Theorem trnsetN 40785

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 40784	. . 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 6871	. 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 6869	. . . 4 ⊢ (𝑑 = 𝐷 → (𝐿‘𝑑) = (𝐿‘𝐷))
12		fveq2 6869	. . . . 5 ⊢ (𝑑 = 𝐷 → (𝑊‘𝑑) = (𝑊‘𝐷))
13		sneq 4594	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → {𝑑} = {𝐷})
14	13	fveq2d 6873	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → ( ⊥ ‘{𝑑}) = ( ⊥ ‘{𝐷}))
15	14	ineq2d 4174	. . . . . . 7 ⊢ (𝑑 = 𝐷 → ((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝑑})) = ((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝐷})))
16	14	ineq2d 4174	. . . . . . 7 ⊢ (𝑑 = 𝐷 → ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝑑})) = ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝐷})))
17	15, 16	eqeq12d 2780	. . . . . 6 ⊢ (𝑑 = 𝐷 → (((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝑑})) = ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝑑})) ↔ ((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝐷})) = ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝐷}))))
18	12, 17	raleqbidv 3338	. . . . 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 3338	. . . 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 3434	. . 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 2764	. . 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 6882	. . . 4 ⊢ (𝐿‘𝐷) ∈ V
23	22	rabex 5297	. . 3 ⊢ {𝑓 ∈ (𝐿‘𝐷) ∣ ∀𝑞 ∈ (𝑊‘𝐷)∀𝑟 ∈ (𝑊‘𝐷)((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝐷})) = ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝐷}))} ∈ V
24	20, 21, 23	fvmpt 6977	. 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 2819	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 399   = wceq 1562   ∈ wcel 2144  ∀wral 3078  {crab 3416   ∩ cin 3905  {csn 4584   ↦ cmpt 5183  ‘cfv 6523  (class class class)co 7398  Atomscatm 39892  PSubSpcpsubsp 40125  +_𝑃cpadd 40424  ⊥_𝑃cpolN 40531  WAtomscwpointsN 40615  PAutcpautN 40616  DilcdilN 40731  TrnctrnN 40732

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-ov 7401  df-trnN 40736

This theorem is referenced by:  istrnN  40786