Mathbox for Norm Megill < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lautset Structured version   Visualization version   GIF version

Theorem lautset 34386
 Description: The set of lattice automorphisms. (Contributed by NM, 11-May-2012.)
Hypotheses
Ref Expression
lautset.b 𝐵 = (Base‘𝐾)
lautset.l = (le‘𝐾)
lautset.i 𝐼 = (LAut‘𝐾)
Assertion
Ref Expression
lautset (𝐾𝐴𝐼 = {𝑓 ∣ (𝑓:𝐵1-1-onto𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑓𝑥) (𝑓𝑦)))})
Distinct variable groups:   𝑥,𝑓,𝑦,𝐵   𝑓,𝐾,𝑥,𝑦   ,𝑓
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑓)   𝐼(𝑥,𝑦,𝑓)   (𝑥,𝑦)

Proof of Theorem lautset
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 elex 3185 . 2 (𝐾𝐴𝐾 ∈ V)
2 lautset.i . . 3 𝐼 = (LAut‘𝐾)
3 fveq2 6103 . . . . . . . . 9 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
4 lautset.b . . . . . . . . 9 𝐵 = (Base‘𝐾)
53, 4syl6eqr 2662 . . . . . . . 8 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
6 f1oeq2 6041 . . . . . . . 8 ((Base‘𝑘) = 𝐵 → (𝑓:(Base‘𝑘)–1-1-onto→(Base‘𝑘) ↔ 𝑓:𝐵1-1-onto→(Base‘𝑘)))
75, 6syl 17 . . . . . . 7 (𝑘 = 𝐾 → (𝑓:(Base‘𝑘)–1-1-onto→(Base‘𝑘) ↔ 𝑓:𝐵1-1-onto→(Base‘𝑘)))
8 f1oeq3 6042 . . . . . . . 8 ((Base‘𝑘) = 𝐵 → (𝑓:𝐵1-1-onto→(Base‘𝑘) ↔ 𝑓:𝐵1-1-onto𝐵))
95, 8syl 17 . . . . . . 7 (𝑘 = 𝐾 → (𝑓:𝐵1-1-onto→(Base‘𝑘) ↔ 𝑓:𝐵1-1-onto𝐵))
107, 9bitrd 267 . . . . . 6 (𝑘 = 𝐾 → (𝑓:(Base‘𝑘)–1-1-onto→(Base‘𝑘) ↔ 𝑓:𝐵1-1-onto𝐵))
11 fveq2 6103 . . . . . . . . . . 11 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
12 lautset.l . . . . . . . . . . 11 = (le‘𝐾)
1311, 12syl6eqr 2662 . . . . . . . . . 10 (𝑘 = 𝐾 → (le‘𝑘) = )
1413breqd 4594 . . . . . . . . 9 (𝑘 = 𝐾 → (𝑥(le‘𝑘)𝑦𝑥 𝑦))
1513breqd 4594 . . . . . . . . 9 (𝑘 = 𝐾 → ((𝑓𝑥)(le‘𝑘)(𝑓𝑦) ↔ (𝑓𝑥) (𝑓𝑦)))
1614, 15bibi12d 334 . . . . . . . 8 (𝑘 = 𝐾 → ((𝑥(le‘𝑘)𝑦 ↔ (𝑓𝑥)(le‘𝑘)(𝑓𝑦)) ↔ (𝑥 𝑦 ↔ (𝑓𝑥) (𝑓𝑦))))
175, 16raleqbidv 3129 . . . . . . 7 (𝑘 = 𝐾 → (∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦 ↔ (𝑓𝑥)(le‘𝑘)(𝑓𝑦)) ↔ ∀𝑦𝐵 (𝑥 𝑦 ↔ (𝑓𝑥) (𝑓𝑦))))
185, 17raleqbidv 3129 . . . . . 6 (𝑘 = 𝐾 → (∀𝑥 ∈ (Base‘𝑘)∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦 ↔ (𝑓𝑥)(le‘𝑘)(𝑓𝑦)) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑓𝑥) (𝑓𝑦))))
1910, 18anbi12d 743 . . . . 5 (𝑘 = 𝐾 → ((𝑓:(Base‘𝑘)–1-1-onto→(Base‘𝑘) ∧ ∀𝑥 ∈ (Base‘𝑘)∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦 ↔ (𝑓𝑥)(le‘𝑘)(𝑓𝑦))) ↔ (𝑓:𝐵1-1-onto𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑓𝑥) (𝑓𝑦)))))
2019abbidv 2728 . . . 4 (𝑘 = 𝐾 → {𝑓 ∣ (𝑓:(Base‘𝑘)–1-1-onto→(Base‘𝑘) ∧ ∀𝑥 ∈ (Base‘𝑘)∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦 ↔ (𝑓𝑥)(le‘𝑘)(𝑓𝑦)))} = {𝑓 ∣ (𝑓:𝐵1-1-onto𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑓𝑥) (𝑓𝑦)))})
21 df-laut 34293 . . . 4 LAut = (𝑘 ∈ V ↦ {𝑓 ∣ (𝑓:(Base‘𝑘)–1-1-onto→(Base‘𝑘) ∧ ∀𝑥 ∈ (Base‘𝑘)∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦 ↔ (𝑓𝑥)(le‘𝑘)(𝑓𝑦)))})
22 fvex 6113 . . . . . . . . 9 (Base‘𝐾) ∈ V
234, 22eqeltri 2684 . . . . . . . 8 𝐵 ∈ V
2423, 23mapval 7756 . . . . . . 7 (𝐵𝑚 𝐵) = {𝑓𝑓:𝐵𝐵}
25 ovex 6577 . . . . . . 7 (𝐵𝑚 𝐵) ∈ V
2624, 25eqeltrri 2685 . . . . . 6 {𝑓𝑓:𝐵𝐵} ∈ V
27 f1of 6050 . . . . . . 7 (𝑓:𝐵1-1-onto𝐵𝑓:𝐵𝐵)
2827ss2abi 3637 . . . . . 6 {𝑓𝑓:𝐵1-1-onto𝐵} ⊆ {𝑓𝑓:𝐵𝐵}
2926, 28ssexi 4731 . . . . 5 {𝑓𝑓:𝐵1-1-onto𝐵} ∈ V
30 simpl 472 . . . . . 6 ((𝑓:𝐵1-1-onto𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑓𝑥) (𝑓𝑦))) → 𝑓:𝐵1-1-onto𝐵)
3130ss2abi 3637 . . . . 5 {𝑓 ∣ (𝑓:𝐵1-1-onto𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑓𝑥) (𝑓𝑦)))} ⊆ {𝑓𝑓:𝐵1-1-onto𝐵}
3229, 31ssexi 4731 . . . 4 {𝑓 ∣ (𝑓:𝐵1-1-onto𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑓𝑥) (𝑓𝑦)))} ∈ V
3320, 21, 32fvmpt 6191 . . 3 (𝐾 ∈ V → (LAut‘𝐾) = {𝑓 ∣ (𝑓:𝐵1-1-onto𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑓𝑥) (𝑓𝑦)))})
342, 33syl5eq 2656 . 2 (𝐾 ∈ V → 𝐼 = {𝑓 ∣ (𝑓:𝐵1-1-onto𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑓𝑥) (𝑓𝑦)))})
351, 34syl 17 1 (𝐾𝐴𝐼 = {𝑓 ∣ (𝑓:𝐵1-1-onto𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑓𝑥) (𝑓𝑦)))})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  ∀wral 2896  Vcvv 3173   class class class wbr 4583  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   ↑𝑚 cmap 7744  Basecbs 15695  lecple 15775  LAutclaut 34289 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-laut 34293 This theorem is referenced by:  islaut  34387
 Copyright terms: Public domain W3C validator