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Theorem ltrniotavalbN 38160
 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 1188 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝐹𝑇) ∧ (𝐹𝑃) = 𝑄) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simpl3 1190 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝐹𝑇) ∧ (𝐹𝑃) = 𝑄) → 𝐹𝑇)
3 simpl2l 1223 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝐹𝑇) ∧ (𝐹𝑃) = 𝑄) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
4 simpl2r 1224 . . . 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 2758 . . . . 5 (𝑓𝑇 (𝑓𝑃) = 𝑄) = (𝑓𝑇 (𝑓𝑃) = 𝑄)
105, 6, 7, 8, 9ltrniotacl 38155 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑓𝑇 (𝑓𝑃) = 𝑄) ∈ 𝑇)
111, 3, 4, 10syl3anc 1368 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝐹𝑇) ∧ (𝐹𝑃) = 𝑄) → (𝑓𝑇 (𝑓𝑃) = 𝑄) ∈ 𝑇)
12 simpr 488 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝐹𝑇) ∧ (𝐹𝑃) = 𝑄) → (𝐹𝑃) = 𝑄)
135, 6, 7, 8, 9ltrniotaval 38157 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ((𝑓𝑇 (𝑓𝑃) = 𝑄)‘𝑃) = 𝑄)
141, 3, 4, 13syl3anc 1368 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝐹𝑇) ∧ (𝐹𝑃) = 𝑄) → ((𝑓𝑇 (𝑓𝑃) = 𝑄)‘𝑃) = 𝑄)
1512, 14eqtr4d 2796 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝐹𝑇) ∧ (𝐹𝑃) = 𝑄) → (𝐹𝑃) = ((𝑓𝑇 (𝑓𝑃) = 𝑄)‘𝑃))
165, 6, 7, 8cdlemd 37783 . . 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 1381 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝐹𝑇) ∧ (𝐹𝑃) = 𝑄) → 𝐹 = (𝑓𝑇 (𝑓𝑃) = 𝑄))
18 fveq1 6657 . . 3 (𝐹 = (𝑓𝑇 (𝑓𝑃) = 𝑄) → (𝐹𝑃) = ((𝑓𝑇 (𝑓𝑃) = 𝑄)‘𝑃))
19 simp1 1133 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝐹𝑇) → (𝐾 ∈ HL ∧ 𝑊𝐻))
20 simp2l 1196 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝐹𝑇) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
21 simp2r 1197 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝐹𝑇) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
2219, 20, 21, 13syl3anc 1368 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝐹𝑇) → ((𝑓𝑇 (𝑓𝑃) = 𝑄)‘𝑃) = 𝑄)
2318, 22sylan9eqr 2815 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝐹𝑇) ∧ 𝐹 = (𝑓𝑇 (𝑓𝑃) = 𝑄)) → (𝐹𝑃) = 𝑄)
2417, 23impbida 800 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 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   class class class wbr 5032  ‘cfv 6335  ℩crio 7107  lecple 16630  Atomscatm 36839  HLchlt 36926  LHypclh 37560  LTrncltrn 37677 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-riotaBAD 36529 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-iin 4886  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7693  df-2nd 7694  df-undef 7949  df-map 8418  df-proset 17604  df-poset 17622  df-plt 17634  df-lub 17650  df-glb 17651  df-join 17652  df-meet 17653  df-p0 17715  df-p1 17716  df-lat 17722  df-clat 17784  df-oposet 36752  df-ol 36754  df-oml 36755  df-covers 36842  df-ats 36843  df-atl 36874  df-cvlat 36898  df-hlat 36927  df-llines 37074  df-lplanes 37075  df-lvols 37076  df-lines 37077  df-psubsp 37079  df-pmap 37080  df-padd 37372  df-lhyp 37564  df-laut 37565  df-ldil 37680  df-ltrn 37681  df-trl 37735 This theorem is referenced by: (None)
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