Theorem ldilset 40745

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 40744	. . . 4 ⊢ (𝐾 ∈ 𝐶 → (LDil‘𝐾) = (𝑤 ∈ 𝐻 ↦ {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑤 → (𝑓‘𝑥) = 𝑥)}))
7	6	fveq1d 6873	. . 3 ⊢ (𝐾 ∈ 𝐶 → ((LDil‘𝐾)‘𝑊) = ((𝑤 ∈ 𝐻 ↦ {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑤 → (𝑓‘𝑥) = 𝑥)})‘𝑊))
8		breq2 5109	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑥 ≤ 𝑤 ↔ 𝑥 ≤ 𝑊))
9	8	imbi1d 344	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((𝑥 ≤ 𝑤 → (𝑓‘𝑥) = 𝑥) ↔ (𝑥 ≤ 𝑊 → (𝑓‘𝑥) = 𝑥)))
10	9	ralbidv 3188	. . . . 5 ⊢ (𝑤 = 𝑊 → (∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑤 → (𝑓‘𝑥) = 𝑥) ↔ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝑓‘𝑥) = 𝑥)))
11	10	rabbidv 3424	. . . 4 ⊢ (𝑤 = 𝑊 → {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑤 → (𝑓‘𝑥) = 𝑥)} = {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝑓‘𝑥) = 𝑥)})
12		eqid 2765	. . . 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 6885	. . . . 5 ⊢ 𝐼 ∈ V
14	13	rabex 5300	. . . 4 ⊢ {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝑓‘𝑥) = 𝑥)} ∈ V
15	11, 12, 14	fvmpt 6979	. . 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 2820	. 2 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝑊 ∈ 𝐻) → ((LDil‘𝐾)‘𝑊) = {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝑓‘𝑥) = 𝑥)})
17	1, 16	eqtrid 2812	1 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝑊 ∈ 𝐻) → 𝐷 = {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝑓‘𝑥) = 𝑥)})

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ∀wral 3079  {crab 3417   class class class wbr 5105   ↦ cmpt 5186  ‘cfv 6525  Basecbs 17259  lecple 17307  LHypclh 40620  LAutclaut 40621  LDilcldil 40736

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ldil 40740

This theorem is referenced by:  isldil  40746