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Theorem tendoex 38968
Description: Generalization of Lemma K of [Crawley] p. 118, cdlemk 38967. TODO: can this be used to shorten uses of cdlemk 38967? (Contributed by NM, 15-Oct-2013.)
Hypotheses
Ref Expression
tendoex.l = (le‘𝐾)
tendoex.h 𝐻 = (LHyp‘𝐾)
tendoex.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
tendoex.r 𝑅 = ((trL‘𝐾)‘𝑊)
tendoex.e 𝐸 = ((TEndo‘𝐾)‘𝑊)
Assertion
Ref Expression
tendoex (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) → ∃𝑢𝐸 (𝑢𝐹) = 𝑁)
Distinct variable groups:   𝑢,𝐸   𝑢,𝐹   𝑢,𝐾   𝑢,𝑁   𝑢,𝑅   𝑢,𝑇   𝑢,𝑊
Allowed substitution hints:   𝐻(𝑢)   (𝑢)

Proof of Theorem tendoex
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 simpl1l 1222 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) ∧ (𝑅𝐹) ∈ (Atoms‘𝐾)) → 𝐾 ∈ HL)
2 hlop 37355 . . . . . . 7 (𝐾 ∈ HL → 𝐾 ∈ OP)
31, 2syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) ∧ (𝑅𝐹) ∈ (Atoms‘𝐾)) → 𝐾 ∈ OP)
4 simpl1 1189 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) ∧ (𝑅𝐹) ∈ (Atoms‘𝐾)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
5 simpl2r 1225 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) ∧ (𝑅𝐹) ∈ (Atoms‘𝐾)) → 𝑁𝑇)
6 eqid 2739 . . . . . . . 8 (Base‘𝐾) = (Base‘𝐾)
7 tendoex.h . . . . . . . 8 𝐻 = (LHyp‘𝐾)
8 tendoex.t . . . . . . . 8 𝑇 = ((LTrn‘𝐾)‘𝑊)
9 tendoex.r . . . . . . . 8 𝑅 = ((trL‘𝐾)‘𝑊)
106, 7, 8, 9trlcl 38157 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑁𝑇) → (𝑅𝑁) ∈ (Base‘𝐾))
114, 5, 10syl2anc 583 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) ∧ (𝑅𝐹) ∈ (Atoms‘𝐾)) → (𝑅𝑁) ∈ (Base‘𝐾))
12 simpr 484 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) ∧ (𝑅𝐹) ∈ (Atoms‘𝐾)) → (𝑅𝐹) ∈ (Atoms‘𝐾))
13 simpl3 1191 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) ∧ (𝑅𝐹) ∈ (Atoms‘𝐾)) → (𝑅𝑁) (𝑅𝐹))
14 tendoex.l . . . . . . 7 = (le‘𝐾)
15 eqid 2739 . . . . . . 7 (0.‘𝐾) = (0.‘𝐾)
16 eqid 2739 . . . . . . 7 (Atoms‘𝐾) = (Atoms‘𝐾)
176, 14, 15, 16leat 37286 . . . . . 6 (((𝐾 ∈ OP ∧ (𝑅𝑁) ∈ (Base‘𝐾) ∧ (𝑅𝐹) ∈ (Atoms‘𝐾)) ∧ (𝑅𝑁) (𝑅𝐹)) → ((𝑅𝑁) = (𝑅𝐹) ∨ (𝑅𝑁) = (0.‘𝐾)))
183, 11, 12, 13, 17syl31anc 1371 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) ∧ (𝑅𝐹) ∈ (Atoms‘𝐾)) → ((𝑅𝑁) = (𝑅𝐹) ∨ (𝑅𝑁) = (0.‘𝐾)))
19 simp3 1136 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) → (𝑅𝑁) (𝑅𝐹))
20 breq2 5082 . . . . . . . . 9 ((𝑅𝐹) = (0.‘𝐾) → ((𝑅𝑁) (𝑅𝐹) ↔ (𝑅𝑁) (0.‘𝐾)))
2119, 20syl5ibcom 244 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) → ((𝑅𝐹) = (0.‘𝐾) → (𝑅𝑁) (0.‘𝐾)))
2221imp 406 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) ∧ (𝑅𝐹) = (0.‘𝐾)) → (𝑅𝑁) (0.‘𝐾))
23 simpl1l 1222 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) ∧ (𝑅𝐹) = (0.‘𝐾)) → 𝐾 ∈ HL)
2423, 2syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) ∧ (𝑅𝐹) = (0.‘𝐾)) → 𝐾 ∈ OP)
25 simpl1 1189 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) ∧ (𝑅𝐹) = (0.‘𝐾)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
26 simpl2r 1225 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) ∧ (𝑅𝐹) = (0.‘𝐾)) → 𝑁𝑇)
2725, 26, 10syl2anc 583 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) ∧ (𝑅𝐹) = (0.‘𝐾)) → (𝑅𝑁) ∈ (Base‘𝐾))
286, 14, 15ople0 37180 . . . . . . . 8 ((𝐾 ∈ OP ∧ (𝑅𝑁) ∈ (Base‘𝐾)) → ((𝑅𝑁) (0.‘𝐾) ↔ (𝑅𝑁) = (0.‘𝐾)))
2924, 27, 28syl2anc 583 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) ∧ (𝑅𝐹) = (0.‘𝐾)) → ((𝑅𝑁) (0.‘𝐾) ↔ (𝑅𝑁) = (0.‘𝐾)))
3022, 29mpbid 231 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) ∧ (𝑅𝐹) = (0.‘𝐾)) → (𝑅𝑁) = (0.‘𝐾))
3130olcd 870 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) ∧ (𝑅𝐹) = (0.‘𝐾)) → ((𝑅𝑁) = (𝑅𝐹) ∨ (𝑅𝑁) = (0.‘𝐾)))
32 simp1 1134 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
33 simp2l 1197 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) → 𝐹𝑇)
3415, 16, 7, 8, 9trlator0 38164 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → ((𝑅𝐹) ∈ (Atoms‘𝐾) ∨ (𝑅𝐹) = (0.‘𝐾)))
3532, 33, 34syl2anc 583 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) → ((𝑅𝐹) ∈ (Atoms‘𝐾) ∨ (𝑅𝐹) = (0.‘𝐾)))
3618, 31, 35mpjaodan 955 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) → ((𝑅𝑁) = (𝑅𝐹) ∨ (𝑅𝑁) = (0.‘𝐾)))
37363expa 1116 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇)) ∧ (𝑅𝑁) (𝑅𝐹)) → ((𝑅𝑁) = (𝑅𝐹) ∨ (𝑅𝑁) = (0.‘𝐾)))
38 eqcom 2746 . . . . 5 ((𝑅𝑁) = (𝑅𝐹) ↔ (𝑅𝐹) = (𝑅𝑁))
39 tendoex.e . . . . . . 7 𝐸 = ((TEndo‘𝐾)‘𝑊)
407, 8, 9, 39cdlemk 38967 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝐹) = (𝑅𝑁)) → ∃𝑢𝐸 (𝑢𝐹) = 𝑁)
41403expa 1116 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇)) ∧ (𝑅𝐹) = (𝑅𝑁)) → ∃𝑢𝐸 (𝑢𝐹) = 𝑁)
4238, 41sylan2b 593 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇)) ∧ (𝑅𝑁) = (𝑅𝐹)) → ∃𝑢𝐸 (𝑢𝐹) = 𝑁)
43 eqid 2739 . . . . . . 7 (𝑇 ↦ ( I ↾ (Base‘𝐾))) = (𝑇 ↦ ( I ↾ (Base‘𝐾)))
446, 7, 8, 39, 43tendo0cl 38783 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (𝑇 ↦ ( I ↾ (Base‘𝐾))) ∈ 𝐸)
4544ad2antrr 722 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇)) ∧ (𝑅𝑁) = (0.‘𝐾)) → (𝑇 ↦ ( I ↾ (Base‘𝐾))) ∈ 𝐸)
46 simplrl 773 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇)) ∧ (𝑅𝑁) = (0.‘𝐾)) → 𝐹𝑇)
4743, 6tendo02 38780 . . . . . . 7 (𝐹𝑇 → ((𝑇 ↦ ( I ↾ (Base‘𝐾)))‘𝐹) = ( I ↾ (Base‘𝐾)))
4846, 47syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇)) ∧ (𝑅𝑁) = (0.‘𝐾)) → ((𝑇 ↦ ( I ↾ (Base‘𝐾)))‘𝐹) = ( I ↾ (Base‘𝐾)))
496, 15, 7, 8, 9trlid0b 38171 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑁𝑇) → (𝑁 = ( I ↾ (Base‘𝐾)) ↔ (𝑅𝑁) = (0.‘𝐾)))
5049adantrl 712 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇)) → (𝑁 = ( I ↾ (Base‘𝐾)) ↔ (𝑅𝑁) = (0.‘𝐾)))
5150biimpar 477 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇)) ∧ (𝑅𝑁) = (0.‘𝐾)) → 𝑁 = ( I ↾ (Base‘𝐾)))
5248, 51eqtr4d 2782 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇)) ∧ (𝑅𝑁) = (0.‘𝐾)) → ((𝑇 ↦ ( I ↾ (Base‘𝐾)))‘𝐹) = 𝑁)
53 fveq1 6767 . . . . . . 7 (𝑢 = (𝑇 ↦ ( I ↾ (Base‘𝐾))) → (𝑢𝐹) = ((𝑇 ↦ ( I ↾ (Base‘𝐾)))‘𝐹))
5453eqeq1d 2741 . . . . . 6 (𝑢 = (𝑇 ↦ ( I ↾ (Base‘𝐾))) → ((𝑢𝐹) = 𝑁 ↔ ((𝑇 ↦ ( I ↾ (Base‘𝐾)))‘𝐹) = 𝑁))
5554rspcev 3560 . . . . 5 (((𝑇 ↦ ( I ↾ (Base‘𝐾))) ∈ 𝐸 ∧ ((𝑇 ↦ ( I ↾ (Base‘𝐾)))‘𝐹) = 𝑁) → ∃𝑢𝐸 (𝑢𝐹) = 𝑁)
5645, 52, 55syl2anc 583 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇)) ∧ (𝑅𝑁) = (0.‘𝐾)) → ∃𝑢𝐸 (𝑢𝐹) = 𝑁)
5742, 56jaodan 954 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇)) ∧ ((𝑅𝑁) = (𝑅𝐹) ∨ (𝑅𝑁) = (0.‘𝐾))) → ∃𝑢𝐸 (𝑢𝐹) = 𝑁)
5837, 57syldan 590 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇)) ∧ (𝑅𝑁) (𝑅𝐹)) → ∃𝑢𝐸 (𝑢𝐹) = 𝑁)
59583impa 1108 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) → ∃𝑢𝐸 (𝑢𝐹) = 𝑁)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wo 843  w3a 1085   = wceq 1541  wcel 2109  wrex 3066   class class class wbr 5078  cmpt 5161   I cid 5487  cres 5590  cfv 6430  Basecbs 16893  lecple 16950  0.cp0 18122  OPcops 37165  Atomscatm 37256  HLchlt 37343  LHypclh 37977  LTrncltrn 38094  trLctrl 38151  TEndoctendo 38745
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-rep 5213  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579  ax-riotaBAD 36946
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-reu 3072  df-rmo 3073  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-iun 4931  df-iin 4932  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-riota 7225  df-ov 7271  df-oprab 7272  df-mpo 7273  df-1st 7817  df-2nd 7818  df-undef 8073  df-map 8591  df-proset 17994  df-poset 18012  df-plt 18029  df-lub 18045  df-glb 18046  df-join 18047  df-meet 18048  df-p0 18124  df-p1 18125  df-lat 18131  df-clat 18198  df-oposet 37169  df-ol 37171  df-oml 37172  df-covers 37259  df-ats 37260  df-atl 37291  df-cvlat 37315  df-hlat 37344  df-llines 37491  df-lplanes 37492  df-lvols 37493  df-lines 37494  df-psubsp 37496  df-pmap 37497  df-padd 37789  df-lhyp 37981  df-laut 37982  df-ldil 38097  df-ltrn 38098  df-trl 38152  df-tendo 38748
This theorem is referenced by:  dva1dim  38978  dihjatcclem4  39414
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