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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
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