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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 39556 | . . 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 499 | . 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 5151 | . . . 4 β’ (π₯ = π β (π₯ β€ π¦ β π β€ π¦)) | |
7 | fveq2 6897 | . . . . 5 β’ (π₯ = π β (πΉβπ₯) = (πΉβπ)) | |
8 | 7 | breq1d 5158 | . . . 4 β’ (π₯ = π β ((πΉβπ₯) β€ (πΉβπ¦) β (πΉβπ) β€ (πΉβπ¦))) |
9 | 6, 8 | bibi12d 345 | . . 3 β’ (π₯ = π β ((π₯ β€ π¦ β (πΉβπ₯) β€ (πΉβπ¦)) β (π β€ π¦ β (πΉβπ) β€ (πΉβπ¦)))) |
10 | breq2 5152 | . . . 4 β’ (π¦ = π β (π β€ π¦ β π β€ π)) | |
11 | fveq2 6897 | . . . . 5 β’ (π¦ = π β (πΉβπ¦) = (πΉβπ)) | |
12 | 11 | breq2d 5160 | . . . 4 β’ (π¦ = π β ((πΉβπ) β€ (πΉβπ¦) β (πΉβπ) β€ (πΉβπ))) |
13 | 10, 12 | bibi12d 345 | . . 3 β’ (π¦ = π β ((π β€ π¦ β (πΉβπ) β€ (πΉβπ¦)) β (π β€ π β (πΉβπ) β€ (πΉβπ)))) |
14 | 9, 13 | rspc2v 3620 | . 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 506 | 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 395 = wceq 1534 β wcel 2099 βwral 3058 class class class wbr 5148 β1-1-ontoβwf1o 6547 βcfv 6548 Basecbs 17180 lecple 17240 LAutclaut 39458 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-map 8847 df-laut 39462 |
This theorem is referenced by: lautcnvle 39562 lautlt 39564 lautj 39566 lautm 39567 lauteq 39568 lautco 39570 ltrnle 39602 |
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