Mathbox for Norm Megill < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  tendoex Structured version   Visualization version   GIF version

Theorem tendoex 36783
 Description: Generalization of Lemma K of [Crawley] p. 118, cdlemk 36782. TODO: can this be used to shorten uses of cdlemk 36782? (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 1279 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) ∧ (𝑅𝐹) ∈ (Atoms‘𝐾)) → 𝐾 ∈ HL)
2 hlop 35170 . . . . . . 7 (𝐾 ∈ HL → 𝐾 ∈ OP)
31, 2syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) ∧ (𝑅𝐹) ∈ (Atoms‘𝐾)) → 𝐾 ∈ OP)
4 simpl1 1228 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) ∧ (𝑅𝐹) ∈ (Atoms‘𝐾)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
5 simpl2r 1285 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) ∧ (𝑅𝐹) ∈ (Atoms‘𝐾)) → 𝑁𝑇)
6 eqid 2760 . . . . . . . 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 35972 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑁𝑇) → (𝑅𝑁) ∈ (Base‘𝐾))
114, 5, 10syl2anc 696 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) ∧ (𝑅𝐹) ∈ (Atoms‘𝐾)) → (𝑅𝑁) ∈ (Base‘𝐾))
12 simpr 479 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) ∧ (𝑅𝐹) ∈ (Atoms‘𝐾)) → (𝑅𝐹) ∈ (Atoms‘𝐾))
13 simpl3 1232 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) ∧ (𝑅𝐹) ∈ (Atoms‘𝐾)) → (𝑅𝑁) (𝑅𝐹))
14 tendoex.l . . . . . . 7 = (le‘𝐾)
15 eqid 2760 . . . . . . 7 (0.‘𝐾) = (0.‘𝐾)
16 eqid 2760 . . . . . . 7 (Atoms‘𝐾) = (Atoms‘𝐾)
176, 14, 15, 16leat 35101 . . . . . 6 (((𝐾 ∈ OP ∧ (𝑅𝑁) ∈ (Base‘𝐾) ∧ (𝑅𝐹) ∈ (Atoms‘𝐾)) ∧ (𝑅𝑁) (𝑅𝐹)) → ((𝑅𝑁) = (𝑅𝐹) ∨ (𝑅𝑁) = (0.‘𝐾)))
183, 11, 12, 13, 17syl31anc 1480 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) ∧ (𝑅𝐹) ∈ (Atoms‘𝐾)) → ((𝑅𝑁) = (𝑅𝐹) ∨ (𝑅𝑁) = (0.‘𝐾)))
19 simp3 1133 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) → (𝑅𝑁) (𝑅𝐹))
20 breq2 4808 . . . . . . . . 9 ((𝑅𝐹) = (0.‘𝐾) → ((𝑅𝑁) (𝑅𝐹) ↔ (𝑅𝑁) (0.‘𝐾)))
2119, 20syl5ibcom 235 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) → ((𝑅𝐹) = (0.‘𝐾) → (𝑅𝑁) (0.‘𝐾)))
2221imp 444 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) ∧ (𝑅𝐹) = (0.‘𝐾)) → (𝑅𝑁) (0.‘𝐾))
23 simpl1l 1279 . . . . . . . . 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 1228 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) ∧ (𝑅𝐹) = (0.‘𝐾)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
26 simpl2r 1285 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) ∧ (𝑅𝐹) = (0.‘𝐾)) → 𝑁𝑇)
2725, 26, 10syl2anc 696 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) ∧ (𝑅𝐹) = (0.‘𝐾)) → (𝑅𝑁) ∈ (Base‘𝐾))
286, 14, 15ople0 34995 . . . . . . . 8 ((𝐾 ∈ OP ∧ (𝑅𝑁) ∈ (Base‘𝐾)) → ((𝑅𝑁) (0.‘𝐾) ↔ (𝑅𝑁) = (0.‘𝐾)))
2924, 27, 28syl2anc 696 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) ∧ (𝑅𝐹) = (0.‘𝐾)) → ((𝑅𝑁) (0.‘𝐾) ↔ (𝑅𝑁) = (0.‘𝐾)))
3022, 29mpbid 222 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) ∧ (𝑅𝐹) = (0.‘𝐾)) → (𝑅𝑁) = (0.‘𝐾))
3130olcd 407 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) ∧ (𝑅𝐹) = (0.‘𝐾)) → ((𝑅𝑁) = (𝑅𝐹) ∨ (𝑅𝑁) = (0.‘𝐾)))
32 simp1 1131 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
33 simp2l 1242 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) → 𝐹𝑇)
3415, 16, 7, 8, 9trlator0 35979 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → ((𝑅𝐹) ∈ (Atoms‘𝐾) ∨ (𝑅𝐹) = (0.‘𝐾)))
3532, 33, 34syl2anc 696 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) → ((𝑅𝐹) ∈ (Atoms‘𝐾) ∨ (𝑅𝐹) = (0.‘𝐾)))
3618, 31, 35mpjaodan 862 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) → ((𝑅𝑁) = (𝑅𝐹) ∨ (𝑅𝑁) = (0.‘𝐾)))
37363expa 1112 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇)) ∧ (𝑅𝑁) (𝑅𝐹)) → ((𝑅𝑁) = (𝑅𝐹) ∨ (𝑅𝑁) = (0.‘𝐾)))
38 eqcom 2767 . . . . 5 ((𝑅𝑁) = (𝑅𝐹) ↔ (𝑅𝐹) = (𝑅𝑁))
39 tendoex.e . . . . . . 7 𝐸 = ((TEndo‘𝐾)‘𝑊)
407, 8, 9, 39cdlemk 36782 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝐹) = (𝑅𝑁)) → ∃𝑢𝐸 (𝑢𝐹) = 𝑁)
41403expa 1112 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇)) ∧ (𝑅𝐹) = (𝑅𝑁)) → ∃𝑢𝐸 (𝑢𝐹) = 𝑁)
4238, 41sylan2b 493 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇)) ∧ (𝑅𝑁) = (𝑅𝐹)) → ∃𝑢𝐸 (𝑢𝐹) = 𝑁)
43 eqid 2760 . . . . . . 7 (𝑇 ↦ ( I ↾ (Base‘𝐾))) = (𝑇 ↦ ( I ↾ (Base‘𝐾)))
446, 7, 8, 39, 43tendo0cl 36598 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (𝑇 ↦ ( I ↾ (Base‘𝐾))) ∈ 𝐸)
4544ad2antrr 764 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇)) ∧ (𝑅𝑁) = (0.‘𝐾)) → (𝑇 ↦ ( I ↾ (Base‘𝐾))) ∈ 𝐸)
46 simplrl 819 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇)) ∧ (𝑅𝑁) = (0.‘𝐾)) → 𝐹𝑇)
4743, 6tendo02 36595 . . . . . . 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 35986 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑁𝑇) → (𝑁 = ( I ↾ (Base‘𝐾)) ↔ (𝑅𝑁) = (0.‘𝐾)))
5049adantrl 754 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇)) → (𝑁 = ( I ↾ (Base‘𝐾)) ↔ (𝑅𝑁) = (0.‘𝐾)))
5150biimpar 503 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇)) ∧ (𝑅𝑁) = (0.‘𝐾)) → 𝑁 = ( I ↾ (Base‘𝐾)))
5248, 51eqtr4d 2797 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇)) ∧ (𝑅𝑁) = (0.‘𝐾)) → ((𝑇 ↦ ( I ↾ (Base‘𝐾)))‘𝐹) = 𝑁)
53 fveq1 6352 . . . . . . 7 (𝑢 = (𝑇 ↦ ( I ↾ (Base‘𝐾))) → (𝑢𝐹) = ((𝑇 ↦ ( I ↾ (Base‘𝐾)))‘𝐹))
5453eqeq1d 2762 . . . . . 6 (𝑢 = (𝑇 ↦ ( I ↾ (Base‘𝐾))) → ((𝑢𝐹) = 𝑁 ↔ ((𝑇 ↦ ( I ↾ (Base‘𝐾)))‘𝐹) = 𝑁))
5554rspcev 3449 . . . . 5 (((𝑇 ↦ ( I ↾ (Base‘𝐾))) ∈ 𝐸 ∧ ((𝑇 ↦ ( I ↾ (Base‘𝐾)))‘𝐹) = 𝑁) → ∃𝑢𝐸 (𝑢𝐹) = 𝑁)
5645, 52, 55syl2anc 696 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇)) ∧ (𝑅𝑁) = (0.‘𝐾)) → ∃𝑢𝐸 (𝑢𝐹) = 𝑁)
5742, 56jaodan 861 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇)) ∧ ((𝑅𝑁) = (𝑅𝐹) ∨ (𝑅𝑁) = (0.‘𝐾))) → ∃𝑢𝐸 (𝑢𝐹) = 𝑁)
5837, 57syldan 488 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇)) ∧ (𝑅𝑁) (𝑅𝐹)) → ∃𝑢𝐸 (𝑢𝐹) = 𝑁)
59583impa 1101 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) → ∃𝑢𝐸 (𝑢𝐹) = 𝑁)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∨ wo 382   ∧ wa 383   ∧ w3a 1072   = wceq 1632   ∈ wcel 2139  ∃wrex 3051   class class class wbr 4804   ↦ cmpt 4881   I cid 5173   ↾ cres 5268  ‘cfv 6049  Basecbs 16079  lecple 16170  0.cp0 17258  OPcops 34980  Atomscatm 35071  HLchlt 35158  LHypclh 35791  LTrncltrn 35908  trLctrl 35966  TEndoctendo 36560 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-riotaBAD 34760 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335  df-undef 7569  df-map 8027  df-preset 17149  df-poset 17167  df-plt 17179  df-lub 17195  df-glb 17196  df-join 17197  df-meet 17198  df-p0 17260  df-p1 17261  df-lat 17267  df-clat 17329  df-oposet 34984  df-ol 34986  df-oml 34987  df-covers 35074  df-ats 35075  df-atl 35106  df-cvlat 35130  df-hlat 35159  df-llines 35305  df-lplanes 35306  df-lvols 35307  df-lines 35308  df-psubsp 35310  df-pmap 35311  df-padd 35603  df-lhyp 35795  df-laut 35796  df-ldil 35911  df-ltrn 35912  df-trl 35967  df-tendo 36563 This theorem is referenced by:  dva1dim  36793  dihjatcclem4  37230
 Copyright terms: Public domain W3C validator