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Theorem istrnN 39630
Description: The predicate "is a translation". (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
trnset.a 𝐴 = (Atomsβ€˜πΎ)
trnset.s 𝑆 = (PSubSpβ€˜πΎ)
trnset.p + = (+π‘ƒβ€˜πΎ)
trnset.o βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)
trnset.w π‘Š = (WAtomsβ€˜πΎ)
trnset.m 𝑀 = (PAutβ€˜πΎ)
trnset.l 𝐿 = (Dilβ€˜πΎ)
trnset.t 𝑇 = (Trnβ€˜πΎ)
Assertion
Ref Expression
istrnN ((𝐾 ∈ 𝐡 ∧ 𝐷 ∈ 𝐴) β†’ (𝐹 ∈ (π‘‡β€˜π·) ↔ (𝐹 ∈ (πΏβ€˜π·) ∧ βˆ€π‘ž ∈ (π‘Šβ€˜π·)βˆ€π‘Ÿ ∈ (π‘Šβ€˜π·)((π‘ž + (πΉβ€˜π‘ž)) ∩ ( βŠ₯ β€˜{𝐷})) = ((π‘Ÿ + (πΉβ€˜π‘Ÿ)) ∩ ( βŠ₯ β€˜{𝐷})))))
Distinct variable groups:   π‘Ÿ,π‘ž,𝐾   π‘Š,π‘ž,π‘Ÿ   𝐷,π‘ž,π‘Ÿ   𝐹,π‘ž,π‘Ÿ
Allowed substitution hints:   𝐴(π‘Ÿ,π‘ž)   𝐡(π‘Ÿ,π‘ž)   + (π‘Ÿ,π‘ž)   𝑆(π‘Ÿ,π‘ž)   𝑇(π‘Ÿ,π‘ž)   𝐿(π‘Ÿ,π‘ž)   𝑀(π‘Ÿ,π‘ž)   βŠ₯ (π‘Ÿ,π‘ž)

Proof of Theorem istrnN
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 trnset.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
2 trnset.s . . . 4 𝑆 = (PSubSpβ€˜πΎ)
3 trnset.p . . . 4 + = (+π‘ƒβ€˜πΎ)
4 trnset.o . . . 4 βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)
5 trnset.w . . . 4 π‘Š = (WAtomsβ€˜πΎ)
6 trnset.m . . . 4 𝑀 = (PAutβ€˜πΎ)
7 trnset.l . . . 4 𝐿 = (Dilβ€˜πΎ)
8 trnset.t . . . 4 𝑇 = (Trnβ€˜πΎ)
91, 2, 3, 4, 5, 6, 7, 8trnsetN 39629 . . 3 ((𝐾 ∈ 𝐡 ∧ 𝐷 ∈ 𝐴) β†’ (π‘‡β€˜π·) = {𝑓 ∈ (πΏβ€˜π·) ∣ βˆ€π‘ž ∈ (π‘Šβ€˜π·)βˆ€π‘Ÿ ∈ (π‘Šβ€˜π·)((π‘ž + (π‘“β€˜π‘ž)) ∩ ( βŠ₯ β€˜{𝐷})) = ((π‘Ÿ + (π‘“β€˜π‘Ÿ)) ∩ ( βŠ₯ β€˜{𝐷}))})
109eleq2d 2815 . 2 ((𝐾 ∈ 𝐡 ∧ 𝐷 ∈ 𝐴) β†’ (𝐹 ∈ (π‘‡β€˜π·) ↔ 𝐹 ∈ {𝑓 ∈ (πΏβ€˜π·) ∣ βˆ€π‘ž ∈ (π‘Šβ€˜π·)βˆ€π‘Ÿ ∈ (π‘Šβ€˜π·)((π‘ž + (π‘“β€˜π‘ž)) ∩ ( βŠ₯ β€˜{𝐷})) = ((π‘Ÿ + (π‘“β€˜π‘Ÿ)) ∩ ( βŠ₯ β€˜{𝐷}))}))
11 fveq1 6896 . . . . . . 7 (𝑓 = 𝐹 β†’ (π‘“β€˜π‘ž) = (πΉβ€˜π‘ž))
1211oveq2d 7436 . . . . . 6 (𝑓 = 𝐹 β†’ (π‘ž + (π‘“β€˜π‘ž)) = (π‘ž + (πΉβ€˜π‘ž)))
1312ineq1d 4211 . . . . 5 (𝑓 = 𝐹 β†’ ((π‘ž + (π‘“β€˜π‘ž)) ∩ ( βŠ₯ β€˜{𝐷})) = ((π‘ž + (πΉβ€˜π‘ž)) ∩ ( βŠ₯ β€˜{𝐷})))
14 fveq1 6896 . . . . . . 7 (𝑓 = 𝐹 β†’ (π‘“β€˜π‘Ÿ) = (πΉβ€˜π‘Ÿ))
1514oveq2d 7436 . . . . . 6 (𝑓 = 𝐹 β†’ (π‘Ÿ + (π‘“β€˜π‘Ÿ)) = (π‘Ÿ + (πΉβ€˜π‘Ÿ)))
1615ineq1d 4211 . . . . 5 (𝑓 = 𝐹 β†’ ((π‘Ÿ + (π‘“β€˜π‘Ÿ)) ∩ ( βŠ₯ β€˜{𝐷})) = ((π‘Ÿ + (πΉβ€˜π‘Ÿ)) ∩ ( βŠ₯ β€˜{𝐷})))
1713, 16eqeq12d 2744 . . . 4 (𝑓 = 𝐹 β†’ (((π‘ž + (π‘“β€˜π‘ž)) ∩ ( βŠ₯ β€˜{𝐷})) = ((π‘Ÿ + (π‘“β€˜π‘Ÿ)) ∩ ( βŠ₯ β€˜{𝐷})) ↔ ((π‘ž + (πΉβ€˜π‘ž)) ∩ ( βŠ₯ β€˜{𝐷})) = ((π‘Ÿ + (πΉβ€˜π‘Ÿ)) ∩ ( βŠ₯ β€˜{𝐷}))))
18172ralbidv 3215 . . 3 (𝑓 = 𝐹 β†’ (βˆ€π‘ž ∈ (π‘Šβ€˜π·)βˆ€π‘Ÿ ∈ (π‘Šβ€˜π·)((π‘ž + (π‘“β€˜π‘ž)) ∩ ( βŠ₯ β€˜{𝐷})) = ((π‘Ÿ + (π‘“β€˜π‘Ÿ)) ∩ ( βŠ₯ β€˜{𝐷})) ↔ βˆ€π‘ž ∈ (π‘Šβ€˜π·)βˆ€π‘Ÿ ∈ (π‘Šβ€˜π·)((π‘ž + (πΉβ€˜π‘ž)) ∩ ( βŠ₯ β€˜{𝐷})) = ((π‘Ÿ + (πΉβ€˜π‘Ÿ)) ∩ ( βŠ₯ β€˜{𝐷}))))
1918elrab 3682 . 2 (𝐹 ∈ {𝑓 ∈ (πΏβ€˜π·) ∣ βˆ€π‘ž ∈ (π‘Šβ€˜π·)βˆ€π‘Ÿ ∈ (π‘Šβ€˜π·)((π‘ž + (π‘“β€˜π‘ž)) ∩ ( βŠ₯ β€˜{𝐷})) = ((π‘Ÿ + (π‘“β€˜π‘Ÿ)) ∩ ( βŠ₯ β€˜{𝐷}))} ↔ (𝐹 ∈ (πΏβ€˜π·) ∧ βˆ€π‘ž ∈ (π‘Šβ€˜π·)βˆ€π‘Ÿ ∈ (π‘Šβ€˜π·)((π‘ž + (πΉβ€˜π‘ž)) ∩ ( βŠ₯ β€˜{𝐷})) = ((π‘Ÿ + (πΉβ€˜π‘Ÿ)) ∩ ( βŠ₯ β€˜{𝐷}))))
2010, 19bitrdi 287 1 ((𝐾 ∈ 𝐡 ∧ 𝐷 ∈ 𝐴) β†’ (𝐹 ∈ (π‘‡β€˜π·) ↔ (𝐹 ∈ (πΏβ€˜π·) ∧ βˆ€π‘ž ∈ (π‘Šβ€˜π·)βˆ€π‘Ÿ ∈ (π‘Šβ€˜π·)((π‘ž + (πΉβ€˜π‘ž)) ∩ ( βŠ₯ β€˜{𝐷})) = ((π‘Ÿ + (πΉβ€˜π‘Ÿ)) ∩ ( βŠ₯ β€˜{𝐷})))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  βˆ€wral 3058  {crab 3429   ∩ cin 3946  {csn 4629  β€˜cfv 6548  (class class class)co 7420  Atomscatm 38735  PSubSpcpsubsp 38969  +𝑃cpadd 39268  βŠ₯𝑃cpolN 39375  WAtomscwpointsN 39459  PAutcpautN 39460  DilcdilN 39575  TrnctrnN 39576
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 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-trnN 39580
This theorem is referenced by: (None)
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