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Theorem dihpN 36451
Description: The value of isomorphism H at the fiducial atom 𝑃 is determined by the vector ⟨0, 𝑆 (the zero translation ltrnid 35247 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 2621 . 2 (0g𝑈) = (0g𝑈)
2 dihp.n . 2 𝑁 = (LSpan‘𝑈)
3 eqid 2621 . 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 36224 . 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 36450 . 2 (𝜑 → (𝐼𝑃) ∈ (LSAtoms‘𝑈))
11 eqid 2621 . . . . . . . . 9 (le‘𝐾) = (le‘𝐾)
12 eqid 2621 . . . . . . . . 9 (Atoms‘𝐾) = (Atoms‘𝐾)
1311, 12, 4, 8lhpocnel2 35131 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (𝑃 ∈ (Atoms‘𝐾) ∧ ¬ 𝑃(le‘𝐾)𝑊))
146, 13syl 17 . . . . . . 7 (𝜑 → (𝑃 ∈ (Atoms‘𝐾) ∧ ¬ 𝑃(le‘𝐾)𝑊))
15 dihp.b . . . . . . . 8 𝐵 = (Base‘𝐾)
16 dihp.t . . . . . . . 8 𝑇 = ((LTrn‘𝐾)‘𝑊)
17 eqid 2621 . . . . . . . 8 (𝑓𝑇 (𝑓𝑃) = 𝑃) = (𝑓𝑇 (𝑓𝑃) = 𝑃)
1815, 11, 12, 4, 16, 17ltrniotaidvalN 35697 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃 ∈ (Atoms‘𝐾) ∧ ¬ 𝑃(le‘𝐾)𝑊)) → (𝑓𝑇 (𝑓𝑃) = 𝑃) = ( I ↾ 𝐵))
196, 14, 18syl2anc 693 . . . . . 6 (𝜑 → (𝑓𝑇 (𝑓𝑃) = 𝑃) = ( I ↾ 𝐵))
2019fveq2d 6193 . . . . 5 (𝜑 → (𝑆‘(𝑓𝑇 (𝑓𝑃) = 𝑃)) = (𝑆‘( I ↾ 𝐵)))
21 dihp.s . . . . . . 7 (𝜑 → (𝑆𝐸𝑆𝑂))
2221simpld 475 . . . . . 6 (𝜑𝑆𝐸)
23 dihp.e . . . . . . 7 𝐸 = ((TEndo‘𝐾)‘𝑊)
2415, 4, 23tendoid 35887 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸) → (𝑆‘( I ↾ 𝐵)) = ( I ↾ 𝐵))
256, 22, 24syl2anc 693 . . . . 5 (𝜑 → (𝑆‘( I ↾ 𝐵)) = ( I ↾ 𝐵))
2620, 25eqtr2d 2656 . . . 4 (𝜑 → ( I ↾ 𝐵) = (𝑆‘(𝑓𝑇 (𝑓𝑃) = 𝑃)))
27 fvex 6199 . . . . . . 7 (Base‘𝐾) ∈ V
2815, 27eqeltri 2696 . . . . . 6 𝐵 ∈ V
29 resiexg 7099 . . . . . 6 (𝐵 ∈ V → ( I ↾ 𝐵) ∈ V)
3028, 29mp1i 13 . . . . 5 (𝜑 → ( I ↾ 𝐵) ∈ V)
31 eqeq1 2625 . . . . . . 7 (𝑔 = ( I ↾ 𝐵) → (𝑔 = (𝑠‘(𝑓𝑇 (𝑓𝑃) = 𝑃)) ↔ ( I ↾ 𝐵) = (𝑠‘(𝑓𝑇 (𝑓𝑃) = 𝑃))))
3231anbi1d 741 . . . . . 6 (𝑔 = ( I ↾ 𝐵) → ((𝑔 = (𝑠‘(𝑓𝑇 (𝑓𝑃) = 𝑃)) ∧ 𝑠𝐸) ↔ (( I ↾ 𝐵) = (𝑠‘(𝑓𝑇 (𝑓𝑃) = 𝑃)) ∧ 𝑠𝐸)))
33 fveq1 6188 . . . . . . . 8 (𝑠 = 𝑆 → (𝑠‘(𝑓𝑇 (𝑓𝑃) = 𝑃)) = (𝑆‘(𝑓𝑇 (𝑓𝑃) = 𝑃)))
3433eqeq2d 2631 . . . . . . 7 (𝑠 = 𝑆 → (( I ↾ 𝐵) = (𝑠‘(𝑓𝑇 (𝑓𝑃) = 𝑃)) ↔ ( I ↾ 𝐵) = (𝑆‘(𝑓𝑇 (𝑓𝑃) = 𝑃))))
35 eleq1 2688 . . . . . . 7 (𝑠 = 𝑆 → (𝑠𝐸𝑆𝐸))
3634, 35anbi12d 747 . . . . . 6 (𝑠 = 𝑆 → ((( I ↾ 𝐵) = (𝑠‘(𝑓𝑇 (𝑓𝑃) = 𝑃)) ∧ 𝑠𝐸) ↔ (( I ↾ 𝐵) = (𝑆‘(𝑓𝑇 (𝑓𝑃) = 𝑃)) ∧ 𝑆𝐸)))
3732, 36opelopabg 4991 . . . . 5 ((( I ↾ 𝐵) ∈ V ∧ 𝑆𝐸) → (⟨( I ↾ 𝐵), 𝑆⟩ ∈ {⟨𝑔, 𝑠⟩ ∣ (𝑔 = (𝑠‘(𝑓𝑇 (𝑓𝑃) = 𝑃)) ∧ 𝑠𝐸)} ↔ (( I ↾ 𝐵) = (𝑆‘(𝑓𝑇 (𝑓𝑃) = 𝑃)) ∧ 𝑆𝐸)))
3830, 22, 37syl2anc 693 . . . 4 (𝜑 → (⟨( I ↾ 𝐵), 𝑆⟩ ∈ {⟨𝑔, 𝑠⟩ ∣ (𝑔 = (𝑠‘(𝑓𝑇 (𝑓𝑃) = 𝑃)) ∧ 𝑠𝐸)} ↔ (( I ↾ 𝐵) = (𝑆‘(𝑓𝑇 (𝑓𝑃) = 𝑃)) ∧ 𝑆𝐸)))
3926, 22, 38mpbir2and 957 . . 3 (𝜑 → ⟨( I ↾ 𝐵), 𝑆⟩ ∈ {⟨𝑔, 𝑠⟩ ∣ (𝑔 = (𝑠‘(𝑓𝑇 (𝑓𝑃) = 𝑃)) ∧ 𝑠𝐸)})
40 eqid 2621 . . . . . 6 ((DIsoC‘𝐾)‘𝑊) = ((DIsoC‘𝐾)‘𝑊)
4111, 12, 4, 40, 9dihvalcqat 36354 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃 ∈ (Atoms‘𝐾) ∧ ¬ 𝑃(le‘𝐾)𝑊)) → (𝐼𝑃) = (((DIsoC‘𝐾)‘𝑊)‘𝑃))
426, 14, 41syl2anc 693 . . . 4 (𝜑 → (𝐼𝑃) = (((DIsoC‘𝐾)‘𝑊)‘𝑃))
4311, 12, 4, 8, 16, 23, 40dicval 36291 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃 ∈ (Atoms‘𝐾) ∧ ¬ 𝑃(le‘𝐾)𝑊)) → (((DIsoC‘𝐾)‘𝑊)‘𝑃) = {⟨𝑔, 𝑠⟩ ∣ (𝑔 = (𝑠‘(𝑓𝑇 (𝑓𝑃) = 𝑃)) ∧ 𝑠𝐸)})
446, 14, 43syl2anc 693 . . . 4 (𝜑 → (((DIsoC‘𝐾)‘𝑊)‘𝑃) = {⟨𝑔, 𝑠⟩ ∣ (𝑔 = (𝑠‘(𝑓𝑇 (𝑓𝑃) = 𝑃)) ∧ 𝑠𝐸)})
4542, 44eqtr2d 2656 . . 3 (𝜑 → {⟨𝑔, 𝑠⟩ ∣ (𝑔 = (𝑠‘(𝑓𝑇 (𝑓𝑃) = 𝑃)) ∧ 𝑠𝐸)} = (𝐼𝑃))
4639, 45eleqtrd 2702 . 2 (𝜑 → ⟨( I ↾ 𝐵), 𝑆⟩ ∈ (𝐼𝑃))
4721simprd 479 . . 3 (𝜑𝑆𝑂)
48 dihp.o . . . . . . . 8 𝑂 = (𝑓𝑇 ↦ ( I ↾ 𝐵))
4915, 4, 16, 5, 1, 48dvh0g 36226 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (0g𝑈) = ⟨( I ↾ 𝐵), 𝑂⟩)
506, 49syl 17 . . . . . 6 (𝜑 → (0g𝑈) = ⟨( I ↾ 𝐵), 𝑂⟩)
5150eqeq2d 2631 . . . . 5 (𝜑 → (⟨( I ↾ 𝐵), 𝑆⟩ = (0g𝑈) ↔ ⟨( I ↾ 𝐵), 𝑆⟩ = ⟨( I ↾ 𝐵), 𝑂⟩))
5228, 29ax-mp 5 . . . . . . 7 ( I ↾ 𝐵) ∈ V
53 fvex 6199 . . . . . . . . . 10 ((LTrn‘𝐾)‘𝑊) ∈ V
5416, 53eqeltri 2696 . . . . . . . . 9 𝑇 ∈ V
5554mptex 6483 . . . . . . . 8 (𝑓𝑇 ↦ ( I ↾ 𝐵)) ∈ V
5648, 55eqeltri 2696 . . . . . . 7 𝑂 ∈ V
5752, 56opth2 4947 . . . . . 6 (⟨( I ↾ 𝐵), 𝑆⟩ = ⟨( I ↾ 𝐵), 𝑂⟩ ↔ (( I ↾ 𝐵) = ( I ↾ 𝐵) ∧ 𝑆 = 𝑂))
5857simprbi 480 . . . . 5 (⟨( I ↾ 𝐵), 𝑆⟩ = ⟨( I ↾ 𝐵), 𝑂⟩ → 𝑆 = 𝑂)
5951, 58syl6bi 243 . . . 4 (𝜑 → (⟨( I ↾ 𝐵), 𝑆⟩ = (0g𝑈) → 𝑆 = 𝑂))
6059necon3d 2814 . . 3 (𝜑 → (𝑆𝑂 → ⟨( I ↾ 𝐵), 𝑆⟩ ≠ (0g𝑈)))
6147, 60mpd 15 . 2 (𝜑 → ⟨( I ↾ 𝐵), 𝑆⟩ ≠ (0g𝑈))
621, 2, 3, 7, 10, 46, 61lsatel 34118 1 (𝜑 → (𝐼𝑃) = (𝑁‘{⟨( I ↾ 𝐵), 𝑆⟩}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1482  wcel 1989  wne 2793  Vcvv 3198  {csn 4175  cop 4181   class class class wbr 4651  {copab 4710  cmpt 4727   I cid 5021  cres 5114  cfv 5886  crio 6607  Basecbs 15851  lecple 15942  occoc 15943  0gc0g 16094  LSpanclspn 18965  LSAtomsclsa 34087  Atomscatm 34376  HLchlt 34463  LHypclh 35096  LTrncltrn 35213  TEndoctendo 35866  DVecHcdvh 36193  DIsoCcdic 36287  DIsoHcdih 36343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946  ax-cnex 9989  ax-resscn 9990  ax-1cn 9991  ax-icn 9992  ax-addcl 9993  ax-addrcl 9994  ax-mulcl 9995  ax-mulrcl 9996  ax-mulcom 9997  ax-addass 9998  ax-mulass 9999  ax-distr 10000  ax-i2m1 10001  ax-1ne0 10002  ax-1rid 10003  ax-rnegex 10004  ax-rrecex 10005  ax-cnre 10006  ax-pre-lttri 10007  ax-pre-lttrn 10008  ax-pre-ltadd 10009  ax-pre-mulgt0 10010  ax-riotaBAD 34065
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-fal 1488  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-nel 2897  df-ral 2916  df-rex 2917  df-reu 2918  df-rmo 2919  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-int 4474  df-iun 4520  df-iin 4521  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-riota 6608  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-om 7063  df-1st 7165  df-2nd 7166  df-tpos 7349  df-undef 7396  df-wrecs 7404  df-recs 7465  df-rdg 7503  df-1o 7557  df-oadd 7561  df-er 7739  df-map 7856  df-en 7953  df-dom 7954  df-sdom 7955  df-fin 7956  df-pnf 10073  df-mnf 10074  df-xr 10075  df-ltxr 10076  df-le 10077  df-sub 10265  df-neg 10266  df-nn 11018  df-2 11076  df-3 11077  df-4 11078  df-5 11079  df-6 11080  df-n0 11290  df-z 11375  df-uz 11685  df-fz 12324  df-struct 15853  df-ndx 15854  df-slot 15855  df-base 15857  df-sets 15858  df-ress 15859  df-plusg 15948  df-mulr 15949  df-sca 15951  df-vsca 15952  df-0g 16096  df-preset 16922  df-poset 16940  df-plt 16952  df-lub 16968  df-glb 16969  df-join 16970  df-meet 16971  df-p0 17033  df-p1 17034  df-lat 17040  df-clat 17102  df-mgm 17236  df-sgrp 17278  df-mnd 17289  df-submnd 17330  df-grp 17419  df-minusg 17420  df-sbg 17421  df-subg 17585  df-cntz 17744  df-lsm 18045  df-cmn 18189  df-abl 18190  df-mgp 18484  df-ur 18496  df-ring 18543  df-oppr 18617  df-dvdsr 18635  df-unit 18636  df-invr 18666  df-dvr 18677  df-drng 18743  df-lmod 18859  df-lss 18927  df-lsp 18966  df-lvec 19097  df-lsatoms 34089  df-oposet 34289  df-ol 34291  df-oml 34292  df-covers 34379  df-ats 34380  df-atl 34411  df-cvlat 34435  df-hlat 34464  df-llines 34610  df-lplanes 34611  df-lvols 34612  df-lines 34613  df-psubsp 34615  df-pmap 34616  df-padd 34908  df-lhyp 35100  df-laut 35101  df-ldil 35216  df-ltrn 35217  df-trl 35272  df-tendo 35869  df-edring 35871  df-disoa 36144  df-dvech 36194  df-dib 36254  df-dic 36288  df-dih 36344
This theorem is referenced by: (None)
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