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Theorem ldilset 35918
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 ((𝐾𝐶𝑊𝐻) → 𝐷 = {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑊 → (𝑓𝑥) = 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.
StepHypRef 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)
62, 3, 4, 5ldilfset 35917 . . . 4 (𝐾𝐶 → (LDil‘𝐾) = (𝑤𝐻 ↦ {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑤 → (𝑓𝑥) = 𝑥)}))
76fveq1d 6335 . . 3 (𝐾𝐶 → ((LDil‘𝐾)‘𝑊) = ((𝑤𝐻 ↦ {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑤 → (𝑓𝑥) = 𝑥)})‘𝑊))
8 breq2 4791 . . . . . . 7 (𝑤 = 𝑊 → (𝑥 𝑤𝑥 𝑊))
98imbi1d 330 . . . . . 6 (𝑤 = 𝑊 → ((𝑥 𝑤 → (𝑓𝑥) = 𝑥) ↔ (𝑥 𝑊 → (𝑓𝑥) = 𝑥)))
109ralbidv 3135 . . . . 5 (𝑤 = 𝑊 → (∀𝑥𝐵 (𝑥 𝑤 → (𝑓𝑥) = 𝑥) ↔ ∀𝑥𝐵 (𝑥 𝑊 → (𝑓𝑥) = 𝑥)))
1110rabbidv 3339 . . . 4 (𝑤 = 𝑊 → {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑤 → (𝑓𝑥) = 𝑥)} = {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑊 → (𝑓𝑥) = 𝑥)})
12 eqid 2771 . . . 4 (𝑤𝐻 ↦ {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑤 → (𝑓𝑥) = 𝑥)}) = (𝑤𝐻 ↦ {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑤 → (𝑓𝑥) = 𝑥)})
135fvexi 6345 . . . . 5 𝐼 ∈ V
1413rabex 4947 . . . 4 {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑊 → (𝑓𝑥) = 𝑥)} ∈ V
1511, 12, 14fvmpt 6426 . . 3 (𝑊𝐻 → ((𝑤𝐻 ↦ {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑤 → (𝑓𝑥) = 𝑥)})‘𝑊) = {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑊 → (𝑓𝑥) = 𝑥)})
167, 15sylan9eq 2825 . 2 ((𝐾𝐶𝑊𝐻) → ((LDil‘𝐾)‘𝑊) = {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑊 → (𝑓𝑥) = 𝑥)})
171, 16syl5eq 2817 1 ((𝐾𝐶𝑊𝐻) → 𝐷 = {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑊 → (𝑓𝑥) = 𝑥)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wral 3061  {crab 3065   class class class wbr 4787  cmpt 4864  cfv 6030  Basecbs 16064  lecple 16156  LHypclh 35793  LAutclaut 35794  LDilcldil 35909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ldil 35913
This theorem is referenced by:  isldil  35919
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