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Theorem dihwN 39802
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 497 . . . . 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 38511 . . . . 5 (π‘Š ∈ 𝐻 β†’ π‘Š ∈ 𝐡)
62, 5syl 17 . . . 4 (πœ‘ β†’ π‘Š ∈ 𝐡)
71simpld 496 . . . . . 6 (πœ‘ β†’ 𝐾 ∈ HL)
87hllatd 37876 . . . . 5 (πœ‘ β†’ 𝐾 ∈ Lat)
9 eqid 2733 . . . . . 6 (leβ€˜πΎ) = (leβ€˜πΎ)
103, 9latref 18338 . . . . 5 ((𝐾 ∈ Lat ∧ π‘Š ∈ 𝐡) β†’ π‘Š(leβ€˜πΎ)π‘Š)
118, 6, 10syl2anc 585 . . . 4 (πœ‘ β†’ π‘Š(leβ€˜πΎ)π‘Š)
126, 11jca 513 . . 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 2733 . . . 4 ((DIsoBβ€˜πΎ)β€˜π‘Š) = ((DIsoBβ€˜πΎ)β€˜π‘Š)
153, 9, 4, 13, 14dihvalb 39750 . . 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 585 . 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 2733 . . . 4 ((DIsoAβ€˜πΎ)β€˜π‘Š) = ((DIsoAβ€˜πΎ)β€˜π‘Š)
203, 9, 4, 17, 18, 19, 14dibval2 39657 . . 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 585 . 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 2733 . . . . . 6 ((trLβ€˜πΎ)β€˜π‘Š) = ((trLβ€˜πΎ)β€˜π‘Š)
233, 9, 4, 17, 22, 19diaval 39545 . . . . 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 585 . . . 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 38697 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑔 ∈ 𝑇) β†’ (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘”)(leβ€˜πΎ)π‘Š)
261, 25sylan 581 . . . . . 6 ((πœ‘ ∧ 𝑔 ∈ 𝑇) β†’ (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘”)(leβ€˜πΎ)π‘Š)
2726ralrimiva 3140 . . . . 5 (πœ‘ β†’ βˆ€π‘” ∈ 𝑇 (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘”)(leβ€˜πΎ)π‘Š)
28 rabid2 3438 . . . . 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 2776 . . 3 (πœ‘ β†’ (((DIsoAβ€˜πΎ)β€˜π‘Š)β€˜π‘Š) = 𝑇)
3130xpeq1d 5666 . 2 (πœ‘ β†’ ((((DIsoAβ€˜πΎ)β€˜π‘Š)β€˜π‘Š) Γ— { 0 }) = (𝑇 Γ— { 0 }))
3216, 21, 313eqtrd 2777 1 (πœ‘ β†’ (πΌβ€˜π‘Š) = (𝑇 Γ— { 0 }))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  {crab 3406  {csn 4590   class class class wbr 5109   ↦ cmpt 5192   I cid 5534   Γ— cxp 5635   β†Ύ cres 5639  β€˜cfv 6500  Basecbs 17091  lecple 17148  Latclat 18328  HLchlt 37862  LHypclh 38497  LTrncltrn 38614  trLctrl 38671  DIsoAcdia 39541  DIsoBcdib 39651  DIsoHcdih 39741
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-map 8773  df-proset 18192  df-poset 18210  df-plt 18227  df-lub 18243  df-glb 18244  df-join 18245  df-meet 18246  df-p0 18322  df-p1 18323  df-lat 18329  df-oposet 37688  df-ol 37690  df-oml 37691  df-covers 37778  df-ats 37779  df-atl 37810  df-cvlat 37834  df-hlat 37863  df-lhyp 38501  df-laut 38502  df-ldil 38617  df-ltrn 38618  df-trl 38672  df-disoa 39542  df-dib 39652  df-dih 39742
This theorem is referenced by: (None)
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