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Theorem trnsetN 34260
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 34259 . . 3 (𝐾𝐵𝑇 = (𝑑𝐴 ↦ {𝑓 ∈ (𝐿𝑑) ∣ ∀𝑞 ∈ (𝑊𝑑)∀𝑟 ∈ (𝑊𝑑)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝑑})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝑑}))}))
109fveq1d 6086 . 2 (𝐾𝐵 → (𝑇𝐷) = ((𝑑𝐴 ↦ {𝑓 ∈ (𝐿𝑑) ∣ ∀𝑞 ∈ (𝑊𝑑)∀𝑟 ∈ (𝑊𝑑)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝑑})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝑑}))})‘𝐷))
11 fveq2 6084 . . . 4 (𝑑 = 𝐷 → (𝐿𝑑) = (𝐿𝐷))
12 fveq2 6084 . . . . 5 (𝑑 = 𝐷 → (𝑊𝑑) = (𝑊𝐷))
13 sneq 4130 . . . . . . . . 9 (𝑑 = 𝐷 → {𝑑} = {𝐷})
1413fveq2d 6088 . . . . . . . 8 (𝑑 = 𝐷 → ( ‘{𝑑}) = ( ‘{𝐷}))
1514ineq2d 3771 . . . . . . 7 (𝑑 = 𝐷 → ((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝑑})) = ((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝐷})))
1614ineq2d 3771 . . . . . . 7 (𝑑 = 𝐷 → ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝑑})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝐷})))
1715, 16eqeq12d 2620 . . . . . 6 (𝑑 = 𝐷 → (((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝑑})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝑑})) ↔ ((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝐷})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝐷}))))
1812, 17raleqbidv 3124 . . . . 5 (𝑑 = 𝐷 → (∀𝑟 ∈ (𝑊𝑑)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝑑})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝑑})) ↔ ∀𝑟 ∈ (𝑊𝐷)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝐷})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝐷}))))
1912, 18raleqbidv 3124 . . . 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 3163 . . 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 2605 . . 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 6094 . . . 4 (𝐿𝐷) ∈ V
2322rabex 4731 . . 3 {𝑓 ∈ (𝐿𝐷) ∣ ∀𝑞 ∈ (𝑊𝐷)∀𝑟 ∈ (𝑊𝐷)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝐷})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝐷}))} ∈ V
2420, 21, 23fvmpt 6172 . 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 2659 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 382   = wceq 1474  wcel 1975  wral 2891  {crab 2895  cin 3534  {csn 4120  cmpt 4633  cfv 5786  (class class class)co 6523  Atomscatm 33367  PSubSpcpsubsp 33599  +𝑃cpadd 33898  𝑃cpolN 34005  WAtomscwpointsN 34089  PAutcpautN 34090  DilcdilN 34205  TrnctrnN 34206
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pr 4824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-reu 2898  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-ov 6526  df-trnN 34210
This theorem is referenced by:  istrnN  34261
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