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Theorem istrnN 36178
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 36177 . . 3 ((𝐾𝐵𝐷𝐴) → (𝑇𝐷) = {𝑓 ∈ (𝐿𝐷) ∣ ∀𝑞 ∈ (𝑊𝐷)∀𝑟 ∈ (𝑊𝐷)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝐷})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝐷}))})
109eleq2d 2864 . 2 ((𝐾𝐵𝐷𝐴) → (𝐹 ∈ (𝑇𝐷) ↔ 𝐹 ∈ {𝑓 ∈ (𝐿𝐷) ∣ ∀𝑞 ∈ (𝑊𝐷)∀𝑟 ∈ (𝑊𝐷)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝐷})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝐷}))}))
11 fveq1 6410 . . . . . . 7 (𝑓 = 𝐹 → (𝑓𝑞) = (𝐹𝑞))
1211oveq2d 6894 . . . . . 6 (𝑓 = 𝐹 → (𝑞 + (𝑓𝑞)) = (𝑞 + (𝐹𝑞)))
1312ineq1d 4011 . . . . 5 (𝑓 = 𝐹 → ((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝐷})) = ((𝑞 + (𝐹𝑞)) ∩ ( ‘{𝐷})))
14 fveq1 6410 . . . . . . 7 (𝑓 = 𝐹 → (𝑓𝑟) = (𝐹𝑟))
1514oveq2d 6894 . . . . . 6 (𝑓 = 𝐹 → (𝑟 + (𝑓𝑟)) = (𝑟 + (𝐹𝑟)))
1615ineq1d 4011 . . . . 5 (𝑓 = 𝐹 → ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝐷})) = ((𝑟 + (𝐹𝑟)) ∩ ( ‘{𝐷})))
1713, 16eqeq12d 2814 . . . 4 (𝑓 = 𝐹 → (((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝐷})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝐷})) ↔ ((𝑞 + (𝐹𝑞)) ∩ ( ‘{𝐷})) = ((𝑟 + (𝐹𝑟)) ∩ ( ‘{𝐷}))))
18172ralbidv 3170 . . 3 (𝑓 = 𝐹 → (∀𝑞 ∈ (𝑊𝐷)∀𝑟 ∈ (𝑊𝐷)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝐷})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝐷})) ↔ ∀𝑞 ∈ (𝑊𝐷)∀𝑟 ∈ (𝑊𝐷)((𝑞 + (𝐹𝑞)) ∩ ( ‘{𝐷})) = ((𝑟 + (𝐹𝑟)) ∩ ( ‘{𝐷}))))
1918elrab 3556 . 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 279 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 198  wa 385   = wceq 1653  wcel 2157  wral 3089  {crab 3093  cin 3768  {csn 4368  cfv 6101  (class class class)co 6878  Atomscatm 35284  PSubSpcpsubsp 35517  +𝑃cpadd 35816  𝑃cpolN 35923  WAtomscwpointsN 36007  PAutcpautN 36008  DilcdilN 36123  TrnctrnN 36124
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-trnN 36128
This theorem is referenced by: (None)
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