Theorem ldilfset 36775

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 3455	. 2 ⊢ (𝐾 ∈ 𝐶 → 𝐾 ∈ V)
2		fveq2 6538	. . . . 5 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3		ldilset.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
4	2, 3	syl6eqr 2849	. . . 4 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5		fveq2 6538	. . . . . 6 ⊢ (𝑘 = 𝐾 → (LAut‘𝑘) = (LAut‘𝐾))
6		ldilset.i	. . . . . 6 ⊢ 𝐼 = (LAut‘𝐾)
7	5, 6	syl6eqr 2849	. . . . 5 ⊢ (𝑘 = 𝐾 → (LAut‘𝑘) = 𝐼)
8		fveq2 6538	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
9		ldilset.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐾)
10	8, 9	syl6eqr 2849	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
11		fveq2 6538	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
12		ldilset.l	. . . . . . . . 9 ⊢ ≤ = (le‘𝐾)
13	11, 12	syl6eqr 2849	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = ≤ )
14	13	breqd 4973	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝑥(le‘𝑘)𝑤 ↔ 𝑥 ≤ 𝑤))
15	14	imbi1d 343	. . . . . 6 ⊢ (𝑘 = 𝐾 → ((𝑥(le‘𝑘)𝑤 → (𝑓‘𝑥) = 𝑥) ↔ (𝑥 ≤ 𝑤 → (𝑓‘𝑥) = 𝑥)))
16	10, 15	raleqbidv 3361	. . . . 5 ⊢ (𝑘 = 𝐾 → (∀𝑥 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑤 → (𝑓‘𝑥) = 𝑥) ↔ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑤 → (𝑓‘𝑥) = 𝑥)))
17	7, 16	rabeqbidv 3430	. . . 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 5045	. . 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 36771	. . 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 6856	. 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 1522   ∈ wcel 2081  ∀wral 3105  {crab 3109  Vcvv 3437   class class class wbr 4962   ↦ cmpt 5041  ‘cfv 6225  Basecbs 16312  lecple 16401  LHypclh 36651  LAutclaut 36652  LDilcldil 36767

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pr 5221

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-ldil 36771

This theorem is referenced by:  ldilset  36776