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Theorem lautcvr 35798
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 2724 . . . 4 (lt‘𝐾) = (lt‘𝐾)
3 lautcvr.i . . . 4 𝐼 = (LAut‘𝐾)
41, 2, 3lautlt 35797 . . 3 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋(lt‘𝐾)𝑌 ↔ (𝐹𝑋)(lt‘𝐾)(𝐹𝑌)))
5 simpll 807 . . . . . . . . 9 (((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) ∧ 𝑤𝐵) → 𝐾𝐴)
6 simplr1 1237 . . . . . . . . 9 (((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) ∧ 𝑤𝐵) → 𝐹𝐼)
7 simplr2 1239 . . . . . . . . 9 (((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) ∧ 𝑤𝐵) → 𝑋𝐵)
8 simpr 479 . . . . . . . . 9 (((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) ∧ 𝑤𝐵) → 𝑤𝐵)
91, 2, 3lautlt 35797 . . . . . . . . 9 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑤𝐵)) → (𝑋(lt‘𝐾)𝑤 ↔ (𝐹𝑋)(lt‘𝐾)(𝐹𝑤)))
105, 6, 7, 8, 9syl13anc 1441 . . . . . . . 8 (((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) ∧ 𝑤𝐵) → (𝑋(lt‘𝐾)𝑤 ↔ (𝐹𝑋)(lt‘𝐾)(𝐹𝑤)))
11 simplr3 1241 . . . . . . . . 9 (((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) ∧ 𝑤𝐵) → 𝑌𝐵)
121, 2, 3lautlt 35797 . . . . . . . . 9 ((𝐾𝐴 ∧ (𝐹𝐼𝑤𝐵𝑌𝐵)) → (𝑤(lt‘𝐾)𝑌 ↔ (𝐹𝑤)(lt‘𝐾)(𝐹𝑌)))
135, 6, 8, 11, 12syl13anc 1441 . . . . . . . 8 (((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) ∧ 𝑤𝐵) → (𝑤(lt‘𝐾)𝑌 ↔ (𝐹𝑤)(lt‘𝐾)(𝐹𝑌)))
1410, 13anbi12d 749 . . . . . . 7 (((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) ∧ 𝑤𝐵) → ((𝑋(lt‘𝐾)𝑤𝑤(lt‘𝐾)𝑌) ↔ ((𝐹𝑋)(lt‘𝐾)(𝐹𝑤) ∧ (𝐹𝑤)(lt‘𝐾)(𝐹𝑌))))
151, 3lautcl 35793 . . . . . . . . 9 (((𝐾𝐴𝐹𝐼) ∧ 𝑤𝐵) → (𝐹𝑤) ∈ 𝐵)
165, 6, 8, 15syl21anc 1438 . . . . . . . 8 (((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) ∧ 𝑤𝐵) → (𝐹𝑤) ∈ 𝐵)
17 breq2 4764 . . . . . . . . . . 11 (𝑧 = (𝐹𝑤) → ((𝐹𝑋)(lt‘𝐾)𝑧 ↔ (𝐹𝑋)(lt‘𝐾)(𝐹𝑤)))
18 breq1 4763 . . . . . . . . . . 11 (𝑧 = (𝐹𝑤) → (𝑧(lt‘𝐾)(𝐹𝑌) ↔ (𝐹𝑤)(lt‘𝐾)(𝐹𝑌)))
1917, 18anbi12d 749 . . . . . . . . . 10 (𝑧 = (𝐹𝑤) → (((𝐹𝑋)(lt‘𝐾)𝑧𝑧(lt‘𝐾)(𝐹𝑌)) ↔ ((𝐹𝑋)(lt‘𝐾)(𝐹𝑤) ∧ (𝐹𝑤)(lt‘𝐾)(𝐹𝑌))))
2019rspcev 3413 . . . . . . . . 9 (((𝐹𝑤) ∈ 𝐵 ∧ ((𝐹𝑋)(lt‘𝐾)(𝐹𝑤) ∧ (𝐹𝑤)(lt‘𝐾)(𝐹𝑌))) → ∃𝑧𝐵 ((𝐹𝑋)(lt‘𝐾)𝑧𝑧(lt‘𝐾)(𝐹𝑌)))
2120ex 449 . . . . . . . 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 3133 . . . . 5 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (∃𝑤𝐵 (𝑋(lt‘𝐾)𝑤𝑤(lt‘𝐾)𝑌) → ∃𝑧𝐵 ((𝐹𝑋)(lt‘𝐾)𝑧𝑧(lt‘𝐾)(𝐹𝑌))))
25 simpll 807 . . . . . . . . . 10 (((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) ∧ 𝑧𝐵) → 𝐾𝐴)
26 simplr1 1237 . . . . . . . . . 10 (((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) ∧ 𝑧𝐵) → 𝐹𝐼)
27 simplr2 1239 . . . . . . . . . 10 (((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) ∧ 𝑧𝐵) → 𝑋𝐵)
281, 3laut1o 35791 . . . . . . . . . . . 12 ((𝐾𝐴𝐹𝐼) → 𝐹:𝐵1-1-onto𝐵)
2925, 26, 28syl2anc 696 . . . . . . . . . . 11 (((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) ∧ 𝑧𝐵) → 𝐹:𝐵1-1-onto𝐵)
30 f1ocnvdm 6655 . . . . . . . . . . 11 ((𝐹:𝐵1-1-onto𝐵𝑧𝐵) → (𝐹𝑧) ∈ 𝐵)
3129, 30sylancom 704 . . . . . . . . . 10 (((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) ∧ 𝑧𝐵) → (𝐹𝑧) ∈ 𝐵)
321, 2, 3lautlt 35797 . . . . . . . . . 10 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵 ∧ (𝐹𝑧) ∈ 𝐵)) → (𝑋(lt‘𝐾)(𝐹𝑧) ↔ (𝐹𝑋)(lt‘𝐾)(𝐹‘(𝐹𝑧))))
3325, 26, 27, 31, 32syl13anc 1441 . . . . . . . . 9 (((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) ∧ 𝑧𝐵) → (𝑋(lt‘𝐾)(𝐹𝑧) ↔ (𝐹𝑋)(lt‘𝐾)(𝐹‘(𝐹𝑧))))
34 f1ocnvfv2 6648 . . . . . . . . . . 11 ((𝐹:𝐵1-1-onto𝐵𝑧𝐵) → (𝐹‘(𝐹𝑧)) = 𝑧)
3529, 34sylancom 704 . . . . . . . . . 10 (((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) ∧ 𝑧𝐵) → (𝐹‘(𝐹𝑧)) = 𝑧)
3635breq2d 4772 . . . . . . . . 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 1241 . . . . . . . . . 10 (((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) ∧ 𝑧𝐵) → 𝑌𝐵)
391, 2, 3lautlt 35797 . . . . . . . . . 10 ((𝐾𝐴 ∧ (𝐹𝐼 ∧ (𝐹𝑧) ∈ 𝐵𝑌𝐵)) → ((𝐹𝑧)(lt‘𝐾)𝑌 ↔ (𝐹‘(𝐹𝑧))(lt‘𝐾)(𝐹𝑌)))
4025, 26, 31, 38, 39syl13anc 1441 . . . . . . . . 9 (((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) ∧ 𝑧𝐵) → ((𝐹𝑧)(lt‘𝐾)𝑌 ↔ (𝐹‘(𝐹𝑧))(lt‘𝐾)(𝐹𝑌)))
4135breq1d 4770 . . . . . . . . 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 749 . . . . . . 7 (((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) ∧ 𝑧𝐵) → (((𝐹𝑋)(lt‘𝐾)𝑧𝑧(lt‘𝐾)(𝐹𝑌)) ↔ (𝑋(lt‘𝐾)(𝐹𝑧) ∧ (𝐹𝑧)(lt‘𝐾)𝑌)))
44 breq2 4764 . . . . . . . . . . 11 (𝑤 = (𝐹𝑧) → (𝑋(lt‘𝐾)𝑤𝑋(lt‘𝐾)(𝐹𝑧)))
45 breq1 4763 . . . . . . . . . . 11 (𝑤 = (𝐹𝑧) → (𝑤(lt‘𝐾)𝑌 ↔ (𝐹𝑧)(lt‘𝐾)𝑌))
4644, 45anbi12d 749 . . . . . . . . . 10 (𝑤 = (𝐹𝑧) → ((𝑋(lt‘𝐾)𝑤𝑤(lt‘𝐾)𝑌) ↔ (𝑋(lt‘𝐾)(𝐹𝑧) ∧ (𝐹𝑧)(lt‘𝐾)𝑌)))
4746rspcev 3413 . . . . . . . . 9 (((𝐹𝑧) ∈ 𝐵 ∧ (𝑋(lt‘𝐾)(𝐹𝑧) ∧ (𝐹𝑧)(lt‘𝐾)𝑌)) → ∃𝑤𝐵 (𝑋(lt‘𝐾)𝑤𝑤(lt‘𝐾)𝑌))
4847ex 449 . . . . . . . 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 3133 . . . . 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 307 . . 3 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (¬ ∃𝑤𝐵 (𝑋(lt‘𝐾)𝑤𝑤(lt‘𝐾)𝑌) ↔ ¬ ∃𝑧𝐵 ((𝐹𝑋)(lt‘𝐾)𝑧𝑧(lt‘𝐾)(𝐹𝑌))))
544, 53anbi12d 749 . 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 34976 . . 3 ((𝐾𝐴𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ (𝑋(lt‘𝐾)𝑌 ∧ ¬ ∃𝑤𝐵 (𝑋(lt‘𝐾)𝑤𝑤(lt‘𝐾)𝑌))))
57563adant3r1 1174 . 2 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋𝐶𝑌 ↔ (𝑋(lt‘𝐾)𝑌 ∧ ¬ ∃𝑤𝐵 (𝑋(lt‘𝐾)𝑤𝑤(lt‘𝐾)𝑌))))
58 simpl 474 . . 3 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝐾𝐴)
59 simpr1 1210 . . . 4 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝐹𝐼)
60 simpr2 1212 . . . 4 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝑋𝐵)
611, 3lautcl 35793 . . . 4 (((𝐾𝐴𝐹𝐼) ∧ 𝑋𝐵) → (𝐹𝑋) ∈ 𝐵)
6258, 59, 60, 61syl21anc 1438 . . 3 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑋) ∈ 𝐵)
63 simpr3 1214 . . . 4 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝑌𝐵)
641, 3lautcl 35793 . . . 4 (((𝐾𝐴𝐹𝐼) ∧ 𝑌𝐵) → (𝐹𝑌) ∈ 𝐵)
6558, 59, 63, 64syl21anc 1438 . . 3 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑌) ∈ 𝐵)
661, 2, 55cvrval 34976 . . 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 1439 . 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 383  w3a 1072   = wceq 1596  wcel 2103  wrex 3015   class class class wbr 4760  ccnv 5217  1-1-ontowf1o 6000  cfv 6001  Basecbs 15980  ltcplt 17063  ccvr 34969  LAutclaut 35691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-map 7976  df-plt 17080  df-covers 34973  df-laut 35695
This theorem is referenced by:  ltrncvr  35839
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