 4lhpbase 38864 . . . . 5 (π‘Š ∈ 𝐻 β†’ π‘Š ∈ 𝐡)
62, 5syl 17 . . . 4 (πœ‘ β†’ π‘Š ∈ 𝐡)
71simpld 495 . . . . . 6 (πœ‘ β†’ 𝐾 ∈ HL)
87hllatd 38229 . . . . 5 (πœ‘ β†’ 𝐾 ∈ Lat)
9 eqid 2732 . . . . . 6 (leβ€˜πΎ) = (leβ€˜πΎ)
103, 9latref 18393 . . . . 5 ((𝐾 ∈ Lat ∧ π‘Š ∈ 𝐡) β†’ π‘Š(leβ€˜πΎ)π‘Š)
118, 6, 10syl2anc 584 . . . 4 (πœ‘ β†’ π‘Š(leβ€˜πΎ)π‘Š)
126, 11jca 512 . . 3 (πœ‘ β†’ (π‘Š ∈ 𝐡 ∧ π‘Š(leβ€˜πΎ)π‘Š))
13 dihw.i . . . 4 𝐼 = ((DIsoHβ€˜πΎ)β€˜π‘Š)
14 eqid 2732 . . . 4 ((DIsoBβ€˜πΎ)β€˜π‘Š) = ((DIsoBβ€˜πΎ)β€˜π‘Š)
153, 9, 4, 13, 14dihvalb 40103 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘Š ∈ 𝐡 ∧ π‘Š(leβ€˜πΎ)π‘Š)) β†’ (πΌβ€˜π‘Š) = (((DIsoBβ€˜πΎ)β€˜π‘Š)β€˜π‘Š))
161, 12, 15syl2anc 584 . 2 (πœ‘ β†’ (πΌβ€˜π‘Š) = (((DIsoBβ€˜πΎ)β€˜π‘Š)β€˜π‘Š))
17 dihw.t . . . 4 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
18 dihw.o . . . 4 0 = (𝑓 ∈ 𝑇 ↦ ( I β†Ύ 𝐡))
19 eqid 2732 . . . 4 ((DIsoAβ€˜πΎ)β€˜π‘Š) = ((DIsoAβ€˜πΎ)β€˜π‘Š)
203, 9, 4, 17, 18, 19, 14dibval2 40010 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘Š ∈ 𝐡 ∧ π‘Š(leβ€˜πΎ)π‘Š)) β†’ (((DIsoBβ€˜πΎ)β€˜π‘Š)β€˜π‘Š) = ((((DIsoAβ€˜πΎ)β€˜π‘Š)β€˜π‘Š) Γ— { 0 }))
211, 12, 20syl2anc 584 . 2 (πœ‘ β†’ (((DIsoBβ€˜πΎ)β€˜π‘Š)β€˜π‘Š) = ((((DIsoAβ€˜πΎ)β€˜π‘Š)β€˜π‘Š) Γ— { 0 }))
22 eqid 2732 . . . . . 6 ((trLβ€˜πΎ)β€˜π‘Š) = ((trLβ€˜πΎ)β€˜π‘Š)
233, 9, 4, 17, 22, 19diaval 39898 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘Š ∈ 𝐡 ∧ π‘Š(leβ€˜πΎ)π‘Š)) β†’ (((DIsoAβ€˜πΎ)β€˜π‘Š)β€˜π‘Š) = {𝑔 ∈ 𝑇 ∣ (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘”)(leβ€˜πΎ)π‘Š})
241, 12, 23syl2anc 584 . . . 4 (πœ‘ β†’ (((DIsoAβ€˜πΎ)β€˜π‘Š)β€˜π‘Š) = {𝑔 ∈ 𝑇 ∣ (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘”)(leβ€˜πΎ)π‘Š})
259, 4, 17, 22trlle 39050 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑔 ∈ 𝑇) β†’ (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘”)(leβ€˜πΎ)π‘Š)
261, 25sylan 580 . . . . . 6 ((πœ‘ ∧ 𝑔 ∈ 𝑇) β†’ (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘”)(leβ€˜πΎ)π‘Š)
2726ralrimiva 3146 . . . . 5 (πœ‘ β†’ βˆ€π‘” ∈ 𝑇 (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘”)(leβ€˜πΎ)π‘Š)
28 rabid2 3464 . . . . 5 (𝑇 = {𝑔 ∈ 𝑇 ∣ (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘”)(leβ€˜πΎ)π‘Š} ↔ βˆ€π‘” ∈ 𝑇 (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘”)(leβ€˜πΎ)π‘Š)
2927, 28sylibr 233 . . . 4 (πœ‘ β†’ 𝑇 = {𝑔 ∈ 𝑇 ∣ (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘”)(leβ€˜πΎ)π‘Š})
3024, 29eqtr4d 2775 . . 3 (πœ‘ β†’ (((DIsoAβ€˜πΎ)β€˜π‘Š)β€˜π‘Š) = 𝑇)
3130xpeq1d 5705 . 2 (πœ‘ β†’ ((((DIsoAβ€˜πΎ)β€˜π‘Š)β€˜π‘Š) Γ— { 0 }) = (𝑇 Γ— { 0 }))
3216, 21, 313eqtrd 2776 1 (πœ‘ β†’ (πΌβ€˜π‘Š) = (𝑇 Γ— { 0 }))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  {crab 3432  {csn 4628   class class class wbr 5148   ↦ cmpt 5231   I cid 5573   Γ— cxp 5674   β†Ύ cres 5678  β€˜cfv 6543  Basecbs 17143  lecple 17203  Latclat 18383  HLchlt 38215  LHypclh 38850  LTrncltrn 38967  trLctrl 39024  DIsoAcdia 39894  DIsoBcdib 40004  DIsoHcdih 40094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  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-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 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-map 8821  df-proset 18247  df-poset 18265  df-plt 18282  df-lub 18298  df-glb 18299  df-join 18300  df-meet 18301  df-p0 18377  df-p1 18378  df-lat 18384  df-oposet 38041  df-ol 38043  df-oml 38044  df-covers 38131  df-ats 38132  df-atl 38163  df-cvlat 38187  df-hlat 38216  df-lhyp 38854  df-laut 38855  df-ldil 38970  df-ltrn 38971  df-trl 39025  df-disoa 39895  df-dib 40005  df-dih 40095
This theorem is referenced by: (None)
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