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Theorem trnsetN 37766
 Description: The set of translations for a fiducial atom 𝐷. (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
trnsetN ((𝐾𝐵𝐷𝐴) → (𝑇𝐷) = {𝑓 ∈ (𝐿𝐷) ∣ ∀𝑞 ∈ (𝑊𝐷)∀𝑟 ∈ (𝑊𝐷)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝐷})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝐷}))})
Distinct variable groups:   𝑓,𝑞,𝑟,𝐾   𝑓,𝐿   𝑊,𝑞,𝑟   𝐷,𝑓,𝑞,𝑟
Allowed substitution hints:   𝐴(𝑓,𝑟,𝑞)   𝐵(𝑓,𝑟,𝑞)   + (𝑓,𝑟,𝑞)   𝑆(𝑓,𝑟,𝑞)   𝑇(𝑓,𝑟,𝑞)   𝐿(𝑟,𝑞)   𝑀(𝑓,𝑟,𝑞)   (𝑓,𝑟,𝑞)   𝑊(𝑓)

Proof of Theorem trnsetN
Dummy variable 𝑑 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, 8trnfsetN 37765 . . 3 (𝐾𝐵𝑇 = (𝑑𝐴 ↦ {𝑓 ∈ (𝐿𝑑) ∣ ∀𝑞 ∈ (𝑊𝑑)∀𝑟 ∈ (𝑊𝑑)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝑑})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝑑}))}))
109fveq1d 6665 . 2 (𝐾𝐵 → (𝑇𝐷) = ((𝑑𝐴 ↦ {𝑓 ∈ (𝐿𝑑) ∣ ∀𝑞 ∈ (𝑊𝑑)∀𝑟 ∈ (𝑊𝑑)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝑑})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝑑}))})‘𝐷))
11 fveq2 6663 . . . 4 (𝑑 = 𝐷 → (𝐿𝑑) = (𝐿𝐷))
12 fveq2 6663 . . . . 5 (𝑑 = 𝐷 → (𝑊𝑑) = (𝑊𝐷))
13 sneq 4535 . . . . . . . . 9 (𝑑 = 𝐷 → {𝑑} = {𝐷})
1413fveq2d 6667 . . . . . . . 8 (𝑑 = 𝐷 → ( ‘{𝑑}) = ( ‘{𝐷}))
1514ineq2d 4119 . . . . . . 7 (𝑑 = 𝐷 → ((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝑑})) = ((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝐷})))
1614ineq2d 4119 . . . . . . 7 (𝑑 = 𝐷 → ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝑑})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝐷})))
1715, 16eqeq12d 2774 . . . . . 6 (𝑑 = 𝐷 → (((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝑑})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝑑})) ↔ ((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝐷})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝐷}))))
1812, 17raleqbidv 3319 . . . . 5 (𝑑 = 𝐷 → (∀𝑟 ∈ (𝑊𝑑)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝑑})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝑑})) ↔ ∀𝑟 ∈ (𝑊𝐷)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝐷})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝐷}))))
1912, 18raleqbidv 3319 . . . 4 (𝑑 = 𝐷 → (∀𝑞 ∈ (𝑊𝑑)∀𝑟 ∈ (𝑊𝑑)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝑑})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝑑})) ↔ ∀𝑞 ∈ (𝑊𝐷)∀𝑟 ∈ (𝑊𝐷)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝐷})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝐷}))))
2011, 19rabeqbidv 3398 . . 3 (𝑑 = 𝐷 → {𝑓 ∈ (𝐿𝑑) ∣ ∀𝑞 ∈ (𝑊𝑑)∀𝑟 ∈ (𝑊𝑑)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝑑})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝑑}))} = {𝑓 ∈ (𝐿𝐷) ∣ ∀𝑞 ∈ (𝑊𝐷)∀𝑟 ∈ (𝑊𝐷)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝐷})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝐷}))})
21 eqid 2758 . . 3 (𝑑𝐴 ↦ {𝑓 ∈ (𝐿𝑑) ∣ ∀𝑞 ∈ (𝑊𝑑)∀𝑟 ∈ (𝑊𝑑)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝑑})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝑑}))}) = (𝑑𝐴 ↦ {𝑓 ∈ (𝐿𝑑) ∣ ∀𝑞 ∈ (𝑊𝑑)∀𝑟 ∈ (𝑊𝑑)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝑑})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝑑}))})
22 fvex 6676 . . . 4 (𝐿𝐷) ∈ V
2322rabex 5206 . . 3 {𝑓 ∈ (𝐿𝐷) ∣ ∀𝑞 ∈ (𝑊𝐷)∀𝑟 ∈ (𝑊𝐷)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝐷})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝐷}))} ∈ V
2420, 21, 23fvmpt 6764 . 2 (𝐷𝐴 → ((𝑑𝐴 ↦ {𝑓 ∈ (𝐿𝑑) ∣ ∀𝑞 ∈ (𝑊𝑑)∀𝑟 ∈ (𝑊𝑑)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝑑})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝑑}))})‘𝐷) = {𝑓 ∈ (𝐿𝐷) ∣ ∀𝑞 ∈ (𝑊𝐷)∀𝑟 ∈ (𝑊𝐷)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝐷})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝐷}))})
2510, 24sylan9eq 2813 1 ((𝐾𝐵𝐷𝐴) → (𝑇𝐷) = {𝑓 ∈ (𝐿𝐷) ∣ ∀𝑞 ∈ (𝑊𝐷)∀𝑟 ∈ (𝑊𝐷)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝐷})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝐷}))})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070  {crab 3074   ∩ cin 3859  {csn 4525   ↦ cmpt 5116  ‘cfv 6340  (class class class)co 7156  Atomscatm 36873  PSubSpcpsubsp 37106  +𝑃cpadd 37405  ⊥𝑃cpolN 37512  WAtomscwpointsN 37596  PAutcpautN 37597  DilcdilN 37712  TrnctrnN 37713 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pr 5302 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7159  df-trnN 37717 This theorem is referenced by:  istrnN  37767
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