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Theorem trnsetN 39661
Description: The set of translations for a fiducial atom 𝐷. (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
trnsetN ((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}))})
Distinct variable groups:   𝑓,π‘ž,π‘Ÿ,𝐾   𝑓,𝐿   π‘Š,π‘ž,π‘Ÿ   𝐷,𝑓,π‘ž,π‘Ÿ
Allowed substitution hints:   𝐴(𝑓,π‘Ÿ,π‘ž)   𝐡(𝑓,π‘Ÿ,π‘ž)   + (𝑓,π‘Ÿ,π‘ž)   𝑆(𝑓,π‘Ÿ,π‘ž)   𝑇(𝑓,π‘Ÿ,π‘ž)   𝐿(π‘Ÿ,π‘ž)   𝑀(𝑓,π‘Ÿ,π‘ž)   βŠ₯ (𝑓,π‘Ÿ,π‘ž)   π‘Š(𝑓)

Proof of Theorem trnsetN
Dummy variable 𝑑 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, 8trnfsetN 39660 . . 3 (𝐾 ∈ 𝐡 β†’ 𝑇 = (𝑑 ∈ 𝐴 ↦ {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}))}))
109fveq1d 6904 . 2 (𝐾 ∈ 𝐡 β†’ (π‘‡β€˜π·) = ((𝑑 ∈ 𝐴 ↦ {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}))})β€˜π·))
11 fveq2 6902 . . . 4 (𝑑 = 𝐷 β†’ (πΏβ€˜π‘‘) = (πΏβ€˜π·))
12 fveq2 6902 . . . . 5 (𝑑 = 𝐷 β†’ (π‘Šβ€˜π‘‘) = (π‘Šβ€˜π·))
13 sneq 4642 . . . . . . . . 9 (𝑑 = 𝐷 β†’ {𝑑} = {𝐷})
1413fveq2d 6906 . . . . . . . 8 (𝑑 = 𝐷 β†’ ( βŠ₯ β€˜{𝑑}) = ( βŠ₯ β€˜{𝐷}))
1514ineq2d 4214 . . . . . . 7 (𝑑 = 𝐷 β†’ ((π‘ž + (π‘“β€˜π‘ž)) ∩ ( βŠ₯ β€˜{𝑑})) = ((π‘ž + (π‘“β€˜π‘ž)) ∩ ( βŠ₯ β€˜{𝐷})))
1614ineq2d 4214 . . . . . . 7 (𝑑 = 𝐷 β†’ ((π‘Ÿ + (π‘“β€˜π‘Ÿ)) ∩ ( βŠ₯ β€˜{𝑑})) = ((π‘Ÿ + (π‘“β€˜π‘Ÿ)) ∩ ( βŠ₯ β€˜{𝐷})))
1715, 16eqeq12d 2744 . . . . . 6 (𝑑 = 𝐷 β†’ (((π‘ž + (π‘“β€˜π‘ž)) ∩ ( βŠ₯ β€˜{𝑑})) = ((π‘Ÿ + (π‘“β€˜π‘Ÿ)) ∩ ( βŠ₯ β€˜{𝑑})) ↔ ((π‘ž + (π‘“β€˜π‘ž)) ∩ ( βŠ₯ β€˜{𝐷})) = ((π‘Ÿ + (π‘“β€˜π‘Ÿ)) ∩ ( βŠ₯ β€˜{𝐷}))))
1812, 17raleqbidv 3340 . . . . 5 (𝑑 = 𝐷 β†’ (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‘Ÿ ∈ (π‘Šβ€˜π·)((π‘ž + (π‘“β€˜π‘ž)) ∩ ( βŠ₯ β€˜{𝐷})) = ((π‘Ÿ + (π‘“β€˜π‘Ÿ)) ∩ ( βŠ₯ β€˜{𝐷}))))
1912, 18raleqbidv 3340 . . . 4 (𝑑 = 𝐷 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‘“β€˜π‘ž)) ∩ ( βŠ₯ β€˜{𝐷})) = ((π‘Ÿ + (π‘“β€˜π‘Ÿ)) ∩ ( βŠ₯ β€˜{𝐷}))))
2011, 19rabeqbidv 3448 . . 3 (𝑑 = 𝐷 β†’ {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‘“β€˜π‘ž)) ∩ ( βŠ₯ β€˜{𝐷})) = ((π‘Ÿ + (π‘“β€˜π‘Ÿ)) ∩ ( βŠ₯ β€˜{𝐷}))})
21 eqid 2728 . . 3 (𝑑 ∈ 𝐴 ↦ {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}))}) = (𝑑 ∈ 𝐴 ↦ {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}))})
22 fvex 6915 . . . 4 (πΏβ€˜π·) ∈ V
2322rabex 5338 . . 3 {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}))} ∈ V
2420, 21, 23fvmpt 7010 . 2 (𝐷 ∈ 𝐴 β†’ ((𝑑 ∈ 𝐴 ↦ {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}))})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}))})
2510, 24sylan9eq 2788 1 ((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}))})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  βˆ€wral 3058  {crab 3430   ∩ cin 3948  {csn 4632   ↦ cmpt 5235  β€˜cfv 6553  (class class class)co 7426  Atomscatm 38767  PSubSpcpsubsp 39001  +𝑃cpadd 39300  βŠ₯𝑃cpolN 39407  WAtomscwpointsN 39491  PAutcpautN 39492  DilcdilN 39607  TrnctrnN 39608
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 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-trnN 39612
This theorem is referenced by:  istrnN  39662
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