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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 39508 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β πΉ:π΅β1-1-ontoβπ΅) |
5 | f1ocnvdm 7279 | . . . 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 39521 | . . 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 7271 | . . . 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 2812 | . 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 280 | 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 395 β§ w3a 1084 = wceq 1533 β wcel 2098 β‘ccnv 5668 β1-1-ontoβwf1o 6536 βcfv 6537 Basecbs 17153 Atomscatm 38646 HLchlt 38733 LHypclh 39368 LTrncltrn 39485 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-map 8824 df-plt 18295 df-glb 18312 df-p0 18390 df-oposet 38559 df-ol 38561 df-oml 38562 df-covers 38649 df-ats 38650 df-hlat 38734 df-lhyp 39372 df-laut 39373 df-ldil 39488 df-ltrn 39489 |
This theorem is referenced by: ltrncnvat 39525 |
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