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Theorem trnfsetN 39668
Description: The mapping from fiducial atom to set of translations. (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
trnfsetN (𝐾 ∈ 𝐢 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}))}))
Distinct variable groups:   𝐴,𝑑   𝑓,𝑑,π‘ž,π‘Ÿ,𝐾   𝑓,𝐿   π‘Š,π‘ž,π‘Ÿ
Allowed substitution hints:   𝐴(𝑓,π‘Ÿ,π‘ž)   𝐢(𝑓,π‘Ÿ,π‘ž,𝑑)   + (𝑓,π‘Ÿ,π‘ž,𝑑)   𝑆(𝑓,π‘Ÿ,π‘ž,𝑑)   𝑇(𝑓,π‘Ÿ,π‘ž,𝑑)   𝐿(π‘Ÿ,π‘ž,𝑑)   𝑀(𝑓,π‘Ÿ,π‘ž,𝑑)   βŠ₯ (𝑓,π‘Ÿ,π‘ž,𝑑)   π‘Š(𝑓,𝑑)

Proof of Theorem trnfsetN
Dummy variable π‘˜ is distinct from all other variables.
StepHypRef Expression
1 elex 3492 . 2 (𝐾 ∈ 𝐢 β†’ 𝐾 ∈ V)
2 trnset.t . . 3 𝑇 = (Trnβ€˜πΎ)
3 fveq2 6902 . . . . . 6 (π‘˜ = 𝐾 β†’ (Atomsβ€˜π‘˜) = (Atomsβ€˜πΎ))
4 trnset.a . . . . . 6 𝐴 = (Atomsβ€˜πΎ)
53, 4eqtr4di 2786 . . . . 5 (π‘˜ = 𝐾 β†’ (Atomsβ€˜π‘˜) = 𝐴)
6 fveq2 6902 . . . . . . . 8 (π‘˜ = 𝐾 β†’ (Dilβ€˜π‘˜) = (Dilβ€˜πΎ))
7 trnset.l . . . . . . . 8 𝐿 = (Dilβ€˜πΎ)
86, 7eqtr4di 2786 . . . . . . 7 (π‘˜ = 𝐾 β†’ (Dilβ€˜π‘˜) = 𝐿)
98fveq1d 6904 . . . . . 6 (π‘˜ = 𝐾 β†’ ((Dilβ€˜π‘˜)β€˜π‘‘) = (πΏβ€˜π‘‘))
10 fveq2 6902 . . . . . . . . 9 (π‘˜ = 𝐾 β†’ (WAtomsβ€˜π‘˜) = (WAtomsβ€˜πΎ))
11 trnset.w . . . . . . . . 9 π‘Š = (WAtomsβ€˜πΎ)
1210, 11eqtr4di 2786 . . . . . . . 8 (π‘˜ = 𝐾 β†’ (WAtomsβ€˜π‘˜) = π‘Š)
1312fveq1d 6904 . . . . . . 7 (π‘˜ = 𝐾 β†’ ((WAtomsβ€˜π‘˜)β€˜π‘‘) = (π‘Šβ€˜π‘‘))
14 fveq2 6902 . . . . . . . . . . . 12 (π‘˜ = 𝐾 β†’ (+π‘ƒβ€˜π‘˜) = (+π‘ƒβ€˜πΎ))
15 trnset.p . . . . . . . . . . . 12 + = (+π‘ƒβ€˜πΎ)
1614, 15eqtr4di 2786 . . . . . . . . . . 11 (π‘˜ = 𝐾 β†’ (+π‘ƒβ€˜π‘˜) = + )
1716oveqd 7443 . . . . . . . . . 10 (π‘˜ = 𝐾 β†’ (π‘ž(+π‘ƒβ€˜π‘˜)(π‘“β€˜π‘ž)) = (π‘ž + (π‘“β€˜π‘ž)))
18 fveq2 6902 . . . . . . . . . . . 12 (π‘˜ = 𝐾 β†’ (βŠ₯π‘ƒβ€˜π‘˜) = (βŠ₯π‘ƒβ€˜πΎ))
19 trnset.o . . . . . . . . . . . 12 βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)
2018, 19eqtr4di 2786 . . . . . . . . . . 11 (π‘˜ = 𝐾 β†’ (βŠ₯π‘ƒβ€˜π‘˜) = βŠ₯ )
2120fveq1d 6904 . . . . . . . . . 10 (π‘˜ = 𝐾 β†’ ((βŠ₯π‘ƒβ€˜π‘˜)β€˜{𝑑}) = ( βŠ₯ β€˜{𝑑}))
2217, 21ineq12d 4215 . . . . . . . . 9 (π‘˜ = 𝐾 β†’ ((π‘ž(+π‘ƒβ€˜π‘˜)(π‘“β€˜π‘ž)) ∩ ((βŠ₯π‘ƒβ€˜π‘˜)β€˜{𝑑})) = ((π‘ž + (π‘“β€˜π‘ž)) ∩ ( βŠ₯ β€˜{𝑑})))
2316oveqd 7443 . . . . . . . . . 10 (π‘˜ = 𝐾 β†’ (π‘Ÿ(+π‘ƒβ€˜π‘˜)(π‘“β€˜π‘Ÿ)) = (π‘Ÿ + (π‘“β€˜π‘Ÿ)))
2423, 21ineq12d 4215 . . . . . . . . 9 (π‘˜ = 𝐾 β†’ ((π‘Ÿ(+π‘ƒβ€˜π‘˜)(π‘“β€˜π‘Ÿ)) ∩ ((βŠ₯π‘ƒβ€˜π‘˜)β€˜{𝑑})) = ((π‘Ÿ + (π‘“β€˜π‘Ÿ)) ∩ ( βŠ₯ β€˜{𝑑})))
2522, 24eqeq12d 2744 . . . . . . . 8 (π‘˜ = 𝐾 β†’ (((π‘ž(+π‘ƒβ€˜π‘˜)(π‘“β€˜π‘ž)) ∩ ((βŠ₯π‘ƒβ€˜π‘˜)β€˜{𝑑})) = ((π‘Ÿ(+π‘ƒβ€˜π‘˜)(π‘“β€˜π‘Ÿ)) ∩ ((βŠ₯π‘ƒβ€˜π‘˜)β€˜{𝑑})) ↔ ((π‘ž + (π‘“β€˜π‘ž)) ∩ ( βŠ₯ β€˜{𝑑})) = ((π‘Ÿ + (π‘“β€˜π‘Ÿ)) ∩ ( βŠ₯ β€˜{𝑑}))))
2613, 25raleqbidv 3340 . . . . . . 7 (π‘˜ = 𝐾 β†’ (βˆ€π‘Ÿ ∈ ((WAtomsβ€˜π‘˜)β€˜π‘‘)((π‘ž(+π‘ƒβ€˜π‘˜)(π‘“β€˜π‘ž)) ∩ ((βŠ₯π‘ƒβ€˜π‘˜)β€˜{𝑑})) = ((π‘Ÿ(+π‘ƒβ€˜π‘˜)(π‘“β€˜π‘Ÿ)) ∩ ((βŠ₯π‘ƒβ€˜π‘˜)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}))))
2713, 26raleqbidv 3340 . . . . . 6 (π‘˜ = 𝐾 β†’ (βˆ€π‘ž ∈ ((WAtomsβ€˜π‘˜)β€˜π‘‘)βˆ€π‘Ÿ ∈ ((WAtomsβ€˜π‘˜)β€˜π‘‘)((π‘ž(+π‘ƒβ€˜π‘˜)(π‘“β€˜π‘ž)) ∩ ((βŠ₯π‘ƒβ€˜π‘˜)β€˜{𝑑})) = ((π‘Ÿ(+π‘ƒβ€˜π‘˜)(π‘“β€˜π‘Ÿ)) ∩ ((βŠ₯π‘ƒβ€˜π‘˜)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}))))
289, 27rabeqbidv 3448 . . . . 5 (π‘˜ = 𝐾 β†’ {𝑓 ∈ ((Dilβ€˜π‘˜)β€˜π‘‘) ∣ βˆ€π‘ž ∈ ((WAtomsβ€˜π‘˜)β€˜π‘‘)βˆ€π‘Ÿ ∈ ((WAtomsβ€˜π‘˜)β€˜π‘‘)((π‘ž(+π‘ƒβ€˜π‘˜)(π‘“β€˜π‘ž)) ∩ ((βŠ₯π‘ƒβ€˜π‘˜)β€˜{𝑑})) = ((π‘Ÿ(+π‘ƒβ€˜π‘˜)(π‘“β€˜π‘Ÿ)) ∩ ((βŠ₯π‘ƒβ€˜π‘˜)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}))})
295, 28mpteq12dv 5243 . . . 4 (π‘˜ = 𝐾 β†’ (𝑑 ∈ (Atomsβ€˜π‘˜) ↦ {𝑓 ∈ ((Dilβ€˜π‘˜)β€˜π‘‘) ∣ βˆ€π‘ž ∈ ((WAtomsβ€˜π‘˜)β€˜π‘‘)βˆ€π‘Ÿ ∈ ((WAtomsβ€˜π‘˜)β€˜π‘‘)((π‘ž(+π‘ƒβ€˜π‘˜)(π‘“β€˜π‘ž)) ∩ ((βŠ₯π‘ƒβ€˜π‘˜)β€˜{𝑑})) = ((π‘Ÿ(+π‘ƒβ€˜π‘˜)(π‘“β€˜π‘Ÿ)) ∩ ((βŠ₯π‘ƒβ€˜π‘˜)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}))}))
30 df-trnN 39620 . . . 4 Trn = (π‘˜ ∈ V ↦ (𝑑 ∈ (Atomsβ€˜π‘˜) ↦ {𝑓 ∈ ((Dilβ€˜π‘˜)β€˜π‘‘) ∣ βˆ€π‘ž ∈ ((WAtomsβ€˜π‘˜)β€˜π‘‘)βˆ€π‘Ÿ ∈ ((WAtomsβ€˜π‘˜)β€˜π‘‘)((π‘ž(+π‘ƒβ€˜π‘˜)(π‘“β€˜π‘ž)) ∩ ((βŠ₯π‘ƒβ€˜π‘˜)β€˜{𝑑})) = ((π‘Ÿ(+π‘ƒβ€˜π‘˜)(π‘“β€˜π‘Ÿ)) ∩ ((βŠ₯π‘ƒβ€˜π‘˜)β€˜{𝑑}))}))
3129, 30, 4mptfvmpt 7246 . . 3 (𝐾 ∈ V β†’ (Trnβ€˜πΎ) = (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}))}))
322, 31eqtrid 2780 . 2 (𝐾 ∈ V 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}))}))
331, 32syl 17 1 (𝐾 ∈ 𝐢 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}))}))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  βˆ€wral 3058  {crab 3430  Vcvv 3473   ∩ cin 3948  {csn 4632   ↦ cmpt 5235  β€˜cfv 6553  (class class class)co 7426  Atomscatm 38775  PSubSpcpsubsp 39009  +𝑃cpadd 39308  βŠ₯𝑃cpolN 39415  WAtomscwpointsN 39499  PAutcpautN 39500  DilcdilN 39615  TrnctrnN 39616
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 39620
This theorem is referenced by:  trnsetN  39669
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