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Theorem dihpN 38476
Description: The value of isomorphism H at the fiducial atom 𝑃 is determined by the vector ⟨0, 𝑆 (the zero translation ltrnid 37275 and a nonzero member of the endomorphism ring). In particular, 𝑆 can be replaced with the ring unit ( I ↾ 𝑇). (Contributed by NM, 26-Aug-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
dihp.b 𝐵 = (Base‘𝐾)
dihp.h 𝐻 = (LHyp‘𝐾)
dihp.p 𝑃 = ((oc‘𝐾)‘𝑊)
dihp.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
dihp.e 𝐸 = ((TEndo‘𝐾)‘𝑊)
dihp.o 𝑂 = (𝑓𝑇 ↦ ( I ↾ 𝐵))
dihp.i 𝐼 = ((DIsoH‘𝐾)‘𝑊)
dihp.u 𝑈 = ((DVecH‘𝐾)‘𝑊)
dihp.n 𝑁 = (LSpan‘𝑈)
dihp.k (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))
dihp.s (𝜑 → (𝑆𝐸𝑆𝑂))
Assertion
Ref Expression
dihpN (𝜑 → (𝐼𝑃) = (𝑁‘{⟨( I ↾ 𝐵), 𝑆⟩}))
Distinct variable groups:   𝐵,𝑓   𝑓,𝐻   𝑓,𝐾   𝑃,𝑓   𝑇,𝑓   𝑓,𝑊
Allowed substitution hints:   𝜑(𝑓)   𝑆(𝑓)   𝑈(𝑓)   𝐸(𝑓)   𝐼(𝑓)   𝑁(𝑓)   𝑂(𝑓)

Proof of Theorem dihpN
Dummy variables 𝑔 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2824 . 2 (0g𝑈) = (0g𝑈)
2 dihp.n . 2 𝑁 = (LSpan‘𝑈)
3 eqid 2824 . 2 (LSAtoms‘𝑈) = (LSAtoms‘𝑈)
4 dihp.h . . 3 𝐻 = (LHyp‘𝐾)
5 dihp.u . . 3 𝑈 = ((DVecH‘𝐾)‘𝑊)
6 dihp.k . . 3 (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))
74, 5, 6dvhlvec 38249 . 2 (𝜑𝑈 ∈ LVec)
8 dihp.p . . 3 𝑃 = ((oc‘𝐾)‘𝑊)
9 dihp.i . . 3 𝐼 = ((DIsoH‘𝐾)‘𝑊)
104, 8, 9, 5, 3, 6dihat 38475 . 2 (𝜑 → (𝐼𝑃) ∈ (LSAtoms‘𝑈))
11 eqid 2824 . . . . . . . 8 (le‘𝐾) = (le‘𝐾)
12 eqid 2824 . . . . . . . 8 (Atoms‘𝐾) = (Atoms‘𝐾)
1311, 12, 4, 8lhpocnel2 37159 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (𝑃 ∈ (Atoms‘𝐾) ∧ ¬ 𝑃(le‘𝐾)𝑊))
14 dihp.b . . . . . . . 8 𝐵 = (Base‘𝐾)
15 dihp.t . . . . . . . 8 𝑇 = ((LTrn‘𝐾)‘𝑊)
16 eqid 2824 . . . . . . . 8 (𝑓𝑇 (𝑓𝑃) = 𝑃) = (𝑓𝑇 (𝑓𝑃) = 𝑃)
1714, 11, 12, 4, 15, 16ltrniotaidvalN 37723 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃 ∈ (Atoms‘𝐾) ∧ ¬ 𝑃(le‘𝐾)𝑊)) → (𝑓𝑇 (𝑓𝑃) = 𝑃) = ( I ↾ 𝐵))
186, 13, 17syl2anc2 587 . . . . . 6 (𝜑 → (𝑓𝑇 (𝑓𝑃) = 𝑃) = ( I ↾ 𝐵))
1918fveq2d 6677 . . . . 5 (𝜑 → (𝑆‘(𝑓𝑇 (𝑓𝑃) = 𝑃)) = (𝑆‘( I ↾ 𝐵)))
20 dihp.s . . . . . . 7 (𝜑 → (𝑆𝐸𝑆𝑂))
2120simpld 497 . . . . . 6 (𝜑𝑆𝐸)
22 dihp.e . . . . . . 7 𝐸 = ((TEndo‘𝐾)‘𝑊)
2314, 4, 22tendoid 37913 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸) → (𝑆‘( I ↾ 𝐵)) = ( I ↾ 𝐵))
246, 21, 23syl2anc 586 . . . . 5 (𝜑 → (𝑆‘( I ↾ 𝐵)) = ( I ↾ 𝐵))
2519, 24eqtr2d 2860 . . . 4 (𝜑 → ( I ↾ 𝐵) = (𝑆‘(𝑓𝑇 (𝑓𝑃) = 𝑃)))
2614fvexi 6687 . . . . . 6 𝐵 ∈ V
27 resiexg 7622 . . . . . 6 (𝐵 ∈ V → ( I ↾ 𝐵) ∈ V)
2826, 27mp1i 13 . . . . 5 (𝜑 → ( I ↾ 𝐵) ∈ V)
29 eqeq1 2828 . . . . . . 7 (𝑔 = ( I ↾ 𝐵) → (𝑔 = (𝑠‘(𝑓𝑇 (𝑓𝑃) = 𝑃)) ↔ ( I ↾ 𝐵) = (𝑠‘(𝑓𝑇 (𝑓𝑃) = 𝑃))))
3029anbi1d 631 . . . . . 6 (𝑔 = ( I ↾ 𝐵) → ((𝑔 = (𝑠‘(𝑓𝑇 (𝑓𝑃) = 𝑃)) ∧ 𝑠𝐸) ↔ (( I ↾ 𝐵) = (𝑠‘(𝑓𝑇 (𝑓𝑃) = 𝑃)) ∧ 𝑠𝐸)))
31 fveq1 6672 . . . . . . . 8 (𝑠 = 𝑆 → (𝑠‘(𝑓𝑇 (𝑓𝑃) = 𝑃)) = (𝑆‘(𝑓𝑇 (𝑓𝑃) = 𝑃)))
3231eqeq2d 2835 . . . . . . 7 (𝑠 = 𝑆 → (( I ↾ 𝐵) = (𝑠‘(𝑓𝑇 (𝑓𝑃) = 𝑃)) ↔ ( I ↾ 𝐵) = (𝑆‘(𝑓𝑇 (𝑓𝑃) = 𝑃))))
33 eleq1 2903 . . . . . . 7 (𝑠 = 𝑆 → (𝑠𝐸𝑆𝐸))
3432, 33anbi12d 632 . . . . . 6 (𝑠 = 𝑆 → ((( I ↾ 𝐵) = (𝑠‘(𝑓𝑇 (𝑓𝑃) = 𝑃)) ∧ 𝑠𝐸) ↔ (( I ↾ 𝐵) = (𝑆‘(𝑓𝑇 (𝑓𝑃) = 𝑃)) ∧ 𝑆𝐸)))
3530, 34opelopabg 5428 . . . . 5 ((( I ↾ 𝐵) ∈ V ∧ 𝑆𝐸) → (⟨( I ↾ 𝐵), 𝑆⟩ ∈ {⟨𝑔, 𝑠⟩ ∣ (𝑔 = (𝑠‘(𝑓𝑇 (𝑓𝑃) = 𝑃)) ∧ 𝑠𝐸)} ↔ (( I ↾ 𝐵) = (𝑆‘(𝑓𝑇 (𝑓𝑃) = 𝑃)) ∧ 𝑆𝐸)))
3628, 21, 35syl2anc 586 . . . 4 (𝜑 → (⟨( I ↾ 𝐵), 𝑆⟩ ∈ {⟨𝑔, 𝑠⟩ ∣ (𝑔 = (𝑠‘(𝑓𝑇 (𝑓𝑃) = 𝑃)) ∧ 𝑠𝐸)} ↔ (( I ↾ 𝐵) = (𝑆‘(𝑓𝑇 (𝑓𝑃) = 𝑃)) ∧ 𝑆𝐸)))
3725, 21, 36mpbir2and 711 . . 3 (𝜑 → ⟨( I ↾ 𝐵), 𝑆⟩ ∈ {⟨𝑔, 𝑠⟩ ∣ (𝑔 = (𝑠‘(𝑓𝑇 (𝑓𝑃) = 𝑃)) ∧ 𝑠𝐸)})
38 eqid 2824 . . . . . 6 ((DIsoC‘𝐾)‘𝑊) = ((DIsoC‘𝐾)‘𝑊)
3911, 12, 4, 38, 9dihvalcqat 38379 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃 ∈ (Atoms‘𝐾) ∧ ¬ 𝑃(le‘𝐾)𝑊)) → (𝐼𝑃) = (((DIsoC‘𝐾)‘𝑊)‘𝑃))
406, 13, 39syl2anc2 587 . . . 4 (𝜑 → (𝐼𝑃) = (((DIsoC‘𝐾)‘𝑊)‘𝑃))
4111, 12, 4, 8, 15, 22, 38dicval 38316 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃 ∈ (Atoms‘𝐾) ∧ ¬ 𝑃(le‘𝐾)𝑊)) → (((DIsoC‘𝐾)‘𝑊)‘𝑃) = {⟨𝑔, 𝑠⟩ ∣ (𝑔 = (𝑠‘(𝑓𝑇 (𝑓𝑃) = 𝑃)) ∧ 𝑠𝐸)})
426, 13, 41syl2anc2 587 . . . 4 (𝜑 → (((DIsoC‘𝐾)‘𝑊)‘𝑃) = {⟨𝑔, 𝑠⟩ ∣ (𝑔 = (𝑠‘(𝑓𝑇 (𝑓𝑃) = 𝑃)) ∧ 𝑠𝐸)})
4340, 42eqtr2d 2860 . . 3 (𝜑 → {⟨𝑔, 𝑠⟩ ∣ (𝑔 = (𝑠‘(𝑓𝑇 (𝑓𝑃) = 𝑃)) ∧ 𝑠𝐸)} = (𝐼𝑃))
4437, 43eleqtrd 2918 . 2 (𝜑 → ⟨( I ↾ 𝐵), 𝑆⟩ ∈ (𝐼𝑃))
4520simprd 498 . . 3 (𝜑𝑆𝑂)
46 dihp.o . . . . . . . 8 𝑂 = (𝑓𝑇 ↦ ( I ↾ 𝐵))
4714, 4, 15, 5, 1, 46dvh0g 38251 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (0g𝑈) = ⟨( I ↾ 𝐵), 𝑂⟩)
486, 47syl 17 . . . . . 6 (𝜑 → (0g𝑈) = ⟨( I ↾ 𝐵), 𝑂⟩)
4948eqeq2d 2835 . . . . 5 (𝜑 → (⟨( I ↾ 𝐵), 𝑆⟩ = (0g𝑈) ↔ ⟨( I ↾ 𝐵), 𝑆⟩ = ⟨( I ↾ 𝐵), 𝑂⟩))
5026, 27ax-mp 5 . . . . . . 7 ( I ↾ 𝐵) ∈ V
5115fvexi 6687 . . . . . . . . 9 𝑇 ∈ V
5251mptex 6989 . . . . . . . 8 (𝑓𝑇 ↦ ( I ↾ 𝐵)) ∈ V
5346, 52eqeltri 2912 . . . . . . 7 𝑂 ∈ V
5450, 53opth2 5375 . . . . . 6 (⟨( I ↾ 𝐵), 𝑆⟩ = ⟨( I ↾ 𝐵), 𝑂⟩ ↔ (( I ↾ 𝐵) = ( I ↾ 𝐵) ∧ 𝑆 = 𝑂))
5554simprbi 499 . . . . 5 (⟨( I ↾ 𝐵), 𝑆⟩ = ⟨( I ↾ 𝐵), 𝑂⟩ → 𝑆 = 𝑂)
5649, 55syl6bi 255 . . . 4 (𝜑 → (⟨( I ↾ 𝐵), 𝑆⟩ = (0g𝑈) → 𝑆 = 𝑂))
5756necon3d 3040 . . 3 (𝜑 → (𝑆𝑂 → ⟨( I ↾ 𝐵), 𝑆⟩ ≠ (0g𝑈)))
5845, 57mpd 15 . 2 (𝜑 → ⟨( I ↾ 𝐵), 𝑆⟩ ≠ (0g𝑈))
591, 2, 3, 7, 10, 44, 58lsatel 36145 1 (𝜑 → (𝐼𝑃) = (𝑁‘{⟨( I ↾ 𝐵), 𝑆⟩}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398   = wceq 1536  wcel 2113  wne 3019  Vcvv 3497  {csn 4570  cop 4576   class class class wbr 5069  {copab 5131  cmpt 5149   I cid 5462  cres 5560  cfv 6358  crio 7116  Basecbs 16486  lecple 16575  occoc 16576  0gc0g 16716  LSpanclspn 19746  LSAtomsclsa 36114  Atomscatm 36403  HLchlt 36490  LHypclh 37124  LTrncltrn 37241  TEndoctendo 37892  DVecHcdvh 38218  DIsoCcdic 38312  DIsoHcdih 38368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464  ax-cnex 10596  ax-resscn 10597  ax-1cn 10598  ax-icn 10599  ax-addcl 10600  ax-addrcl 10601  ax-mulcl 10602  ax-mulrcl 10603  ax-mulcom 10604  ax-addass 10605  ax-mulass 10606  ax-distr 10607  ax-i2m1 10608  ax-1ne0 10609  ax-1rid 10610  ax-rnegex 10611  ax-rrecex 10612  ax-cnre 10613  ax-pre-lttri 10614  ax-pre-lttrn 10615  ax-pre-ltadd 10616  ax-pre-mulgt0 10617  ax-riotaBAD 36093
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-fal 1549  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-nel 3127  df-ral 3146  df-rex 3147  df-reu 3148  df-rmo 3149  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-int 4880  df-iun 4924  df-iin 4925  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-pred 6151  df-ord 6197  df-on 6198  df-lim 6199  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-riota 7117  df-ov 7162  df-oprab 7163  df-mpo 7164  df-om 7584  df-1st 7692  df-2nd 7693  df-tpos 7895  df-undef 7942  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-1o 8105  df-oadd 8109  df-er 8292  df-map 8411  df-en 8513  df-dom 8514  df-sdom 8515  df-fin 8516  df-pnf 10680  df-mnf 10681  df-xr 10682  df-ltxr 10683  df-le 10684  df-sub 10875  df-neg 10876  df-nn 11642  df-2 11703  df-3 11704  df-4 11705  df-5 11706  df-6 11707  df-n0 11901  df-z 11985  df-uz 12247  df-fz 12896  df-struct 16488  df-ndx 16489  df-slot 16490  df-base 16492  df-sets 16493  df-ress 16494  df-plusg 16581  df-mulr 16582  df-sca 16584  df-vsca 16585  df-0g 16718  df-proset 17541  df-poset 17559  df-plt 17571  df-lub 17587  df-glb 17588  df-join 17589  df-meet 17590  df-p0 17652  df-p1 17653  df-lat 17659  df-clat 17721  df-mgm 17855  df-sgrp 17904  df-mnd 17915  df-submnd 17960  df-grp 18109  df-minusg 18110  df-sbg 18111  df-subg 18279  df-cntz 18450  df-lsm 18764  df-cmn 18911  df-abl 18912  df-mgp 19243  df-ur 19255  df-ring 19302  df-oppr 19376  df-dvdsr 19394  df-unit 19395  df-invr 19425  df-dvr 19436  df-drng 19507  df-lmod 19639  df-lss 19707  df-lsp 19747  df-lvec 19878  df-lsatoms 36116  df-oposet 36316  df-ol 36318  df-oml 36319  df-covers 36406  df-ats 36407  df-atl 36438  df-cvlat 36462  df-hlat 36491  df-llines 36638  df-lplanes 36639  df-lvols 36640  df-lines 36641  df-psubsp 36643  df-pmap 36644  df-padd 36936  df-lhyp 37128  df-laut 37129  df-ldil 37244  df-ltrn 37245  df-trl 37299  df-tendo 37895  df-edring 37897  df-disoa 38169  df-dvech 38219  df-dib 38279  df-dic 38313  df-dih 38369
This theorem is referenced by: (None)
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