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Theorem ltrniotavalbN 39994
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 1189 . . 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 1191 . . 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 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†’ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š))
4 simpl2r 1225 . . . 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 2727 . . . . 5 (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘ƒ) = 𝑄) = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘ƒ) = 𝑄)
105, 6, 7, 8, 9ltrniotacl 39989 . . . 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 1369 . . 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 484 . . . 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 39991 . . . . 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 1369 . . . 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 2770 . . 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 39617 . . 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 1382 . 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 6890 . . 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 1134 . . . 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 1197 . . . 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 1198 . . . 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 1369 . . 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 2789 . 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 800 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 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   class class class wbr 5142  β€˜cfv 6542  β„©crio 7369  lecple 17231  Atomscatm 38672  HLchlt 38759  LHypclh 39394  LTrncltrn 39511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-riotaBAD 38362
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-iin 4994  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988  df-undef 8272  df-map 8838  df-proset 18278  df-poset 18296  df-plt 18313  df-lub 18329  df-glb 18330  df-join 18331  df-meet 18332  df-p0 18408  df-p1 18409  df-lat 18415  df-clat 18482  df-oposet 38585  df-ol 38587  df-oml 38588  df-covers 38675  df-ats 38676  df-atl 38707  df-cvlat 38731  df-hlat 38760  df-llines 38908  df-lplanes 38909  df-lvols 38910  df-lines 38911  df-psubsp 38913  df-pmap 38914  df-padd 39206  df-lhyp 39398  df-laut 39399  df-ldil 39514  df-ltrn 39515  df-trl 39569
This theorem is referenced by: (None)
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