Theorem trnsetN 40958

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 40957	. . 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 6883	. 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 6881	. . . 4 ⊢ (𝑑 = 𝐷 → (𝐿‘𝑑) = (𝐿‘𝐷))
12		fveq2 6881	. . . . 5 ⊢ (𝑑 = 𝐷 → (𝑊‘𝑑) = (𝑊‘𝐷))
13		sneq 4598	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → {𝑑} = {𝐷})
14	13	fveq2d 6885	. . . . . . . 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 2778	. . . . . 6 ⊢ (𝑑 = 𝐷 → (((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝑑})) = ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝑑})) ↔ ((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝐷})) = ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝐷}))))
18	12, 17	raleqbidv 3337	. . . . 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 3337	. . . 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 3433	. . 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 2762	. . 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 6894	. . . 4 ⊢ (𝐿‘𝐷) ∈ V
23	22	rabex 5308	. . 3 ⊢ {𝑓 ∈ (𝐿‘𝐷) ∣ ∀𝑞 ∈ (𝑊‘𝐷)∀𝑟 ∈ (𝑊‘𝐷)((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝐷})) = ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝐷}))} ∈ V
24	20, 21, 23	fvmpt 6989	. 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 2817	1 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴) → (𝑇‘𝐷) = {𝑓 ∈ (𝐿‘𝐷) ∣ ∀𝑞 ∈ (𝑊‘𝐷)∀𝑟 ∈ (𝑊‘𝐷)((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝐷})) = ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝐷}))})

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   = wceq 1569   ∈ wcel 2142  ∀wral 3078  {crab 3415   ∩ cin 3903  {csn 4588   ↦ cmpt 5191  ‘cfv 6536  (class class class)co 7412  Atomscatm 40065  PSubSpcpsubsp 40298  +_𝑃cpadd 40597  ⊥_𝑃cpolN 40704  WAtomscwpointsN 40788  PAutcpautN 40789  DilcdilN 40904  TrnctrnN 40905

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7415  df-trnN 40909

This theorem is used by:  istrnN  40959