Theorem trnsetN 40426

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	⊢ + = (+𝑃‘𝐾)
trnset.o	⊢ ⊥ = (⊥𝑃‘𝐾)
trnset.w	⊢ 𝑊 = (WAtoms‘𝐾)
trnset.m	⊢ 𝑀 = (PAut‘𝐾)
trnset.l	⊢ 𝐿 = (Dil‘𝐾)
trnset.t	⊢ 𝑇 = (Trn‘𝐾)

Assertion

Ref	Expression
trnsetN	⊢ ((𝐾 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴) → (𝑇‘𝐷) = {𝑓 ∈ (𝐿‘𝐷) ∣ ∀𝑞 ∈ (𝑊‘𝐷)∀𝑟 ∈ (𝑊‘𝐷)((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝐷})) = ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{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.

Step	Hyp	Ref	Expression
1		trnset.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
2		trnset.s	. . . 4 ⊢ 𝑆 = (PSubSp‘𝐾)
3		trnset.p	. . . 4 ⊢ + = (+𝑃‘𝐾)
4		trnset.o	. . . 4 ⊢ ⊥ = (⊥𝑃‘𝐾)
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)
9	1, 2, 3, 4, 5, 6, 7, 8	trnfsetN 40425	. . 3 ⊢ (𝐾 ∈ 𝐵 → 𝑇 = (𝑑 ∈ 𝐴 ↦ {𝑓 ∈ (𝐿‘𝑑) ∣ ∀𝑞 ∈ (𝑊‘𝑑)∀𝑟 ∈ (𝑊‘𝑑)((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝑑})) = ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝑑}))}))
10	9	fveq1d 6836	. 2 ⊢ (𝐾 ∈ 𝐵 → (𝑇‘𝐷) = ((𝑑 ∈ 𝐴 ↦ {𝑓 ∈ (𝐿‘𝑑) ∣ ∀𝑞 ∈ (𝑊‘𝑑)∀𝑟 ∈ (𝑊‘𝑑)((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝑑})) = ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝑑}))})‘𝐷))
11		fveq2 6834	. . . 4 ⊢ (𝑑 = 𝐷 → (𝐿‘𝑑) = (𝐿‘𝐷))
12		fveq2 6834	. . . . 5 ⊢ (𝑑 = 𝐷 → (𝑊‘𝑑) = (𝑊‘𝐷))
13		sneq 4590	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → {𝑑} = {𝐷})
14	13	fveq2d 6838	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → ( ⊥ ‘{𝑑}) = ( ⊥ ‘{𝐷}))
15	14	ineq2d 4172	. . . . . . 7 ⊢ (𝑑 = 𝐷 → ((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝑑})) = ((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝐷})))
16	14	ineq2d 4172	. . . . . . 7 ⊢ (𝑑 = 𝐷 → ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝑑})) = ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝐷})))
17	15, 16	eqeq12d 2752	. . . . . 6 ⊢ (𝑑 = 𝐷 → (((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝑑})) = ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝑑})) ↔ ((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝐷})) = ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝐷}))))
18	12, 17	raleqbidv 3316	. . . . 5 ⊢ (𝑑 = 𝐷 → (∀𝑟 ∈ (𝑊‘𝑑)((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝑑})) = ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝑑})) ↔ ∀𝑟 ∈ (𝑊‘𝐷)((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝐷})) = ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝐷}))))
19	12, 18	raleqbidv 3316	. . . 4 ⊢ (𝑑 = 𝐷 → (∀𝑞 ∈ (𝑊‘𝑑)∀𝑟 ∈ (𝑊‘𝑑)((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝑑})) = ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝑑})) ↔ ∀𝑞 ∈ (𝑊‘𝐷)∀𝑟 ∈ (𝑊‘𝐷)((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝐷})) = ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝐷}))))
20	11, 19	rabeqbidv 3417	. . 3 ⊢ (𝑑 = 𝐷 → {𝑓 ∈ (𝐿‘𝑑) ∣ ∀𝑞 ∈ (𝑊‘𝑑)∀𝑟 ∈ (𝑊‘𝑑)((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝑑})) = ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝑑}))} = {𝑓 ∈ (𝐿‘𝐷) ∣ ∀𝑞 ∈ (𝑊‘𝐷)∀𝑟 ∈ (𝑊‘𝐷)((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝐷})) = ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝐷}))})
21		eqid 2736	. . 3 ⊢ (𝑑 ∈ 𝐴 ↦ {𝑓 ∈ (𝐿‘𝑑) ∣ ∀𝑞 ∈ (𝑊‘𝑑)∀𝑟 ∈ (𝑊‘𝑑)((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝑑})) = ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝑑}))}) = (𝑑 ∈ 𝐴 ↦ {𝑓 ∈ (𝐿‘𝑑) ∣ ∀𝑞 ∈ (𝑊‘𝑑)∀𝑟 ∈ (𝑊‘𝑑)((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝑑})) = ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝑑}))})
22		fvex 6847	. . . 4 ⊢ (𝐿‘𝐷) ∈ V
23	22	rabex 5284	. . 3 ⊢ {𝑓 ∈ (𝐿‘𝐷) ∣ ∀𝑞 ∈ (𝑊‘𝐷)∀𝑟 ∈ (𝑊‘𝐷)((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝐷})) = ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝐷}))} ∈ V
24	20, 21, 23	fvmpt 6941	. 2 ⊢ (𝐷 ∈ 𝐴 → ((𝑑 ∈ 𝐴 ↦ {𝑓 ∈ (𝐿‘𝑑) ∣ ∀𝑞 ∈ (𝑊‘𝑑)∀𝑟 ∈ (𝑊‘𝑑)((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝑑})) = ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝑑}))})‘𝐷) = {𝑓 ∈ (𝐿‘𝐷) ∣ ∀𝑞 ∈ (𝑊‘𝐷)∀𝑟 ∈ (𝑊‘𝐷)((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝐷})) = ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝐷}))})
25	10, 24	sylan9eq 2791	1 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴) → (𝑇‘𝐷) = {𝑓 ∈ (𝐿‘𝐷) ∣ ∀𝑞 ∈ (𝑊‘𝐷)∀𝑟 ∈ (𝑊‘𝐷)((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝐷})) = ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝐷}))})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3051  {crab 3399   ∩ cin 3900  {csn 4580   ↦ cmpt 5179  ‘cfv 6492  (class class class)co 7358  Atomscatm 39533  PSubSpcpsubsp 39766  +_𝑃cpadd 40065  ⊥_𝑃cpolN 40172  WAtomscwpointsN 40256  PAutcpautN 40257  DilcdilN 40372  TrnctrnN 40373

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-trnN 40377

This theorem is referenced by:  istrnN  40427