Theorem ldilset 40608

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 40607	. . . 4 ⊢ (𝐾 ∈ 𝐶 → (LDil‘𝐾) = (𝑤 ∈ 𝐻 ↦ {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑤 → (𝑓‘𝑥) = 𝑥)}))
7	6	fveq1d 6836	. . 3 ⊢ (𝐾 ∈ 𝐶 → ((LDil‘𝐾)‘𝑊) = ((𝑤 ∈ 𝐻 ↦ {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑤 → (𝑓‘𝑥) = 𝑥)})‘𝑊))
8		breq2 5083	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑥 ≤ 𝑤 ↔ 𝑥 ≤ 𝑊))
9	8	imbi1d 342	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((𝑥 ≤ 𝑤 → (𝑓‘𝑥) = 𝑥) ↔ (𝑥 ≤ 𝑊 → (𝑓‘𝑥) = 𝑥)))
10	9	ralbidv 3163	. . . . 5 ⊢ (𝑤 = 𝑊 → (∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑤 → (𝑓‘𝑥) = 𝑥) ↔ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝑓‘𝑥) = 𝑥)))
11	10	rabbidv 3399	. . . 4 ⊢ (𝑤 = 𝑊 → {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑤 → (𝑓‘𝑥) = 𝑥)} = {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝑓‘𝑥) = 𝑥)})
12		eqid 2740	. . . 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 6848	. . . . 5 ⊢ 𝐼 ∈ V
14	13	rabex 5274	. . . 4 ⊢ {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝑓‘𝑥) = 𝑥)} ∈ V
15	11, 12, 14	fvmpt 6942	. . 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 2795	. 2 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝑊 ∈ 𝐻) → ((LDil‘𝐾)‘𝑊) = {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝑓‘𝑥) = 𝑥)})
17	1, 16	eqtrid 2787	1 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝑊 ∈ 𝐻) → 𝐷 = {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝑓‘𝑥) = 𝑥)})

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3054  {crab 3392   class class class wbr 5079   ↦ cmpt 5160  ‘cfv 6492  Basecbs 17177  lecple 17225  LHypclh 40483  LAutclaut 40484  LDilcldil 40599

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ldil 40603

This theorem is referenced by:  isldil  40609