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Theorem istrnN 37353
Description: The predicate "is a translation". (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
istrnN ((𝐾𝐵𝐷𝐴) → (𝐹 ∈ (𝑇𝐷) ↔ (𝐹 ∈ (𝐿𝐷) ∧ ∀𝑞 ∈ (𝑊𝐷)∀𝑟 ∈ (𝑊𝐷)((𝑞 + (𝐹𝑞)) ∩ ( ‘{𝐷})) = ((𝑟 + (𝐹𝑟)) ∩ ( ‘{𝐷})))))
Distinct variable groups:   𝑟,𝑞,𝐾   𝑊,𝑞,𝑟   𝐷,𝑞,𝑟   𝐹,𝑞,𝑟
Allowed substitution hints:   𝐴(𝑟,𝑞)   𝐵(𝑟,𝑞)   + (𝑟,𝑞)   𝑆(𝑟,𝑞)   𝑇(𝑟,𝑞)   𝐿(𝑟,𝑞)   𝑀(𝑟,𝑞)   (𝑟,𝑞)

Proof of Theorem istrnN
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 trnset.a . . . 4 𝐴 = (Atoms‘𝐾)
2 trnset.s . . . 4 𝑆 = (PSubSp‘𝐾)
3 trnset.p . . . 4 + = (+𝑃𝐾)
4 trnset.o . . . 4 = (⊥𝑃𝐾)
5 trnset.w . . . 4 𝑊 = (WAtoms‘𝐾)
6 trnset.m . . . 4 𝑀 = (PAut‘𝐾)
7 trnset.l . . . 4 𝐿 = (Dil‘𝐾)
8 trnset.t . . . 4 𝑇 = (Trn‘𝐾)
91, 2, 3, 4, 5, 6, 7, 8trnsetN 37352 . . 3 ((𝐾𝐵𝐷𝐴) → (𝑇𝐷) = {𝑓 ∈ (𝐿𝐷) ∣ ∀𝑞 ∈ (𝑊𝐷)∀𝑟 ∈ (𝑊𝐷)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝐷})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝐷}))})
109eleq2d 2901 . 2 ((𝐾𝐵𝐷𝐴) → (𝐹 ∈ (𝑇𝐷) ↔ 𝐹 ∈ {𝑓 ∈ (𝐿𝐷) ∣ ∀𝑞 ∈ (𝑊𝐷)∀𝑟 ∈ (𝑊𝐷)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝐷})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝐷}))}))
11 fveq1 6650 . . . . . . 7 (𝑓 = 𝐹 → (𝑓𝑞) = (𝐹𝑞))
1211oveq2d 7154 . . . . . 6 (𝑓 = 𝐹 → (𝑞 + (𝑓𝑞)) = (𝑞 + (𝐹𝑞)))
1312ineq1d 4171 . . . . 5 (𝑓 = 𝐹 → ((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝐷})) = ((𝑞 + (𝐹𝑞)) ∩ ( ‘{𝐷})))
14 fveq1 6650 . . . . . . 7 (𝑓 = 𝐹 → (𝑓𝑟) = (𝐹𝑟))
1514oveq2d 7154 . . . . . 6 (𝑓 = 𝐹 → (𝑟 + (𝑓𝑟)) = (𝑟 + (𝐹𝑟)))
1615ineq1d 4171 . . . . 5 (𝑓 = 𝐹 → ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝐷})) = ((𝑟 + (𝐹𝑟)) ∩ ( ‘{𝐷})))
1713, 16eqeq12d 2840 . . . 4 (𝑓 = 𝐹 → (((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝐷})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝐷})) ↔ ((𝑞 + (𝐹𝑞)) ∩ ( ‘{𝐷})) = ((𝑟 + (𝐹𝑟)) ∩ ( ‘{𝐷}))))
18172ralbidv 3193 . . 3 (𝑓 = 𝐹 → (∀𝑞 ∈ (𝑊𝐷)∀𝑟 ∈ (𝑊𝐷)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝐷})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝐷})) ↔ ∀𝑞 ∈ (𝑊𝐷)∀𝑟 ∈ (𝑊𝐷)((𝑞 + (𝐹𝑞)) ∩ ( ‘{𝐷})) = ((𝑟 + (𝐹𝑟)) ∩ ( ‘{𝐷}))))
1918elrab 3665 . 2 (𝐹 ∈ {𝑓 ∈ (𝐿𝐷) ∣ ∀𝑞 ∈ (𝑊𝐷)∀𝑟 ∈ (𝑊𝐷)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝐷})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝐷}))} ↔ (𝐹 ∈ (𝐿𝐷) ∧ ∀𝑞 ∈ (𝑊𝐷)∀𝑟 ∈ (𝑊𝐷)((𝑞 + (𝐹𝑞)) ∩ ( ‘{𝐷})) = ((𝑟 + (𝐹𝑟)) ∩ ( ‘{𝐷}))))
2010, 19syl6bb 290 1 ((𝐾𝐵𝐷𝐴) → (𝐹 ∈ (𝑇𝐷) ↔ (𝐹 ∈ (𝐿𝐷) ∧ ∀𝑞 ∈ (𝑊𝐷)∀𝑟 ∈ (𝑊𝐷)((𝑞 + (𝐹𝑞)) ∩ ( ‘{𝐷})) = ((𝑟 + (𝐹𝑟)) ∩ ( ‘{𝐷})))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wral 3132  {crab 3136  cin 3917  {csn 4548  cfv 6336  (class class class)co 7138  Atomscatm 36459  PSubSpcpsubsp 36692  +𝑃cpadd 36991  𝑃cpolN 37098  WAtomscwpointsN 37182  PAutcpautN 37183  DilcdilN 37298  TrnctrnN 37299
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5171  ax-sep 5184  ax-nul 5191  ax-pr 5311
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-reu 3139  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-iun 4902  df-br 5048  df-opab 5110  df-mpt 5128  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-ov 7141  df-trnN 37303
This theorem is referenced by: (None)
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