Theorem lautcnv 37230

Description: The converse of a lattice automorphism is a lattice automorphism. (Contributed by NM, 10-May-2013.)

Hypothesis

Ref	Expression
lautcnv.i	⊢ 𝐼 = (LAut‘𝐾)

Assertion

Ref	Expression
lautcnv	⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) → ◡𝐹 ∈ 𝐼)

Proof of Theorem lautcnv

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2824	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		lautcnv.i	. . . 4 ⊢ 𝐼 = (LAut‘𝐾)
3	1, 2	laut1o 37225	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
4		f1ocnv 6630	. . 3 ⊢ (𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) → ◡𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
5	3, 4	syl 17	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) → ◡𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
6		eqid 2824	. . . 4 ⊢ (le‘𝐾) = (le‘𝐾)
7	1, 6, 2	lautcnvle 37229	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(le‘𝐾)𝑦 ↔ (◡𝐹‘𝑥)(le‘𝐾)(◡𝐹‘𝑦)))
8	7	ralrimivva 3194	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) → ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑦 ↔ (◡𝐹‘𝑥)(le‘𝐾)(◡𝐹‘𝑦)))
9	1, 6, 2	islaut 37223	. . 3 ⊢ (𝐾 ∈ 𝑉 → (◡𝐹 ∈ 𝐼 ↔ (◡𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑦 ↔ (◡𝐹‘𝑥)(le‘𝐾)(◡𝐹‘𝑦)))))
10	9	adantr 483	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) → (◡𝐹 ∈ 𝐼 ↔ (◡𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑦 ↔ (◡𝐹‘𝑥)(le‘𝐾)(◡𝐹‘𝑦)))))
11	5, 8, 10	mpbir2and 711	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) → ◡𝐹 ∈ 𝐼)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1536   ∈ wcel 2113  ∀wral 3141   class class class wbr 5069  ^◡ccnv 5557  –1-1-onto→wf1o 6357  ‘cfv 6358  Basecbs 16486  lecple 16575  LAutclaut 37125

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-reu 3148  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-ov 7162  df-oprab 7163  df-mpo 7164  df-map 8411  df-laut 37129

This theorem is referenced by:  ldilcnv  37255