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Theorem istrnN 39856
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 39855 . . 3 ((𝐾𝐵𝐷𝐴) → (𝑇𝐷) = {𝑓 ∈ (𝐿𝐷) ∣ ∀𝑞 ∈ (𝑊𝐷)∀𝑟 ∈ (𝑊𝐷)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝐷})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝐷}))})
109eleq2d 2812 . 2 ((𝐾𝐵𝐷𝐴) → (𝐹 ∈ (𝑇𝐷) ↔ 𝐹 ∈ {𝑓 ∈ (𝐿𝐷) ∣ ∀𝑞 ∈ (𝑊𝐷)∀𝑟 ∈ (𝑊𝐷)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝐷})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝐷}))}))
11 fveq1 6900 . . . . . . 7 (𝑓 = 𝐹 → (𝑓𝑞) = (𝐹𝑞))
1211oveq2d 7440 . . . . . 6 (𝑓 = 𝐹 → (𝑞 + (𝑓𝑞)) = (𝑞 + (𝐹𝑞)))
1312ineq1d 4212 . . . . 5 (𝑓 = 𝐹 → ((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝐷})) = ((𝑞 + (𝐹𝑞)) ∩ ( ‘{𝐷})))
14 fveq1 6900 . . . . . . 7 (𝑓 = 𝐹 → (𝑓𝑟) = (𝐹𝑟))
1514oveq2d 7440 . . . . . 6 (𝑓 = 𝐹 → (𝑟 + (𝑓𝑟)) = (𝑟 + (𝐹𝑟)))
1615ineq1d 4212 . . . . 5 (𝑓 = 𝐹 → ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝐷})) = ((𝑟 + (𝐹𝑟)) ∩ ( ‘{𝐷})))
1713, 16eqeq12d 2742 . . . 4 (𝑓 = 𝐹 → (((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝐷})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝐷})) ↔ ((𝑞 + (𝐹𝑞)) ∩ ( ‘{𝐷})) = ((𝑟 + (𝐹𝑟)) ∩ ( ‘{𝐷}))))
18172ralbidv 3209 . . 3 (𝑓 = 𝐹 → (∀𝑞 ∈ (𝑊𝐷)∀𝑟 ∈ (𝑊𝐷)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝐷})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝐷})) ↔ ∀𝑞 ∈ (𝑊𝐷)∀𝑟 ∈ (𝑊𝐷)((𝑞 + (𝐹𝑞)) ∩ ( ‘{𝐷})) = ((𝑟 + (𝐹𝑟)) ∩ ( ‘{𝐷}))))
1918elrab 3681 . 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, 19bitrdi 286 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 205  wa 394   = wceq 1534  wcel 2099  wral 3051  {crab 3419  cin 3946  {csn 4633  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: (None)
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