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Theorem trnfsetN 37450
 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 3462 . 2 (𝐾𝐶𝐾 ∈ V)
2 trnset.t . . 3 𝑇 = (Trn‘𝐾)
3 fveq2 6649 . . . . . 6 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4 trnset.a . . . . . 6 𝐴 = (Atoms‘𝐾)
53, 4eqtr4di 2854 . . . . 5 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
6 fveq2 6649 . . . . . . . 8 (𝑘 = 𝐾 → (Dil‘𝑘) = (Dil‘𝐾))
7 trnset.l . . . . . . . 8 𝐿 = (Dil‘𝐾)
86, 7eqtr4di 2854 . . . . . . 7 (𝑘 = 𝐾 → (Dil‘𝑘) = 𝐿)
98fveq1d 6651 . . . . . 6 (𝑘 = 𝐾 → ((Dil‘𝑘)‘𝑑) = (𝐿𝑑))
10 fveq2 6649 . . . . . . . . 9 (𝑘 = 𝐾 → (WAtoms‘𝑘) = (WAtoms‘𝐾))
11 trnset.w . . . . . . . . 9 𝑊 = (WAtoms‘𝐾)
1210, 11eqtr4di 2854 . . . . . . . 8 (𝑘 = 𝐾 → (WAtoms‘𝑘) = 𝑊)
1312fveq1d 6651 . . . . . . 7 (𝑘 = 𝐾 → ((WAtoms‘𝑘)‘𝑑) = (𝑊𝑑))
14 fveq2 6649 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (+𝑃𝑘) = (+𝑃𝐾))
15 trnset.p . . . . . . . . . . . 12 + = (+𝑃𝐾)
1614, 15eqtr4di 2854 . . . . . . . . . . 11 (𝑘 = 𝐾 → (+𝑃𝑘) = + )
1716oveqd 7156 . . . . . . . . . 10 (𝑘 = 𝐾 → (𝑞(+𝑃𝑘)(𝑓𝑞)) = (𝑞 + (𝑓𝑞)))
18 fveq2 6649 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (⊥𝑃𝑘) = (⊥𝑃𝐾))
19 trnset.o . . . . . . . . . . . 12 = (⊥𝑃𝐾)
2018, 19eqtr4di 2854 . . . . . . . . . . 11 (𝑘 = 𝐾 → (⊥𝑃𝑘) = )
2120fveq1d 6651 . . . . . . . . . 10 (𝑘 = 𝐾 → ((⊥𝑃𝑘)‘{𝑑}) = ( ‘{𝑑}))
2217, 21ineq12d 4143 . . . . . . . . 9 (𝑘 = 𝐾 → ((𝑞(+𝑃𝑘)(𝑓𝑞)) ∩ ((⊥𝑃𝑘)‘{𝑑})) = ((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝑑})))
2316oveqd 7156 . . . . . . . . . 10 (𝑘 = 𝐾 → (𝑟(+𝑃𝑘)(𝑓𝑟)) = (𝑟 + (𝑓𝑟)))
2423, 21ineq12d 4143 . . . . . . . . 9 (𝑘 = 𝐾 → ((𝑟(+𝑃𝑘)(𝑓𝑟)) ∩ ((⊥𝑃𝑘)‘{𝑑})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝑑})))
2522, 24eqeq12d 2817 . . . . . . . 8 (𝑘 = 𝐾 → (((𝑞(+𝑃𝑘)(𝑓𝑞)) ∩ ((⊥𝑃𝑘)‘{𝑑})) = ((𝑟(+𝑃𝑘)(𝑓𝑟)) ∩ ((⊥𝑃𝑘)‘{𝑑})) ↔ ((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝑑})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝑑}))))
2613, 25raleqbidv 3357 . . . . . . 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 3357 . . . . . 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 3436 . . . . 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 5118 . . . 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 37402 . . . 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 6972 . . 3 (𝐾 ∈ V → (Trn‘𝐾) = (𝑑𝐴 ↦ {𝑓 ∈ (𝐿𝑑) ∣ ∀𝑞 ∈ (𝑊𝑑)∀𝑟 ∈ (𝑊𝑑)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝑑})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝑑}))}))
322, 31syl5eq 2848 . 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 1538   ∈ wcel 2112  ∀wral 3109  {crab 3113  Vcvv 3444   ∩ cin 3883  {csn 4528   ↦ cmpt 5113  ‘cfv 6328  (class class class)co 7139  Atomscatm 36558  PSubSpcpsubsp 36791  +𝑃cpadd 37090  ⊥𝑃cpolN 37197  WAtomscwpointsN 37281  PAutcpautN 37282  DilcdilN 37397  TrnctrnN 37398 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-trnN 37402 This theorem is referenced by:  trnsetN  37451
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