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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lautle | Structured version Visualization version GIF version |
Description: Less-than or equal property of a lattice automorphism. (Contributed by NM, 19-May-2012.) |
Ref | Expression |
---|---|
lautset.b | β’ π΅ = (BaseβπΎ) |
lautset.l | β’ β€ = (leβπΎ) |
lautset.i | β’ πΌ = (LAutβπΎ) |
Ref | Expression |
---|---|
lautle | β’ (((πΎ β π β§ πΉ β πΌ) β§ (π β π΅ β§ π β π΅)) β (π β€ π β (πΉβπ) β€ (πΉβπ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lautset.b | . . . 4 β’ π΅ = (BaseβπΎ) | |
2 | lautset.l | . . . 4 β’ β€ = (leβπΎ) | |
3 | lautset.i | . . . 4 β’ πΌ = (LAutβπΎ) | |
4 | 1, 2, 3 | islaut 38575 | . . 3 β’ (πΎ β π β (πΉ β πΌ β (πΉ:π΅β1-1-ontoβπ΅ β§ βπ₯ β π΅ βπ¦ β π΅ (π₯ β€ π¦ β (πΉβπ₯) β€ (πΉβπ¦))))) |
5 | 4 | simplbda 501 | . 2 β’ ((πΎ β π β§ πΉ β πΌ) β βπ₯ β π΅ βπ¦ β π΅ (π₯ β€ π¦ β (πΉβπ₯) β€ (πΉβπ¦))) |
6 | breq1 5113 | . . . 4 β’ (π₯ = π β (π₯ β€ π¦ β π β€ π¦)) | |
7 | fveq2 6847 | . . . . 5 β’ (π₯ = π β (πΉβπ₯) = (πΉβπ)) | |
8 | 7 | breq1d 5120 | . . . 4 β’ (π₯ = π β ((πΉβπ₯) β€ (πΉβπ¦) β (πΉβπ) β€ (πΉβπ¦))) |
9 | 6, 8 | bibi12d 346 | . . 3 β’ (π₯ = π β ((π₯ β€ π¦ β (πΉβπ₯) β€ (πΉβπ¦)) β (π β€ π¦ β (πΉβπ) β€ (πΉβπ¦)))) |
10 | breq2 5114 | . . . 4 β’ (π¦ = π β (π β€ π¦ β π β€ π)) | |
11 | fveq2 6847 | . . . . 5 β’ (π¦ = π β (πΉβπ¦) = (πΉβπ)) | |
12 | 11 | breq2d 5122 | . . . 4 β’ (π¦ = π β ((πΉβπ) β€ (πΉβπ¦) β (πΉβπ) β€ (πΉβπ))) |
13 | 10, 12 | bibi12d 346 | . . 3 β’ (π¦ = π β ((π β€ π¦ β (πΉβπ) β€ (πΉβπ¦)) β (π β€ π β (πΉβπ) β€ (πΉβπ)))) |
14 | 9, 13 | rspc2v 3593 | . 2 β’ ((π β π΅ β§ π β π΅) β (βπ₯ β π΅ βπ¦ β π΅ (π₯ β€ π¦ β (πΉβπ₯) β€ (πΉβπ¦)) β (π β€ π β (πΉβπ) β€ (πΉβπ)))) |
15 | 5, 14 | mpan9 508 | 1 β’ (((πΎ β π β§ πΉ β πΌ) β§ (π β π΅ β§ π β π΅)) β (π β€ π β (πΉβπ) β€ (πΉβπ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 βwral 3065 class class class wbr 5110 β1-1-ontoβwf1o 6500 βcfv 6501 Basecbs 17090 lecple 17147 LAutclaut 38477 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 df-map 8774 df-laut 38481 |
This theorem is referenced by: lautcnvle 38581 lautlt 38583 lautj 38585 lautm 38586 lauteq 38587 lautco 38589 ltrnle 38621 |
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