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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ltrncnvatb | Structured version Visualization version GIF version |
Description: The converse of the lattice translation of an atom is an atom. (Contributed by NM, 2-Jun-2012.) |
Ref | Expression |
---|---|
ltrnatb.b | β’ π΅ = (BaseβπΎ) |
ltrnatb.a | β’ π΄ = (AtomsβπΎ) |
ltrnatb.h | β’ π» = (LHypβπΎ) |
ltrnatb.t | β’ π = ((LTrnβπΎ)βπ) |
Ref | Expression |
---|---|
ltrncnvatb | β’ (((πΎ β HL β§ π β π») β§ πΉ β π β§ π β π΅) β (π β π΄ β (β‘πΉβπ) β π΄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltrnatb.b | . . . . 5 β’ π΅ = (BaseβπΎ) | |
2 | ltrnatb.h | . . . . 5 β’ π» = (LHypβπΎ) | |
3 | ltrnatb.t | . . . . 5 β’ π = ((LTrnβπΎ)βπ) | |
4 | 1, 2, 3 | ltrn1o 39652 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β πΉ:π΅β1-1-ontoβπ΅) |
5 | f1ocnvdm 7289 | . . . 4 β’ ((πΉ:π΅β1-1-ontoβπ΅ β§ π β π΅) β (β‘πΉβπ) β π΅) | |
6 | 4, 5 | stoic3 1770 | . . 3 β’ (((πΎ β HL β§ π β π») β§ πΉ β π β§ π β π΅) β (β‘πΉβπ) β π΅) |
7 | ltrnatb.a | . . . 4 β’ π΄ = (AtomsβπΎ) | |
8 | 1, 7, 2, 3 | ltrnatb 39665 | . . 3 β’ (((πΎ β HL β§ π β π») β§ πΉ β π β§ (β‘πΉβπ) β π΅) β ((β‘πΉβπ) β π΄ β (πΉβ(β‘πΉβπ)) β π΄)) |
9 | 6, 8 | syld3an3 1406 | . 2 β’ (((πΎ β HL β§ π β π») β§ πΉ β π β§ π β π΅) β ((β‘πΉβπ) β π΄ β (πΉβ(β‘πΉβπ)) β π΄)) |
10 | f1ocnvfv2 7281 | . . . 4 β’ ((πΉ:π΅β1-1-ontoβπ΅ β§ π β π΅) β (πΉβ(β‘πΉβπ)) = π) | |
11 | 4, 10 | stoic3 1770 | . . 3 β’ (((πΎ β HL β§ π β π») β§ πΉ β π β§ π β π΅) β (πΉβ(β‘πΉβπ)) = π) |
12 | 11 | eleq1d 2810 | . 2 β’ (((πΎ β HL β§ π β π») β§ πΉ β π β§ π β π΅) β ((πΉβ(β‘πΉβπ)) β π΄ β π β π΄)) |
13 | 9, 12 | bitr2d 279 | 1 β’ (((πΎ β HL β§ π β π») β§ πΉ β π β§ π β π΅) β (π β π΄ β (β‘πΉβπ) β π΄)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 β‘ccnv 5671 β1-1-ontoβwf1o 6541 βcfv 6542 Basecbs 17177 Atomscatm 38790 HLchlt 38877 LHypclh 39512 LTrncltrn 39629 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-map 8843 df-plt 18319 df-glb 18336 df-p0 18414 df-oposet 38703 df-ol 38705 df-oml 38706 df-covers 38793 df-ats 38794 df-hlat 38878 df-lhyp 39516 df-laut 39517 df-ldil 39632 df-ltrn 39633 |
This theorem is referenced by: ltrncnvat 39669 |
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