Theorem trnfsetN 41213

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 3471	. 2 ⊢ (𝐾 ∈ 𝐶 → 𝐾 ∈ V)
2		trnset.t	. . 3 ⊢ 𝑇 = (Trn‘𝐾)
3		fveq2 6885	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4		trnset.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
5	3, 4	eqtr4di 2813	. . . . 5 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
6		fveq2 6885	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (Dil‘𝑘) = (Dil‘𝐾))
7		trnset.l	. . . . . . . 8 ⊢ 𝐿 = (Dil‘𝐾)
8	6, 7	eqtr4di 2813	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (Dil‘𝑘) = 𝐿)
9	8	fveq1d 6887	. . . . . 6 ⊢ (𝑘 = 𝐾 → ((Dil‘𝑘)‘𝑑) = (𝐿‘𝑑))
10		fveq2 6885	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (WAtoms‘𝑘) = (WAtoms‘𝐾))
11		trnset.w	. . . . . . . . 9 ⊢ 𝑊 = (WAtoms‘𝐾)
12	10, 11	eqtr4di 2813	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (WAtoms‘𝑘) = 𝑊)
13	12	fveq1d 6887	. . . . . . 7 ⊢ (𝑘 = 𝐾 → ((WAtoms‘𝑘)‘𝑑) = (𝑊‘𝑑))
14		fveq2 6885	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → (+𝑃‘𝑘) = (+𝑃‘𝐾))
15		trnset.p	. . . . . . . . . . . 12 ⊢ + = (+𝑃‘𝐾)
16	14, 15	eqtr4di 2813	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (+𝑃‘𝑘) = + )
17	16	oveqd 7437	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (𝑞(+𝑃‘𝑘)(𝑓‘𝑞)) = (𝑞 + (𝑓‘𝑞)))
18		fveq2 6885	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → (⊥𝑃‘𝑘) = (⊥𝑃‘𝐾))
19		trnset.o	. . . . . . . . . . . 12 ⊢ ⊥ = (⊥𝑃‘𝐾)
20	18, 19	eqtr4di 2813	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (⊥𝑃‘𝑘) = ⊥ )
21	20	fveq1d 6887	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → ((⊥𝑃‘𝑘)‘{𝑑}) = ( ⊥ ‘{𝑑}))
22	17, 21	ineq12d 4118	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → ((𝑞(+𝑃‘𝑘)(𝑓‘𝑞)) ∩ ((⊥𝑃‘𝑘)‘{𝑑})) = ((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝑑})))
23	16	oveqd 7437	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (𝑟(+𝑃‘𝑘)(𝑓‘𝑟)) = (𝑟 + (𝑓‘𝑟)))
24	23, 21	ineq12d 4118	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → ((𝑟(+𝑃‘𝑘)(𝑓‘𝑟)) ∩ ((⊥𝑃‘𝑘)‘{𝑑})) = ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝑑})))
25	22, 24	eqeq12d 2776	. . . . . . . 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 3334	. . . . . . 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 3334	. . . . . 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 3429	. . . . 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 5191	. . . 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 41165	. . . 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	29, 30, 4	mptfvmpt 7234	. . 3 ⊢ (𝐾 ∈ V → (Trn‘𝐾) = (𝑑 ∈ 𝐴 ↦ {𝑓 ∈ (𝐿‘𝑑) ∣ ∀𝑞 ∈ (𝑊‘𝑑)∀𝑟 ∈ (𝑊‘𝑑)((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝑑})) = ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝑑}))}))
32	2, 31	eqtrid 2807	. 2 ⊢ (𝐾 ∈ V → 𝑇 = (𝑑 ∈ 𝐴 ↦ {𝑓 ∈ (𝐿‘𝑑) ∣ ∀𝑞 ∈ (𝑊‘𝑑)∀𝑟 ∈ (𝑊‘𝑑)((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝑑})) = ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝑑}))}))
33	1, 32	syl 18	1 ⊢ (𝐾 ∈ 𝐶 → 𝑇 = (𝑑 ∈ 𝐴 ↦ {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   = wceq 1569   ∈ wcel 2144  ∀wral 3076  {crab 3412  Vcvv 3450   ∩ cin 4089  {csn 4583   ↦ cmpt 5185  ‘cfv 6538  (class class class)co 7420  Atomscatm 40321  PSubSpcpsubsp 40554  +_𝑃cpadd 40853  ⊥_𝑃cpolN 40960  WAtomscwpointsN 41044  PAutcpautN 41045  DilcdilN 41160  TrnctrnN 41161

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-trnN 41165

This theorem is used by:  trnsetN  41214