Theorem ldilset 40479

Description: The set of lattice dilations for a fiducial co-atom 𝑊. (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)
ldilset.d	⊢ 𝐷 = ((LDil‘𝐾)‘𝑊)

Assertion

Ref	Expression
ldilset	⊢ ((𝐾 ∈ 𝐶 ∧ 𝑊 ∈ 𝐻) → 𝐷 = {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝑓‘𝑥) = 𝑥)})

Distinct variable groups:   𝑥,𝐵   𝑓,𝐼   𝑥,𝑓,𝐾   𝑓,𝑊,𝑥

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

Proof of Theorem ldilset

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ldilset.d	. 2 ⊢ 𝐷 = ((LDil‘𝐾)‘𝑊)
2		ldilset.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
3		ldilset.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
4		ldilset.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
5		ldilset.i	. . . . 5 ⊢ 𝐼 = (LAut‘𝐾)
6	2, 3, 4, 5	ldilfset 40478	. . . 4 ⊢ (𝐾 ∈ 𝐶 → (LDil‘𝐾) = (𝑤 ∈ 𝐻 ↦ {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑤 → (𝑓‘𝑥) = 𝑥)}))
7	6	fveq1d 6844	. . 3 ⊢ (𝐾 ∈ 𝐶 → ((LDil‘𝐾)‘𝑊) = ((𝑤 ∈ 𝐻 ↦ {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑤 → (𝑓‘𝑥) = 𝑥)})‘𝑊))
8		breq2 5104	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑥 ≤ 𝑤 ↔ 𝑥 ≤ 𝑊))
9	8	imbi1d 341	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((𝑥 ≤ 𝑤 → (𝑓‘𝑥) = 𝑥) ↔ (𝑥 ≤ 𝑊 → (𝑓‘𝑥) = 𝑥)))
10	9	ralbidv 3161	. . . . 5 ⊢ (𝑤 = 𝑊 → (∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑤 → (𝑓‘𝑥) = 𝑥) ↔ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝑓‘𝑥) = 𝑥)))
11	10	rabbidv 3408	. . . 4 ⊢ (𝑤 = 𝑊 → {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑤 → (𝑓‘𝑥) = 𝑥)} = {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝑓‘𝑥) = 𝑥)})
12		eqid 2737	. . . 4 ⊢ (𝑤 ∈ 𝐻 ↦ {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑤 → (𝑓‘𝑥) = 𝑥)}) = (𝑤 ∈ 𝐻 ↦ {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑤 → (𝑓‘𝑥) = 𝑥)})
13	5	fvexi 6856	. . . . 5 ⊢ 𝐼 ∈ V
14	13	rabex 5286	. . . 4 ⊢ {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝑓‘𝑥) = 𝑥)} ∈ V
15	11, 12, 14	fvmpt 6949	. . 3 ⊢ (𝑊 ∈ 𝐻 → ((𝑤 ∈ 𝐻 ↦ {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑤 → (𝑓‘𝑥) = 𝑥)})‘𝑊) = {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝑓‘𝑥) = 𝑥)})
16	7, 15	sylan9eq 2792	. 2 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝑊 ∈ 𝐻) → ((LDil‘𝐾)‘𝑊) = {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝑓‘𝑥) = 𝑥)})
17	1, 16	eqtrid 2784	1 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝑊 ∈ 𝐻) → 𝐷 = {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝑓‘𝑥) = 𝑥)})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  {crab 3401   class class class wbr 5100   ↦ cmpt 5181  ‘cfv 6500  Basecbs 17148  lecple 17196  LHypclh 40354  LAutclaut 40355  LDilcldil 40470

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ldil 40474

This theorem is referenced by:  isldil  40480