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Theorem tendoex 37991
Description: Generalization of Lemma K of [Crawley] p. 118, cdlemk 37990. TODO: can this be used to shorten uses of cdlemk 37990? (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 1216 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) ∧ (𝑅𝐹) ∈ (Atoms‘𝐾)) → 𝐾 ∈ HL)
2 hlop 36378 . . . . . . 7 (𝐾 ∈ HL → 𝐾 ∈ OP)
31, 2syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) ∧ (𝑅𝐹) ∈ (Atoms‘𝐾)) → 𝐾 ∈ OP)
4 simpl1 1183 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) ∧ (𝑅𝐹) ∈ (Atoms‘𝐾)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
5 simpl2r 1219 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) ∧ (𝑅𝐹) ∈ (Atoms‘𝐾)) → 𝑁𝑇)
6 eqid 2818 . . . . . . . 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 37180 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑁𝑇) → (𝑅𝑁) ∈ (Base‘𝐾))
114, 5, 10syl2anc 584 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) ∧ (𝑅𝐹) ∈ (Atoms‘𝐾)) → (𝑅𝑁) ∈ (Base‘𝐾))
12 simpr 485 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) ∧ (𝑅𝐹) ∈ (Atoms‘𝐾)) → (𝑅𝐹) ∈ (Atoms‘𝐾))
13 simpl3 1185 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) ∧ (𝑅𝐹) ∈ (Atoms‘𝐾)) → (𝑅𝑁) (𝑅𝐹))
14 tendoex.l . . . . . . 7 = (le‘𝐾)
15 eqid 2818 . . . . . . 7 (0.‘𝐾) = (0.‘𝐾)
16 eqid 2818 . . . . . . 7 (Atoms‘𝐾) = (Atoms‘𝐾)
176, 14, 15, 16leat 36309 . . . . . 6 (((𝐾 ∈ OP ∧ (𝑅𝑁) ∈ (Base‘𝐾) ∧ (𝑅𝐹) ∈ (Atoms‘𝐾)) ∧ (𝑅𝑁) (𝑅𝐹)) → ((𝑅𝑁) = (𝑅𝐹) ∨ (𝑅𝑁) = (0.‘𝐾)))
183, 11, 12, 13, 17syl31anc 1365 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) ∧ (𝑅𝐹) ∈ (Atoms‘𝐾)) → ((𝑅𝑁) = (𝑅𝐹) ∨ (𝑅𝑁) = (0.‘𝐾)))
19 simp3 1130 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) → (𝑅𝑁) (𝑅𝐹))
20 breq2 5061 . . . . . . . . 9 ((𝑅𝐹) = (0.‘𝐾) → ((𝑅𝑁) (𝑅𝐹) ↔ (𝑅𝑁) (0.‘𝐾)))
2119, 20syl5ibcom 246 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) → ((𝑅𝐹) = (0.‘𝐾) → (𝑅𝑁) (0.‘𝐾)))
2221imp 407 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) ∧ (𝑅𝐹) = (0.‘𝐾)) → (𝑅𝑁) (0.‘𝐾))
23 simpl1l 1216 . . . . . . . . 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 1183 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) ∧ (𝑅𝐹) = (0.‘𝐾)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
26 simpl2r 1219 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) ∧ (𝑅𝐹) = (0.‘𝐾)) → 𝑁𝑇)
2725, 26, 10syl2anc 584 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) ∧ (𝑅𝐹) = (0.‘𝐾)) → (𝑅𝑁) ∈ (Base‘𝐾))
286, 14, 15ople0 36203 . . . . . . . 8 ((𝐾 ∈ OP ∧ (𝑅𝑁) ∈ (Base‘𝐾)) → ((𝑅𝑁) (0.‘𝐾) ↔ (𝑅𝑁) = (0.‘𝐾)))
2924, 27, 28syl2anc 584 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) ∧ (𝑅𝐹) = (0.‘𝐾)) → ((𝑅𝑁) (0.‘𝐾) ↔ (𝑅𝑁) = (0.‘𝐾)))
3022, 29mpbid 233 . . . . . 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 1128 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
33 simp2l 1191 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) → 𝐹𝑇)
3415, 16, 7, 8, 9trlator0 37187 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → ((𝑅𝐹) ∈ (Atoms‘𝐾) ∨ (𝑅𝐹) = (0.‘𝐾)))
3532, 33, 34syl2anc 584 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) → ((𝑅𝐹) ∈ (Atoms‘𝐾) ∨ (𝑅𝐹) = (0.‘𝐾)))
3618, 31, 35mpjaodan 952 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) → ((𝑅𝑁) = (𝑅𝐹) ∨ (𝑅𝑁) = (0.‘𝐾)))
37363expa 1110 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇)) ∧ (𝑅𝑁) (𝑅𝐹)) → ((𝑅𝑁) = (𝑅𝐹) ∨ (𝑅𝑁) = (0.‘𝐾)))
38 eqcom 2825 . . . . 5 ((𝑅𝑁) = (𝑅𝐹) ↔ (𝑅𝐹) = (𝑅𝑁))
39 tendoex.e . . . . . . 7 𝐸 = ((TEndo‘𝐾)‘𝑊)
407, 8, 9, 39cdlemk 37990 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝐹) = (𝑅𝑁)) → ∃𝑢𝐸 (𝑢𝐹) = 𝑁)
41403expa 1110 . . . . 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 2818 . . . . . . 7 (𝑇 ↦ ( I ↾ (Base‘𝐾))) = (𝑇 ↦ ( I ↾ (Base‘𝐾)))
446, 7, 8, 39, 43tendo0cl 37806 . . . . . 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 37803 . . . . . . 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 37194 . . . . . . . 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 478 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇)) ∧ (𝑅𝑁) = (0.‘𝐾)) → 𝑁 = ( I ↾ (Base‘𝐾)))
5248, 51eqtr4d 2856 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇)) ∧ (𝑅𝑁) = (0.‘𝐾)) → ((𝑇 ↦ ( I ↾ (Base‘𝐾)))‘𝐹) = 𝑁)
53 fveq1 6662 . . . . . . 7 (𝑢 = (𝑇 ↦ ( I ↾ (Base‘𝐾))) → (𝑢𝐹) = ((𝑇 ↦ ( I ↾ (Base‘𝐾)))‘𝐹))
5453eqeq1d 2820 . . . . . 6 (𝑢 = (𝑇 ↦ ( I ↾ (Base‘𝐾))) → ((𝑢𝐹) = 𝑁 ↔ ((𝑇 ↦ ( I ↾ (Base‘𝐾)))‘𝐹) = 𝑁))
5554rspcev 3620 . . . . 5 (((𝑇 ↦ ( I ↾ (Base‘𝐾))) ∈ 𝐸 ∧ ((𝑇 ↦ ( I ↾ (Base‘𝐾)))‘𝐹) = 𝑁) → ∃𝑢𝐸 (𝑢𝐹) = 𝑁)
5645, 52, 55syl2anc 584 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇)) ∧ (𝑅𝑁) = (0.‘𝐾)) → ∃𝑢𝐸 (𝑢𝐹) = 𝑁)
5742, 56jaodan 951 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇)) ∧ ((𝑅𝑁) = (𝑅𝐹) ∨ (𝑅𝑁) = (0.‘𝐾))) → ∃𝑢𝐸 (𝑢𝐹) = 𝑁)
5837, 57syldan 591 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇)) ∧ (𝑅𝑁) (𝑅𝐹)) → ∃𝑢𝐸 (𝑢𝐹) = 𝑁)
59583impa 1102 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) → ∃𝑢𝐸 (𝑢𝐹) = 𝑁)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wo 841  w3a 1079   = wceq 1528  wcel 2105  wrex 3136   class class class wbr 5057  cmpt 5137   I cid 5452  cres 5550  cfv 6348  Basecbs 16471  lecple 16560  0.cp0 17635  OPcops 36188  Atomscatm 36279  HLchlt 36366  LHypclh 37000  LTrncltrn 37117  trLctrl 37174  TEndoctendo 37768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-riotaBAD 35969
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-undef 7928  df-map 8397  df-proset 17526  df-poset 17544  df-plt 17556  df-lub 17572  df-glb 17573  df-join 17574  df-meet 17575  df-p0 17637  df-p1 17638  df-lat 17644  df-clat 17706  df-oposet 36192  df-ol 36194  df-oml 36195  df-covers 36282  df-ats 36283  df-atl 36314  df-cvlat 36338  df-hlat 36367  df-llines 36514  df-lplanes 36515  df-lvols 36516  df-lines 36517  df-psubsp 36519  df-pmap 36520  df-padd 36812  df-lhyp 37004  df-laut 37005  df-ldil 37120  df-ltrn 37121  df-trl 37175  df-tendo 37771
This theorem is referenced by:  dva1dim  38001  dihjatcclem4  38437
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