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Theorem lautcvr 34855
 Description: Covering property of a lattice automorphism. (Contributed by NM, 20-May-2012.)
Hypotheses
Ref Expression
lautcvr.b 𝐵 = (Base‘𝐾)
lautcvr.c 𝐶 = ( ⋖ ‘𝐾)
lautcvr.i 𝐼 = (LAut‘𝐾)
Assertion
Ref Expression
lautcvr ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋𝐶𝑌 ↔ (𝐹𝑋)𝐶(𝐹𝑌)))

Proof of Theorem lautcvr
Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lautcvr.b . . . 4 𝐵 = (Base‘𝐾)
2 eqid 2621 . . . 4 (lt‘𝐾) = (lt‘𝐾)
3 lautcvr.i . . . 4 𝐼 = (LAut‘𝐾)
41, 2, 3lautlt 34854 . . 3 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋(lt‘𝐾)𝑌 ↔ (𝐹𝑋)(lt‘𝐾)(𝐹𝑌)))
5 simpll 789 . . . . . . . . 9 (((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) ∧ 𝑤𝐵) → 𝐾𝐴)
6 simplr1 1101 . . . . . . . . 9 (((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) ∧ 𝑤𝐵) → 𝐹𝐼)
7 simplr2 1102 . . . . . . . . 9 (((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) ∧ 𝑤𝐵) → 𝑋𝐵)
8 simpr 477 . . . . . . . . 9 (((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) ∧ 𝑤𝐵) → 𝑤𝐵)
91, 2, 3lautlt 34854 . . . . . . . . 9 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑤𝐵)) → (𝑋(lt‘𝐾)𝑤 ↔ (𝐹𝑋)(lt‘𝐾)(𝐹𝑤)))
105, 6, 7, 8, 9syl13anc 1325 . . . . . . . 8 (((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) ∧ 𝑤𝐵) → (𝑋(lt‘𝐾)𝑤 ↔ (𝐹𝑋)(lt‘𝐾)(𝐹𝑤)))
11 simplr3 1103 . . . . . . . . 9 (((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) ∧ 𝑤𝐵) → 𝑌𝐵)
121, 2, 3lautlt 34854 . . . . . . . . 9 ((𝐾𝐴 ∧ (𝐹𝐼𝑤𝐵𝑌𝐵)) → (𝑤(lt‘𝐾)𝑌 ↔ (𝐹𝑤)(lt‘𝐾)(𝐹𝑌)))
135, 6, 8, 11, 12syl13anc 1325 . . . . . . . 8 (((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) ∧ 𝑤𝐵) → (𝑤(lt‘𝐾)𝑌 ↔ (𝐹𝑤)(lt‘𝐾)(𝐹𝑌)))
1410, 13anbi12d 746 . . . . . . 7 (((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) ∧ 𝑤𝐵) → ((𝑋(lt‘𝐾)𝑤𝑤(lt‘𝐾)𝑌) ↔ ((𝐹𝑋)(lt‘𝐾)(𝐹𝑤) ∧ (𝐹𝑤)(lt‘𝐾)(𝐹𝑌))))
151, 3lautcl 34850 . . . . . . . . 9 (((𝐾𝐴𝐹𝐼) ∧ 𝑤𝐵) → (𝐹𝑤) ∈ 𝐵)
165, 6, 8, 15syl21anc 1322 . . . . . . . 8 (((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) ∧ 𝑤𝐵) → (𝐹𝑤) ∈ 𝐵)
17 breq2 4617 . . . . . . . . . . 11 (𝑧 = (𝐹𝑤) → ((𝐹𝑋)(lt‘𝐾)𝑧 ↔ (𝐹𝑋)(lt‘𝐾)(𝐹𝑤)))
18 breq1 4616 . . . . . . . . . . 11 (𝑧 = (𝐹𝑤) → (𝑧(lt‘𝐾)(𝐹𝑌) ↔ (𝐹𝑤)(lt‘𝐾)(𝐹𝑌)))
1917, 18anbi12d 746 . . . . . . . . . 10 (𝑧 = (𝐹𝑤) → (((𝐹𝑋)(lt‘𝐾)𝑧𝑧(lt‘𝐾)(𝐹𝑌)) ↔ ((𝐹𝑋)(lt‘𝐾)(𝐹𝑤) ∧ (𝐹𝑤)(lt‘𝐾)(𝐹𝑌))))
2019rspcev 3295 . . . . . . . . 9 (((𝐹𝑤) ∈ 𝐵 ∧ ((𝐹𝑋)(lt‘𝐾)(𝐹𝑤) ∧ (𝐹𝑤)(lt‘𝐾)(𝐹𝑌))) → ∃𝑧𝐵 ((𝐹𝑋)(lt‘𝐾)𝑧𝑧(lt‘𝐾)(𝐹𝑌)))
2120ex 450 . . . . . . . 8 ((𝐹𝑤) ∈ 𝐵 → (((𝐹𝑋)(lt‘𝐾)(𝐹𝑤) ∧ (𝐹𝑤)(lt‘𝐾)(𝐹𝑌)) → ∃𝑧𝐵 ((𝐹𝑋)(lt‘𝐾)𝑧𝑧(lt‘𝐾)(𝐹𝑌))))
2216, 21syl 17 . . . . . . 7 (((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) ∧ 𝑤𝐵) → (((𝐹𝑋)(lt‘𝐾)(𝐹𝑤) ∧ (𝐹𝑤)(lt‘𝐾)(𝐹𝑌)) → ∃𝑧𝐵 ((𝐹𝑋)(lt‘𝐾)𝑧𝑧(lt‘𝐾)(𝐹𝑌))))
2314, 22sylbid 230 . . . . . 6 (((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) ∧ 𝑤𝐵) → ((𝑋(lt‘𝐾)𝑤𝑤(lt‘𝐾)𝑌) → ∃𝑧𝐵 ((𝐹𝑋)(lt‘𝐾)𝑧𝑧(lt‘𝐾)(𝐹𝑌))))
2423rexlimdva 3024 . . . . 5 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (∃𝑤𝐵 (𝑋(lt‘𝐾)𝑤𝑤(lt‘𝐾)𝑌) → ∃𝑧𝐵 ((𝐹𝑋)(lt‘𝐾)𝑧𝑧(lt‘𝐾)(𝐹𝑌))))
25 simpll 789 . . . . . . . . . 10 (((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) ∧ 𝑧𝐵) → 𝐾𝐴)
26 simplr1 1101 . . . . . . . . . 10 (((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) ∧ 𝑧𝐵) → 𝐹𝐼)
27 simplr2 1102 . . . . . . . . . 10 (((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) ∧ 𝑧𝐵) → 𝑋𝐵)
281, 3laut1o 34848 . . . . . . . . . . . 12 ((𝐾𝐴𝐹𝐼) → 𝐹:𝐵1-1-onto𝐵)
2925, 26, 28syl2anc 692 . . . . . . . . . . 11 (((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) ∧ 𝑧𝐵) → 𝐹:𝐵1-1-onto𝐵)
30 f1ocnvdm 6494 . . . . . . . . . . 11 ((𝐹:𝐵1-1-onto𝐵𝑧𝐵) → (𝐹𝑧) ∈ 𝐵)
3129, 30sylancom 700 . . . . . . . . . 10 (((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) ∧ 𝑧𝐵) → (𝐹𝑧) ∈ 𝐵)
321, 2, 3lautlt 34854 . . . . . . . . . 10 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵 ∧ (𝐹𝑧) ∈ 𝐵)) → (𝑋(lt‘𝐾)(𝐹𝑧) ↔ (𝐹𝑋)(lt‘𝐾)(𝐹‘(𝐹𝑧))))
3325, 26, 27, 31, 32syl13anc 1325 . . . . . . . . 9 (((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) ∧ 𝑧𝐵) → (𝑋(lt‘𝐾)(𝐹𝑧) ↔ (𝐹𝑋)(lt‘𝐾)(𝐹‘(𝐹𝑧))))
34 f1ocnvfv2 6487 . . . . . . . . . . 11 ((𝐹:𝐵1-1-onto𝐵𝑧𝐵) → (𝐹‘(𝐹𝑧)) = 𝑧)
3529, 34sylancom 700 . . . . . . . . . 10 (((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) ∧ 𝑧𝐵) → (𝐹‘(𝐹𝑧)) = 𝑧)
3635breq2d 4625 . . . . . . . . 9 (((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) ∧ 𝑧𝐵) → ((𝐹𝑋)(lt‘𝐾)(𝐹‘(𝐹𝑧)) ↔ (𝐹𝑋)(lt‘𝐾)𝑧))
3733, 36bitr2d 269 . . . . . . . 8 (((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) ∧ 𝑧𝐵) → ((𝐹𝑋)(lt‘𝐾)𝑧𝑋(lt‘𝐾)(𝐹𝑧)))
38 simplr3 1103 . . . . . . . . . 10 (((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) ∧ 𝑧𝐵) → 𝑌𝐵)
391, 2, 3lautlt 34854 . . . . . . . . . 10 ((𝐾𝐴 ∧ (𝐹𝐼 ∧ (𝐹𝑧) ∈ 𝐵𝑌𝐵)) → ((𝐹𝑧)(lt‘𝐾)𝑌 ↔ (𝐹‘(𝐹𝑧))(lt‘𝐾)(𝐹𝑌)))
4025, 26, 31, 38, 39syl13anc 1325 . . . . . . . . 9 (((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) ∧ 𝑧𝐵) → ((𝐹𝑧)(lt‘𝐾)𝑌 ↔ (𝐹‘(𝐹𝑧))(lt‘𝐾)(𝐹𝑌)))
4135breq1d 4623 . . . . . . . . 9 (((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) ∧ 𝑧𝐵) → ((𝐹‘(𝐹𝑧))(lt‘𝐾)(𝐹𝑌) ↔ 𝑧(lt‘𝐾)(𝐹𝑌)))
4240, 41bitr2d 269 . . . . . . . 8 (((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) ∧ 𝑧𝐵) → (𝑧(lt‘𝐾)(𝐹𝑌) ↔ (𝐹𝑧)(lt‘𝐾)𝑌))
4337, 42anbi12d 746 . . . . . . 7 (((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) ∧ 𝑧𝐵) → (((𝐹𝑋)(lt‘𝐾)𝑧𝑧(lt‘𝐾)(𝐹𝑌)) ↔ (𝑋(lt‘𝐾)(𝐹𝑧) ∧ (𝐹𝑧)(lt‘𝐾)𝑌)))
44 breq2 4617 . . . . . . . . . . 11 (𝑤 = (𝐹𝑧) → (𝑋(lt‘𝐾)𝑤𝑋(lt‘𝐾)(𝐹𝑧)))
45 breq1 4616 . . . . . . . . . . 11 (𝑤 = (𝐹𝑧) → (𝑤(lt‘𝐾)𝑌 ↔ (𝐹𝑧)(lt‘𝐾)𝑌))
4644, 45anbi12d 746 . . . . . . . . . 10 (𝑤 = (𝐹𝑧) → ((𝑋(lt‘𝐾)𝑤𝑤(lt‘𝐾)𝑌) ↔ (𝑋(lt‘𝐾)(𝐹𝑧) ∧ (𝐹𝑧)(lt‘𝐾)𝑌)))
4746rspcev 3295 . . . . . . . . 9 (((𝐹𝑧) ∈ 𝐵 ∧ (𝑋(lt‘𝐾)(𝐹𝑧) ∧ (𝐹𝑧)(lt‘𝐾)𝑌)) → ∃𝑤𝐵 (𝑋(lt‘𝐾)𝑤𝑤(lt‘𝐾)𝑌))
4847ex 450 . . . . . . . 8 ((𝐹𝑧) ∈ 𝐵 → ((𝑋(lt‘𝐾)(𝐹𝑧) ∧ (𝐹𝑧)(lt‘𝐾)𝑌) → ∃𝑤𝐵 (𝑋(lt‘𝐾)𝑤𝑤(lt‘𝐾)𝑌)))
4931, 48syl 17 . . . . . . 7 (((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) ∧ 𝑧𝐵) → ((𝑋(lt‘𝐾)(𝐹𝑧) ∧ (𝐹𝑧)(lt‘𝐾)𝑌) → ∃𝑤𝐵 (𝑋(lt‘𝐾)𝑤𝑤(lt‘𝐾)𝑌)))
5043, 49sylbid 230 . . . . . 6 (((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) ∧ 𝑧𝐵) → (((𝐹𝑋)(lt‘𝐾)𝑧𝑧(lt‘𝐾)(𝐹𝑌)) → ∃𝑤𝐵 (𝑋(lt‘𝐾)𝑤𝑤(lt‘𝐾)𝑌)))
5150rexlimdva 3024 . . . . 5 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (∃𝑧𝐵 ((𝐹𝑋)(lt‘𝐾)𝑧𝑧(lt‘𝐾)(𝐹𝑌)) → ∃𝑤𝐵 (𝑋(lt‘𝐾)𝑤𝑤(lt‘𝐾)𝑌)))
5224, 51impbid 202 . . . 4 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (∃𝑤𝐵 (𝑋(lt‘𝐾)𝑤𝑤(lt‘𝐾)𝑌) ↔ ∃𝑧𝐵 ((𝐹𝑋)(lt‘𝐾)𝑧𝑧(lt‘𝐾)(𝐹𝑌))))
5352notbid 308 . . 3 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (¬ ∃𝑤𝐵 (𝑋(lt‘𝐾)𝑤𝑤(lt‘𝐾)𝑌) ↔ ¬ ∃𝑧𝐵 ((𝐹𝑋)(lt‘𝐾)𝑧𝑧(lt‘𝐾)(𝐹𝑌))))
544, 53anbi12d 746 . 2 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝑋(lt‘𝐾)𝑌 ∧ ¬ ∃𝑤𝐵 (𝑋(lt‘𝐾)𝑤𝑤(lt‘𝐾)𝑌)) ↔ ((𝐹𝑋)(lt‘𝐾)(𝐹𝑌) ∧ ¬ ∃𝑧𝐵 ((𝐹𝑋)(lt‘𝐾)𝑧𝑧(lt‘𝐾)(𝐹𝑌)))))
55 lautcvr.c . . . 4 𝐶 = ( ⋖ ‘𝐾)
561, 2, 55cvrval 34033 . . 3 ((𝐾𝐴𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ (𝑋(lt‘𝐾)𝑌 ∧ ¬ ∃𝑤𝐵 (𝑋(lt‘𝐾)𝑤𝑤(lt‘𝐾)𝑌))))
57563adant3r1 1271 . 2 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋𝐶𝑌 ↔ (𝑋(lt‘𝐾)𝑌 ∧ ¬ ∃𝑤𝐵 (𝑋(lt‘𝐾)𝑤𝑤(lt‘𝐾)𝑌))))
58 simpl 473 . . 3 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝐾𝐴)
59 simpr1 1065 . . . 4 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝐹𝐼)
60 simpr2 1066 . . . 4 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝑋𝐵)
611, 3lautcl 34850 . . . 4 (((𝐾𝐴𝐹𝐼) ∧ 𝑋𝐵) → (𝐹𝑋) ∈ 𝐵)
6258, 59, 60, 61syl21anc 1322 . . 3 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑋) ∈ 𝐵)
63 simpr3 1067 . . . 4 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝑌𝐵)
641, 3lautcl 34850 . . . 4 (((𝐾𝐴𝐹𝐼) ∧ 𝑌𝐵) → (𝐹𝑌) ∈ 𝐵)
6558, 59, 63, 64syl21anc 1322 . . 3 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑌) ∈ 𝐵)
661, 2, 55cvrval 34033 . . 3 ((𝐾𝐴 ∧ (𝐹𝑋) ∈ 𝐵 ∧ (𝐹𝑌) ∈ 𝐵) → ((𝐹𝑋)𝐶(𝐹𝑌) ↔ ((𝐹𝑋)(lt‘𝐾)(𝐹𝑌) ∧ ¬ ∃𝑧𝐵 ((𝐹𝑋)(lt‘𝐾)𝑧𝑧(lt‘𝐾)(𝐹𝑌)))))
6758, 62, 65, 66syl3anc 1323 . 2 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝐹𝑋)𝐶(𝐹𝑌) ↔ ((𝐹𝑋)(lt‘𝐾)(𝐹𝑌) ∧ ¬ ∃𝑧𝐵 ((𝐹𝑋)(lt‘𝐾)𝑧𝑧(lt‘𝐾)(𝐹𝑌)))))
6854, 57, 673bitr4d 300 1 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋𝐶𝑌 ↔ (𝐹𝑋)𝐶(𝐹𝑌)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987  ∃wrex 2908   class class class wbr 4613  ◡ccnv 5073  –1-1-onto→wf1o 5846  ‘cfv 5847  Basecbs 15781  ltcplt 16862   ⋖ ccvr 34026  LAutclaut 34748 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-map 7804  df-plt 16879  df-covers 34030  df-laut 34752 This theorem is referenced by:  ltrncvr  34896
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