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Theorem ltrniotavalbN 37588
Description: Value of the unique translation specified by a value. (Contributed by NM, 10-Mar-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
ltrniotavalb.l = (le‘𝐾)
ltrniotavalb.a 𝐴 = (Atoms‘𝐾)
ltrniotavalb.h 𝐻 = (LHyp‘𝐾)
ltrniotavalb.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
Assertion
Ref Expression
ltrniotavalbN (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝐹𝑇) → ((𝐹𝑃) = 𝑄𝐹 = (𝑓𝑇 (𝑓𝑃) = 𝑄)))
Distinct variable groups:   ,𝑓   𝐴,𝑓   𝑓,𝐻   𝑓,𝐾   𝑃,𝑓   𝑄,𝑓   𝑇,𝑓   𝑓,𝑊
Allowed substitution hint:   𝐹(𝑓)

Proof of Theorem ltrniotavalbN
StepHypRef Expression
1 simpl1 1185 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝐹𝑇) ∧ (𝐹𝑃) = 𝑄) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simpl3 1187 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝐹𝑇) ∧ (𝐹𝑃) = 𝑄) → 𝐹𝑇)
3 simpl2l 1220 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝐹𝑇) ∧ (𝐹𝑃) = 𝑄) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
4 simpl2r 1221 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝐹𝑇) ∧ (𝐹𝑃) = 𝑄) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
5 ltrniotavalb.l . . . . 5 = (le‘𝐾)
6 ltrniotavalb.a . . . . 5 𝐴 = (Atoms‘𝐾)
7 ltrniotavalb.h . . . . 5 𝐻 = (LHyp‘𝐾)
8 ltrniotavalb.t . . . . 5 𝑇 = ((LTrn‘𝐾)‘𝑊)
9 eqid 2824 . . . . 5 (𝑓𝑇 (𝑓𝑃) = 𝑄) = (𝑓𝑇 (𝑓𝑃) = 𝑄)
105, 6, 7, 8, 9ltrniotacl 37583 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑓𝑇 (𝑓𝑃) = 𝑄) ∈ 𝑇)
111, 3, 4, 10syl3anc 1365 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝐹𝑇) ∧ (𝐹𝑃) = 𝑄) → (𝑓𝑇 (𝑓𝑃) = 𝑄) ∈ 𝑇)
12 simpr 485 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝐹𝑇) ∧ (𝐹𝑃) = 𝑄) → (𝐹𝑃) = 𝑄)
135, 6, 7, 8, 9ltrniotaval 37585 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ((𝑓𝑇 (𝑓𝑃) = 𝑄)‘𝑃) = 𝑄)
141, 3, 4, 13syl3anc 1365 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝐹𝑇) ∧ (𝐹𝑃) = 𝑄) → ((𝑓𝑇 (𝑓𝑃) = 𝑄)‘𝑃) = 𝑄)
1512, 14eqtr4d 2863 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝐹𝑇) ∧ (𝐹𝑃) = 𝑄) → (𝐹𝑃) = ((𝑓𝑇 (𝑓𝑃) = 𝑄)‘𝑃))
165, 6, 7, 8cdlemd 37211 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑓𝑇 (𝑓𝑃) = 𝑄) ∈ 𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑃) = ((𝑓𝑇 (𝑓𝑃) = 𝑄)‘𝑃)) → 𝐹 = (𝑓𝑇 (𝑓𝑃) = 𝑄))
171, 2, 11, 3, 15, 16syl311anc 1378 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝐹𝑇) ∧ (𝐹𝑃) = 𝑄) → 𝐹 = (𝑓𝑇 (𝑓𝑃) = 𝑄))
18 fveq1 6665 . . 3 (𝐹 = (𝑓𝑇 (𝑓𝑃) = 𝑄) → (𝐹𝑃) = ((𝑓𝑇 (𝑓𝑃) = 𝑄)‘𝑃))
19 simp1 1130 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝐹𝑇) → (𝐾 ∈ HL ∧ 𝑊𝐻))
20 simp2l 1193 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝐹𝑇) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
21 simp2r 1194 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝐹𝑇) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
2219, 20, 21, 13syl3anc 1365 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝐹𝑇) → ((𝑓𝑇 (𝑓𝑃) = 𝑄)‘𝑃) = 𝑄)
2318, 22sylan9eqr 2882 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝐹𝑇) ∧ 𝐹 = (𝑓𝑇 (𝑓𝑃) = 𝑄)) → (𝐹𝑃) = 𝑄)
2417, 23impbida 797 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝐹𝑇) → ((𝐹𝑃) = 𝑄𝐹 = (𝑓𝑇 (𝑓𝑃) = 𝑄)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wcel 2106   class class class wbr 5062  cfv 6351  crio 7108  lecple 16564  Atomscatm 36267  HLchlt 36354  LHypclh 36988  LTrncltrn 37105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-13 2385  ax-ext 2796  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-riotaBAD 35957
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-iun 4918  df-iin 4919  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-1st 7683  df-2nd 7684  df-undef 7933  df-map 8401  df-proset 17530  df-poset 17548  df-plt 17560  df-lub 17576  df-glb 17577  df-join 17578  df-meet 17579  df-p0 17641  df-p1 17642  df-lat 17648  df-clat 17710  df-oposet 36180  df-ol 36182  df-oml 36183  df-covers 36270  df-ats 36271  df-atl 36302  df-cvlat 36326  df-hlat 36355  df-llines 36502  df-lplanes 36503  df-lvols 36504  df-lines 36505  df-psubsp 36507  df-pmap 36508  df-padd 36800  df-lhyp 36992  df-laut 36993  df-ldil 37108  df-ltrn 37109  df-trl 37163
This theorem is referenced by: (None)
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