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Theorem trnfsetN 39854
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 = (Atoms‘𝐾)
trnset.s 𝑆 = (PSubSp‘𝐾)
trnset.p + = (+𝑃𝐾)
trnset.o = (⊥𝑃𝐾)
trnset.w 𝑊 = (WAtoms‘𝐾)
trnset.m 𝑀 = (PAut‘𝐾)
trnset.l 𝐿 = (Dil‘𝐾)
trnset.t 𝑇 = (Trn‘𝐾)
Assertion
Ref Expression
trnfsetN (𝐾𝐶𝑇 = (𝑑𝐴 ↦ {𝑓 ∈ (𝐿𝑑) ∣ ∀𝑞 ∈ (𝑊𝑑)∀𝑟 ∈ (𝑊𝑑)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝑑})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝑑}))}))
Distinct variable groups:   𝐴,𝑑   𝑓,𝑑,𝑞,𝑟,𝐾   𝑓,𝐿   𝑊,𝑞,𝑟
Allowed substitution hints:   𝐴(𝑓,𝑟,𝑞)   𝐶(𝑓,𝑟,𝑞,𝑑)   + (𝑓,𝑟,𝑞,𝑑)   𝑆(𝑓,𝑟,𝑞,𝑑)   𝑇(𝑓,𝑟,𝑞,𝑑)   𝐿(𝑟,𝑞,𝑑)   𝑀(𝑓,𝑟,𝑞,𝑑)   (𝑓,𝑟,𝑞,𝑑)   𝑊(𝑓,𝑑)

Proof of Theorem trnfsetN
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 elex 3482 . 2 (𝐾𝐶𝐾 ∈ V)
2 trnset.t . . 3 𝑇 = (Trn‘𝐾)
3 fveq2 6901 . . . . . 6 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4 trnset.a . . . . . 6 𝐴 = (Atoms‘𝐾)
53, 4eqtr4di 2784 . . . . 5 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
6 fveq2 6901 . . . . . . . 8 (𝑘 = 𝐾 → (Dil‘𝑘) = (Dil‘𝐾))
7 trnset.l . . . . . . . 8 𝐿 = (Dil‘𝐾)
86, 7eqtr4di 2784 . . . . . . 7 (𝑘 = 𝐾 → (Dil‘𝑘) = 𝐿)
98fveq1d 6903 . . . . . 6 (𝑘 = 𝐾 → ((Dil‘𝑘)‘𝑑) = (𝐿𝑑))
10 fveq2 6901 . . . . . . . . 9 (𝑘 = 𝐾 → (WAtoms‘𝑘) = (WAtoms‘𝐾))
11 trnset.w . . . . . . . . 9 𝑊 = (WAtoms‘𝐾)
1210, 11eqtr4di 2784 . . . . . . . 8 (𝑘 = 𝐾 → (WAtoms‘𝑘) = 𝑊)
1312fveq1d 6903 . . . . . . 7 (𝑘 = 𝐾 → ((WAtoms‘𝑘)‘𝑑) = (𝑊𝑑))
14 fveq2 6901 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (+𝑃𝑘) = (+𝑃𝐾))
15 trnset.p . . . . . . . . . . . 12 + = (+𝑃𝐾)
1614, 15eqtr4di 2784 . . . . . . . . . . 11 (𝑘 = 𝐾 → (+𝑃𝑘) = + )
1716oveqd 7441 . . . . . . . . . 10 (𝑘 = 𝐾 → (𝑞(+𝑃𝑘)(𝑓𝑞)) = (𝑞 + (𝑓𝑞)))
18 fveq2 6901 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (⊥𝑃𝑘) = (⊥𝑃𝐾))
19 trnset.o . . . . . . . . . . . 12 = (⊥𝑃𝐾)
2018, 19eqtr4di 2784 . . . . . . . . . . 11 (𝑘 = 𝐾 → (⊥𝑃𝑘) = )
2120fveq1d 6903 . . . . . . . . . 10 (𝑘 = 𝐾 → ((⊥𝑃𝑘)‘{𝑑}) = ( ‘{𝑑}))
2217, 21ineq12d 4214 . . . . . . . . 9 (𝑘 = 𝐾 → ((𝑞(+𝑃𝑘)(𝑓𝑞)) ∩ ((⊥𝑃𝑘)‘{𝑑})) = ((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝑑})))
2316oveqd 7441 . . . . . . . . . 10 (𝑘 = 𝐾 → (𝑟(+𝑃𝑘)(𝑓𝑟)) = (𝑟 + (𝑓𝑟)))
2423, 21ineq12d 4214 . . . . . . . . 9 (𝑘 = 𝐾 → ((𝑟(+𝑃𝑘)(𝑓𝑟)) ∩ ((⊥𝑃𝑘)‘{𝑑})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝑑})))
2522, 24eqeq12d 2742 . . . . . . . 8 (𝑘 = 𝐾 → (((𝑞(+𝑃𝑘)(𝑓𝑞)) ∩ ((⊥𝑃𝑘)‘{𝑑})) = ((𝑟(+𝑃𝑘)(𝑓𝑟)) ∩ ((⊥𝑃𝑘)‘{𝑑})) ↔ ((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝑑})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝑑}))))
2613, 25raleqbidv 3330 . . . . . . 7 (𝑘 = 𝐾 → (∀𝑟 ∈ ((WAtoms‘𝑘)‘𝑑)((𝑞(+𝑃𝑘)(𝑓𝑞)) ∩ ((⊥𝑃𝑘)‘{𝑑})) = ((𝑟(+𝑃𝑘)(𝑓𝑟)) ∩ ((⊥𝑃𝑘)‘{𝑑})) ↔ ∀𝑟 ∈ (𝑊𝑑)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝑑})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝑑}))))
2713, 26raleqbidv 3330 . . . . . 6 (𝑘 = 𝐾 → (∀𝑞 ∈ ((WAtoms‘𝑘)‘𝑑)∀𝑟 ∈ ((WAtoms‘𝑘)‘𝑑)((𝑞(+𝑃𝑘)(𝑓𝑞)) ∩ ((⊥𝑃𝑘)‘{𝑑})) = ((𝑟(+𝑃𝑘)(𝑓𝑟)) ∩ ((⊥𝑃𝑘)‘{𝑑})) ↔ ∀𝑞 ∈ (𝑊𝑑)∀𝑟 ∈ (𝑊𝑑)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝑑})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝑑}))))
289, 27rabeqbidv 3437 . . . . 5 (𝑘 = 𝐾 → {𝑓 ∈ ((Dil‘𝑘)‘𝑑) ∣ ∀𝑞 ∈ ((WAtoms‘𝑘)‘𝑑)∀𝑟 ∈ ((WAtoms‘𝑘)‘𝑑)((𝑞(+𝑃𝑘)(𝑓𝑞)) ∩ ((⊥𝑃𝑘)‘{𝑑})) = ((𝑟(+𝑃𝑘)(𝑓𝑟)) ∩ ((⊥𝑃𝑘)‘{𝑑}))} = {𝑓 ∈ (𝐿𝑑) ∣ ∀𝑞 ∈ (𝑊𝑑)∀𝑟 ∈ (𝑊𝑑)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝑑})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝑑}))})
295, 28mpteq12dv 5244 . . . 4 (𝑘 = 𝐾 → (𝑑 ∈ (Atoms‘𝑘) ↦ {𝑓 ∈ ((Dil‘𝑘)‘𝑑) ∣ ∀𝑞 ∈ ((WAtoms‘𝑘)‘𝑑)∀𝑟 ∈ ((WAtoms‘𝑘)‘𝑑)((𝑞(+𝑃𝑘)(𝑓𝑞)) ∩ ((⊥𝑃𝑘)‘{𝑑})) = ((𝑟(+𝑃𝑘)(𝑓𝑟)) ∩ ((⊥𝑃𝑘)‘{𝑑}))}) = (𝑑𝐴 ↦ {𝑓 ∈ (𝐿𝑑) ∣ ∀𝑞 ∈ (𝑊𝑑)∀𝑟 ∈ (𝑊𝑑)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝑑})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝑑}))}))
30 df-trnN 39806 . . . 4 Trn = (𝑘 ∈ V ↦ (𝑑 ∈ (Atoms‘𝑘) ↦ {𝑓 ∈ ((Dil‘𝑘)‘𝑑) ∣ ∀𝑞 ∈ ((WAtoms‘𝑘)‘𝑑)∀𝑟 ∈ ((WAtoms‘𝑘)‘𝑑)((𝑞(+𝑃𝑘)(𝑓𝑞)) ∩ ((⊥𝑃𝑘)‘{𝑑})) = ((𝑟(+𝑃𝑘)(𝑓𝑟)) ∩ ((⊥𝑃𝑘)‘{𝑑}))}))
3129, 30, 4mptfvmpt 7245 . . 3 (𝐾 ∈ V → (Trn‘𝐾) = (𝑑𝐴 ↦ {𝑓 ∈ (𝐿𝑑) ∣ ∀𝑞 ∈ (𝑊𝑑)∀𝑟 ∈ (𝑊𝑑)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝑑})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝑑}))}))
322, 31eqtrid 2778 . 2 (𝐾 ∈ V → 𝑇 = (𝑑𝐴 ↦ {𝑓 ∈ (𝐿𝑑) ∣ ∀𝑞 ∈ (𝑊𝑑)∀𝑟 ∈ (𝑊𝑑)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝑑})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝑑}))}))
331, 32syl 17 1 (𝐾𝐶𝑇 = (𝑑𝐴 ↦ {𝑓 ∈ (𝐿𝑑) ∣ ∀𝑞 ∈ (𝑊𝑑)∀𝑟 ∈ (𝑊𝑑)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝑑})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝑑}))}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  wral 3051  {crab 3419  Vcvv 3462  cin 3946  {csn 4633  cmpt 5236  cfv 6554  (class class class)co 7424  Atomscatm 38961  PSubSpcpsubsp 39195  +𝑃cpadd 39494  𝑃cpolN 39601  WAtomscwpointsN 39685  PAutcpautN 39686  DilcdilN 39801  TrnctrnN 39802
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 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  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 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-ov 7427  df-trnN 39806
This theorem is referenced by:  trnsetN  39855
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