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Theorem trnfsetN 36318
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 3414 . 2 (𝐾𝐶𝐾 ∈ V)
2 trnset.t . . 3 𝑇 = (Trn‘𝐾)
3 fveq2 6448 . . . . . 6 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4 trnset.a . . . . . 6 𝐴 = (Atoms‘𝐾)
53, 4syl6eqr 2832 . . . . 5 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
6 fveq2 6448 . . . . . . . 8 (𝑘 = 𝐾 → (Dil‘𝑘) = (Dil‘𝐾))
7 trnset.l . . . . . . . 8 𝐿 = (Dil‘𝐾)
86, 7syl6eqr 2832 . . . . . . 7 (𝑘 = 𝐾 → (Dil‘𝑘) = 𝐿)
98fveq1d 6450 . . . . . 6 (𝑘 = 𝐾 → ((Dil‘𝑘)‘𝑑) = (𝐿𝑑))
10 fveq2 6448 . . . . . . . . 9 (𝑘 = 𝐾 → (WAtoms‘𝑘) = (WAtoms‘𝐾))
11 trnset.w . . . . . . . . 9 𝑊 = (WAtoms‘𝐾)
1210, 11syl6eqr 2832 . . . . . . . 8 (𝑘 = 𝐾 → (WAtoms‘𝑘) = 𝑊)
1312fveq1d 6450 . . . . . . 7 (𝑘 = 𝐾 → ((WAtoms‘𝑘)‘𝑑) = (𝑊𝑑))
14 fveq2 6448 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (+𝑃𝑘) = (+𝑃𝐾))
15 trnset.p . . . . . . . . . . . 12 + = (+𝑃𝐾)
1614, 15syl6eqr 2832 . . . . . . . . . . 11 (𝑘 = 𝐾 → (+𝑃𝑘) = + )
1716oveqd 6941 . . . . . . . . . 10 (𝑘 = 𝐾 → (𝑞(+𝑃𝑘)(𝑓𝑞)) = (𝑞 + (𝑓𝑞)))
18 fveq2 6448 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (⊥𝑃𝑘) = (⊥𝑃𝐾))
19 trnset.o . . . . . . . . . . . 12 = (⊥𝑃𝐾)
2018, 19syl6eqr 2832 . . . . . . . . . . 11 (𝑘 = 𝐾 → (⊥𝑃𝑘) = )
2120fveq1d 6450 . . . . . . . . . 10 (𝑘 = 𝐾 → ((⊥𝑃𝑘)‘{𝑑}) = ( ‘{𝑑}))
2217, 21ineq12d 4038 . . . . . . . . 9 (𝑘 = 𝐾 → ((𝑞(+𝑃𝑘)(𝑓𝑞)) ∩ ((⊥𝑃𝑘)‘{𝑑})) = ((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝑑})))
2316oveqd 6941 . . . . . . . . . 10 (𝑘 = 𝐾 → (𝑟(+𝑃𝑘)(𝑓𝑟)) = (𝑟 + (𝑓𝑟)))
2423, 21ineq12d 4038 . . . . . . . . 9 (𝑘 = 𝐾 → ((𝑟(+𝑃𝑘)(𝑓𝑟)) ∩ ((⊥𝑃𝑘)‘{𝑑})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝑑})))
2522, 24eqeq12d 2793 . . . . . . . 8 (𝑘 = 𝐾 → (((𝑞(+𝑃𝑘)(𝑓𝑞)) ∩ ((⊥𝑃𝑘)‘{𝑑})) = ((𝑟(+𝑃𝑘)(𝑓𝑟)) ∩ ((⊥𝑃𝑘)‘{𝑑})) ↔ ((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝑑})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝑑}))))
2613, 25raleqbidv 3326 . . . . . . 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 3326 . . . . . 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 3392 . . . . 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 4971 . . . 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 36270 . . . 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 6764 . . 3 (𝐾 ∈ V → (Trn‘𝐾) = (𝑑𝐴 ↦ {𝑓 ∈ (𝐿𝑑) ∣ ∀𝑞 ∈ (𝑊𝑑)∀𝑟 ∈ (𝑊𝑑)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝑑})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝑑}))}))
322, 31syl5eq 2826 . 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 1601  wcel 2107  wral 3090  {crab 3094  Vcvv 3398  cin 3791  {csn 4398  cmpt 4967  cfv 6137  (class class class)co 6924  Atomscatm 35426  PSubSpcpsubsp 35659  +𝑃cpadd 35958  𝑃cpolN 36065  WAtomscwpointsN 36149  PAutcpautN 36150  DilcdilN 36265  TrnctrnN 36266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pr 5140
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-ov 6927  df-trnN 36270
This theorem is referenced by:  trnsetN  36319
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