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Theorem istrnN 34462
 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 34461 . . 3 ((𝐾𝐵𝐷𝐴) → (𝑇𝐷) = {𝑓 ∈ (𝐿𝐷) ∣ ∀𝑞 ∈ (𝑊𝐷)∀𝑟 ∈ (𝑊𝐷)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝐷})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝐷}))})
109eleq2d 2673 . 2 ((𝐾𝐵𝐷𝐴) → (𝐹 ∈ (𝑇𝐷) ↔ 𝐹 ∈ {𝑓 ∈ (𝐿𝐷) ∣ ∀𝑞 ∈ (𝑊𝐷)∀𝑟 ∈ (𝑊𝐷)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝐷})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝐷}))}))
11 fveq1 6102 . . . . . . 7 (𝑓 = 𝐹 → (𝑓𝑞) = (𝐹𝑞))
1211oveq2d 6565 . . . . . 6 (𝑓 = 𝐹 → (𝑞 + (𝑓𝑞)) = (𝑞 + (𝐹𝑞)))
1312ineq1d 3775 . . . . 5 (𝑓 = 𝐹 → ((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝐷})) = ((𝑞 + (𝐹𝑞)) ∩ ( ‘{𝐷})))
14 fveq1 6102 . . . . . . 7 (𝑓 = 𝐹 → (𝑓𝑟) = (𝐹𝑟))
1514oveq2d 6565 . . . . . 6 (𝑓 = 𝐹 → (𝑟 + (𝑓𝑟)) = (𝑟 + (𝐹𝑟)))
1615ineq1d 3775 . . . . 5 (𝑓 = 𝐹 → ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝐷})) = ((𝑟 + (𝐹𝑟)) ∩ ( ‘{𝐷})))
1713, 16eqeq12d 2625 . . . 4 (𝑓 = 𝐹 → (((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝐷})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝐷})) ↔ ((𝑞 + (𝐹𝑞)) ∩ ( ‘{𝐷})) = ((𝑟 + (𝐹𝑟)) ∩ ( ‘{𝐷}))))
18172ralbidv 2972 . . 3 (𝑓 = 𝐹 → (∀𝑞 ∈ (𝑊𝐷)∀𝑟 ∈ (𝑊𝐷)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝐷})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝐷})) ↔ ∀𝑞 ∈ (𝑊𝐷)∀𝑟 ∈ (𝑊𝐷)((𝑞 + (𝐹𝑞)) ∩ ( ‘{𝐷})) = ((𝑟 + (𝐹𝑟)) ∩ ( ‘{𝐷}))))
1918elrab 3331 . 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 275 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 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900   ∩ cin 3539  {csn 4125  ‘cfv 5804  (class class class)co 6549  Atomscatm 33568  PSubSpcpsubsp 33800  +𝑃cpadd 34099  ⊥𝑃cpolN 34206  WAtomscwpointsN 34290  PAutcpautN 34291  DilcdilN 34406  TrnctrnN 34407 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-trnN 34411 This theorem is referenced by: (None)
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