Theorem ldilfset 40097

Description: The mapping from fiducial co-atom 𝑤 to its set of lattice dilations. (Contributed by NM, 11-May-2012.)

Hypotheses

Ref	Expression
ldilset.b	⊢ 𝐵 = (Base‘𝐾)
ldilset.l	⊢ ≤ = (le‘𝐾)
ldilset.h	⊢ 𝐻 = (LHyp‘𝐾)
ldilset.i	⊢ 𝐼 = (LAut‘𝐾)

Assertion

Ref	Expression
ldilfset	⊢ (𝐾 ∈ 𝐶 → (LDil‘𝐾) = (𝑤 ∈ 𝐻 ↦ {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑤 → (𝑓‘𝑥) = 𝑥)}))

Distinct variable groups:   𝑥,𝐵   𝑤,𝐻   𝑓,𝐼   𝑤,𝑓,𝑥,𝐾

Allowed substitution hints:   𝐵(𝑤,𝑓)   𝐶(𝑥,𝑤,𝑓)   𝐻(𝑥,𝑓)   𝐼(𝑥,𝑤)   ≤ (𝑥,𝑤,𝑓)

Proof of Theorem ldilfset

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3471	. 2 ⊢ (𝐾 ∈ 𝐶 → 𝐾 ∈ V)
2		fveq2 6860	. . . . 5 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3		ldilset.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
4	2, 3	eqtr4di 2783	. . . 4 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5		fveq2 6860	. . . . . 6 ⊢ (𝑘 = 𝐾 → (LAut‘𝑘) = (LAut‘𝐾))
6		ldilset.i	. . . . . 6 ⊢ 𝐼 = (LAut‘𝐾)
7	5, 6	eqtr4di 2783	. . . . 5 ⊢ (𝑘 = 𝐾 → (LAut‘𝑘) = 𝐼)
8		fveq2 6860	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
9		ldilset.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐾)
10	8, 9	eqtr4di 2783	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
11		fveq2 6860	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
12		ldilset.l	. . . . . . . . 9 ⊢ ≤ = (le‘𝐾)
13	11, 12	eqtr4di 2783	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = ≤ )
14	13	breqd 5120	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝑥(le‘𝑘)𝑤 ↔ 𝑥 ≤ 𝑤))
15	14	imbi1d 341	. . . . . 6 ⊢ (𝑘 = 𝐾 → ((𝑥(le‘𝑘)𝑤 → (𝑓‘𝑥) = 𝑥) ↔ (𝑥 ≤ 𝑤 → (𝑓‘𝑥) = 𝑥)))
16	10, 15	raleqbidv 3321	. . . . 5 ⊢ (𝑘 = 𝐾 → (∀𝑥 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑤 → (𝑓‘𝑥) = 𝑥) ↔ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑤 → (𝑓‘𝑥) = 𝑥)))
17	7, 16	rabeqbidv 3427	. . . 4 ⊢ (𝑘 = 𝐾 → {𝑓 ∈ (LAut‘𝑘) ∣ ∀𝑥 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑤 → (𝑓‘𝑥) = 𝑥)} = {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑤 → (𝑓‘𝑥) = 𝑥)})
18	4, 17	mpteq12dv 5196	. . 3 ⊢ (𝑘 = 𝐾 → (𝑤 ∈ (LHyp‘𝑘) ↦ {𝑓 ∈ (LAut‘𝑘) ∣ ∀𝑥 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑤 → (𝑓‘𝑥) = 𝑥)}) = (𝑤 ∈ 𝐻 ↦ {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑤 → (𝑓‘𝑥) = 𝑥)}))
19		df-ldil 40093	. . 3 ⊢ LDil = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {𝑓 ∈ (LAut‘𝑘) ∣ ∀𝑥 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑤 → (𝑓‘𝑥) = 𝑥)}))
20	18, 19, 3	mptfvmpt 7204	. 2 ⊢ (𝐾 ∈ V → (LDil‘𝐾) = (𝑤 ∈ 𝐻 ↦ {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑤 → (𝑓‘𝑥) = 𝑥)}))
21	1, 20	syl 17	1 ⊢ (𝐾 ∈ 𝐶 → (LDil‘𝐾) = (𝑤 ∈ 𝐻 ↦ {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑤 → (𝑓‘𝑥) = 𝑥)}))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ∀wral 3045  {crab 3408  Vcvv 3450   class class class wbr 5109   ↦ cmpt 5190  ‘cfv 6513  Basecbs 17185  lecple 17233  LHypclh 39973  LAutclaut 39974  LDilcldil 40089

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5236  ax-sep 5253  ax-nul 5263  ax-pr 5389

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-ldil 40093

This theorem is referenced by:  ldilset  40098