Theorem trnfsetN 35442

Description: The mapping from fiducial atom to set of translations. (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
trnfsetN	⊢ (𝐾 ∈ 𝐶 → 𝑇 = (𝑑 ∈ 𝐴 ↦ {𝑓 ∈ (𝐿‘𝑑) ∣ ∀𝑞 ∈ (𝑊‘𝑑)∀𝑟 ∈ (𝑊‘𝑑)((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝑑})) = ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝑑}))}))

Distinct variable groups:   𝐴,𝑑   𝑓,𝑑,𝑞,𝑟,𝐾   𝑓,𝐿   𝑊,𝑞,𝑟

Allowed substitution hints:   𝐴(𝑓,𝑟,𝑞)   𝐶(𝑓,𝑟,𝑞,𝑑)   + (𝑓,𝑟,𝑞,𝑑)   𝑆(𝑓,𝑟,𝑞,𝑑)   𝑇(𝑓,𝑟,𝑞,𝑑)   𝐿(𝑟,𝑞,𝑑)   𝑀(𝑓,𝑟,𝑞,𝑑)   ⊥ (𝑓,𝑟,𝑞,𝑑)   𝑊(𝑓,𝑑)

Proof of Theorem trnfsetN

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. 2 ⊢ (𝐾 ∈ 𝐶 → 𝐾 ∈ V)
2		trnset.t	. . 3 ⊢ 𝑇 = (Trn‘𝐾)
3		fveq2 6191	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4		trnset.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
5	3, 4	syl6eqr 2674	. . . . 5 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
6		fveq2 6191	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (Dil‘𝑘) = (Dil‘𝐾))
7		trnset.l	. . . . . . . 8 ⊢ 𝐿 = (Dil‘𝐾)
8	6, 7	syl6eqr 2674	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (Dil‘𝑘) = 𝐿)
9	8	fveq1d 6193	. . . . . 6 ⊢ (𝑘 = 𝐾 → ((Dil‘𝑘)‘𝑑) = (𝐿‘𝑑))
10		fveq2 6191	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (WAtoms‘𝑘) = (WAtoms‘𝐾))
11		trnset.w	. . . . . . . . 9 ⊢ 𝑊 = (WAtoms‘𝐾)
12	10, 11	syl6eqr 2674	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (WAtoms‘𝑘) = 𝑊)
13	12	fveq1d 6193	. . . . . . 7 ⊢ (𝑘 = 𝐾 → ((WAtoms‘𝑘)‘𝑑) = (𝑊‘𝑑))
14		fveq2 6191	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → (+𝑃‘𝑘) = (+𝑃‘𝐾))
15		trnset.p	. . . . . . . . . . . 12 ⊢ + = (+𝑃‘𝐾)
16	14, 15	syl6eqr 2674	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (+𝑃‘𝑘) = + )
17	16	oveqd 6667	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (𝑞(+𝑃‘𝑘)(𝑓‘𝑞)) = (𝑞 + (𝑓‘𝑞)))
18		fveq2 6191	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → (⊥𝑃‘𝑘) = (⊥𝑃‘𝐾))
19		trnset.o	. . . . . . . . . . . 12 ⊢ ⊥ = (⊥𝑃‘𝐾)
20	18, 19	syl6eqr 2674	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (⊥𝑃‘𝑘) = ⊥ )
21	20	fveq1d 6193	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → ((⊥𝑃‘𝑘)‘{𝑑}) = ( ⊥ ‘{𝑑}))
22	17, 21	ineq12d 3815	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → ((𝑞(+𝑃‘𝑘)(𝑓‘𝑞)) ∩ ((⊥𝑃‘𝑘)‘{𝑑})) = ((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝑑})))
23	16	oveqd 6667	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (𝑟(+𝑃‘𝑘)(𝑓‘𝑟)) = (𝑟 + (𝑓‘𝑟)))
24	23, 21	ineq12d 3815	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → ((𝑟(+𝑃‘𝑘)(𝑓‘𝑟)) ∩ ((⊥𝑃‘𝑘)‘{𝑑})) = ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝑑})))
25	22, 24	eqeq12d 2637	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (((𝑞(+𝑃‘𝑘)(𝑓‘𝑞)) ∩ ((⊥𝑃‘𝑘)‘{𝑑})) = ((𝑟(+𝑃‘𝑘)(𝑓‘𝑟)) ∩ ((⊥𝑃‘𝑘)‘{𝑑})) ↔ ((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝑑})) = ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝑑}))))
26	13, 25	raleqbidv 3152	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (∀𝑟 ∈ ((WAtoms‘𝑘)‘𝑑)((𝑞(+𝑃‘𝑘)(𝑓‘𝑞)) ∩ ((⊥𝑃‘𝑘)‘{𝑑})) = ((𝑟(+𝑃‘𝑘)(𝑓‘𝑟)) ∩ ((⊥𝑃‘𝑘)‘{𝑑})) ↔ ∀𝑟 ∈ (𝑊‘𝑑)((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝑑})) = ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝑑}))))
27	13, 26	raleqbidv 3152	. . . . . 6 ⊢ (𝑘 = 𝐾 → (∀𝑞 ∈ ((WAtoms‘𝑘)‘𝑑)∀𝑟 ∈ ((WAtoms‘𝑘)‘𝑑)((𝑞(+𝑃‘𝑘)(𝑓‘𝑞)) ∩ ((⊥𝑃‘𝑘)‘{𝑑})) = ((𝑟(+𝑃‘𝑘)(𝑓‘𝑟)) ∩ ((⊥𝑃‘𝑘)‘{𝑑})) ↔ ∀𝑞 ∈ (𝑊‘𝑑)∀𝑟 ∈ (𝑊‘𝑑)((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝑑})) = ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝑑}))))
28	9, 27	rabeqbidv 3195	. . . . 5 ⊢ (𝑘 = 𝐾 → {𝑓 ∈ ((Dil‘𝑘)‘𝑑) ∣ ∀𝑞 ∈ ((WAtoms‘𝑘)‘𝑑)∀𝑟 ∈ ((WAtoms‘𝑘)‘𝑑)((𝑞(+𝑃‘𝑘)(𝑓‘𝑞)) ∩ ((⊥𝑃‘𝑘)‘{𝑑})) = ((𝑟(+𝑃‘𝑘)(𝑓‘𝑟)) ∩ ((⊥𝑃‘𝑘)‘{𝑑}))} = {𝑓 ∈ (𝐿‘𝑑) ∣ ∀𝑞 ∈ (𝑊‘𝑑)∀𝑟 ∈ (𝑊‘𝑑)((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝑑})) = ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝑑}))})
29	5, 28	mpteq12dv 4733	. . . 4 ⊢ (𝑘 = 𝐾 → (𝑑 ∈ (Atoms‘𝑘) ↦ {𝑓 ∈ ((Dil‘𝑘)‘𝑑) ∣ ∀𝑞 ∈ ((WAtoms‘𝑘)‘𝑑)∀𝑟 ∈ ((WAtoms‘𝑘)‘𝑑)((𝑞(+𝑃‘𝑘)(𝑓‘𝑞)) ∩ ((⊥𝑃‘𝑘)‘{𝑑})) = ((𝑟(+𝑃‘𝑘)(𝑓‘𝑟)) ∩ ((⊥𝑃‘𝑘)‘{𝑑}))}) = (𝑑 ∈ 𝐴 ↦ {𝑓 ∈ (𝐿‘𝑑) ∣ ∀𝑞 ∈ (𝑊‘𝑑)∀𝑟 ∈ (𝑊‘𝑑)((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝑑})) = ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝑑}))}))
30		df-trnN 35393	. . . 4 ⊢ Trn = (𝑘 ∈ V ↦ (𝑑 ∈ (Atoms‘𝑘) ↦ {𝑓 ∈ ((Dil‘𝑘)‘𝑑) ∣ ∀𝑞 ∈ ((WAtoms‘𝑘)‘𝑑)∀𝑟 ∈ ((WAtoms‘𝑘)‘𝑑)((𝑞(+𝑃‘𝑘)(𝑓‘𝑞)) ∩ ((⊥𝑃‘𝑘)‘{𝑑})) = ((𝑟(+𝑃‘𝑘)(𝑓‘𝑟)) ∩ ((⊥𝑃‘𝑘)‘{𝑑}))}))
31		fvex 6201	. . . . . 6 ⊢ (Atoms‘𝐾) ∈ V
32	4, 31	eqeltri 2697	. . . . 5 ⊢ 𝐴 ∈ V
33	32	mptex 6486	. . . 4 ⊢ (𝑑 ∈ 𝐴 ↦ {𝑓 ∈ (𝐿‘𝑑) ∣ ∀𝑞 ∈ (𝑊‘𝑑)∀𝑟 ∈ (𝑊‘𝑑)((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝑑})) = ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝑑}))}) ∈ V
34	29, 30, 33	fvmpt 6282	. . 3 ⊢ (𝐾 ∈ V → (Trn‘𝐾) = (𝑑 ∈ 𝐴 ↦ {𝑓 ∈ (𝐿‘𝑑) ∣ ∀𝑞 ∈ (𝑊‘𝑑)∀𝑟 ∈ (𝑊‘𝑑)((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝑑})) = ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝑑}))}))
35	2, 34	syl5eq 2668	. 2 ⊢ (𝐾 ∈ V → 𝑇 = (𝑑 ∈ 𝐴 ↦ {𝑓 ∈ (𝐿‘𝑑) ∣ ∀𝑞 ∈ (𝑊‘𝑑)∀𝑟 ∈ (𝑊‘𝑑)((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝑑})) = ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝑑}))}))
36	1, 35	syl 17	1 ⊢ (𝐾 ∈ 𝐶 → 𝑇 = (𝑑 ∈ 𝐴 ↦ {𝑓 ∈ (𝐿‘𝑑) ∣ ∀𝑞 ∈ (𝑊‘𝑑)∀𝑟 ∈ (𝑊‘𝑑)((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝑑})) = ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝑑}))}))

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  ∀wral 2912  {crab 2916  Vcvv 3200   ∩ cin 3573  {csn 4177   ↦ cmpt 4729  ‘cfv 5888  (class class class)co 6650  Atomscatm 34550  PSubSpcpsubsp 34782  +_𝑃cpadd 35081  ⊥_𝑃cpolN 35188  WAtomscwpointsN 35272  PAutcpautN 35273  DilcdilN 35388  TrnctrnN 35389

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-trnN 35393

This theorem is referenced by:  trnsetN  35443