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

Theorem ltrniotavalbN 40109
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 ≀ = (leβ€˜πΎ)
ltrniotavalb.a 𝐴 = (Atomsβ€˜πΎ)
ltrniotavalb.h 𝐻 = (LHypβ€˜πΎ)
ltrniotavalb.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
ltrniotavalbN (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ 𝐹 ∈ 𝑇) β†’ ((πΉβ€˜π‘ƒ) = 𝑄 ↔ 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘ƒ) = 𝑄)))
Distinct variable groups:   ≀ ,𝑓   𝐴,𝑓   𝑓,𝐻   𝑓,𝐾   𝑃,𝑓   𝑄,𝑓   𝑇,𝑓   𝑓,π‘Š
Allowed substitution hint:   𝐹(𝑓)

Proof of Theorem ltrniotavalbN
StepHypRef Expression
1 simpl1 1188 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ 𝐹 ∈ 𝑇) ∧ (πΉβ€˜π‘ƒ) = 𝑄) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
2 simpl3 1190 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ 𝐹 ∈ 𝑇) ∧ (πΉβ€˜π‘ƒ) = 𝑄) β†’ 𝐹 ∈ 𝑇)
3 simpl2l 1223 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ 𝐹 ∈ 𝑇) ∧ (πΉβ€˜π‘ƒ) = 𝑄) β†’ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š))
4 simpl2r 1224 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ 𝐹 ∈ 𝑇) ∧ (πΉβ€˜π‘ƒ) = 𝑄) β†’ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š))
5 ltrniotavalb.l . . . . 5 ≀ = (leβ€˜πΎ)
6 ltrniotavalb.a . . . . 5 𝐴 = (Atomsβ€˜πΎ)
7 ltrniotavalb.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
8 ltrniotavalb.t . . . . 5 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
9 eqid 2725 . . . . 5 (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘ƒ) = 𝑄) = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘ƒ) = 𝑄)
105, 6, 7, 8, 9ltrniotacl 40104 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) β†’ (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘ƒ) = 𝑄) ∈ 𝑇)
111, 3, 4, 10syl3anc 1368 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ 𝐹 ∈ 𝑇) ∧ (πΉβ€˜π‘ƒ) = 𝑄) β†’ (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘ƒ) = 𝑄) ∈ 𝑇)
12 simpr 483 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ 𝐹 ∈ 𝑇) ∧ (πΉβ€˜π‘ƒ) = 𝑄) β†’ (πΉβ€˜π‘ƒ) = 𝑄)
135, 6, 7, 8, 9ltrniotaval 40106 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) β†’ ((℩𝑓 ∈ 𝑇 (π‘“β€˜π‘ƒ) = 𝑄)β€˜π‘ƒ) = 𝑄)
141, 3, 4, 13syl3anc 1368 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ 𝐹 ∈ 𝑇) ∧ (πΉβ€˜π‘ƒ) = 𝑄) β†’ ((℩𝑓 ∈ 𝑇 (π‘“β€˜π‘ƒ) = 𝑄)β€˜π‘ƒ) = 𝑄)
1512, 14eqtr4d 2768 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ 𝐹 ∈ 𝑇) ∧ (πΉβ€˜π‘ƒ) = 𝑄) β†’ (πΉβ€˜π‘ƒ) = ((℩𝑓 ∈ 𝑇 (π‘“β€˜π‘ƒ) = 𝑄)β€˜π‘ƒ))
165, 6, 7, 8cdlemd 39732 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘ƒ) = 𝑄) ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (πΉβ€˜π‘ƒ) = ((℩𝑓 ∈ 𝑇 (π‘“β€˜π‘ƒ) = 𝑄)β€˜π‘ƒ)) β†’ 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘ƒ) = 𝑄))
171, 2, 11, 3, 15, 16syl311anc 1381 . 2 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ 𝐹 ∈ 𝑇) ∧ (πΉβ€˜π‘ƒ) = 𝑄) β†’ 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘ƒ) = 𝑄))
18 fveq1 6889 . . 3 (𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘ƒ) = 𝑄) β†’ (πΉβ€˜π‘ƒ) = ((℩𝑓 ∈ 𝑇 (π‘“β€˜π‘ƒ) = 𝑄)β€˜π‘ƒ))
19 simp1 1133 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ 𝐹 ∈ 𝑇) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
20 simp2l 1196 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ 𝐹 ∈ 𝑇) β†’ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š))
21 simp2r 1197 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ 𝐹 ∈ 𝑇) β†’ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š))
2219, 20, 21, 13syl3anc 1368 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ 𝐹 ∈ 𝑇) β†’ ((℩𝑓 ∈ 𝑇 (π‘“β€˜π‘ƒ) = 𝑄)β€˜π‘ƒ) = 𝑄)
2318, 22sylan9eqr 2787 . 2 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ 𝐹 ∈ 𝑇) ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘ƒ) = 𝑄)) β†’ (πΉβ€˜π‘ƒ) = 𝑄)
2417, 23impbida 799 1 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ 𝐹 ∈ 𝑇) β†’ ((πΉβ€˜π‘ƒ) = 𝑄 ↔ 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘ƒ) = 𝑄)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   class class class wbr 5144  β€˜cfv 6543  β„©crio 7368  lecple 17234  Atomscatm 38787  HLchlt 38874  LHypclh 39509  LTrncltrn 39626
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735  ax-riotaBAD 38477
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-iin 4995  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7987  df-2nd 7988  df-undef 8272  df-map 8840  df-proset 18281  df-poset 18299  df-plt 18316  df-lub 18332  df-glb 18333  df-join 18334  df-meet 18335  df-p0 18411  df-p1 18412  df-lat 18418  df-clat 18485  df-oposet 38700  df-ol 38702  df-oml 38703  df-covers 38790  df-ats 38791  df-atl 38822  df-cvlat 38846  df-hlat 38875  df-llines 39023  df-lplanes 39024  df-lvols 39025  df-lines 39026  df-psubsp 39028  df-pmap 39029  df-padd 39321  df-lhyp 39513  df-laut 39514  df-ldil 39629  df-ltrn 39630  df-trl 39684
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator