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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dia11N | Structured version Visualization version GIF version |
Description: The partial isomorphism A for a lattice πΎ is one-to-one in the region under co-atom π. Part of Lemma M of [Crawley] p. 120 line 28. (Contributed by NM, 25-Nov-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dia11.b | β’ π΅ = (BaseβπΎ) |
dia11.l | β’ β€ = (leβπΎ) |
dia11.h | β’ π» = (LHypβπΎ) |
dia11.i | β’ πΌ = ((DIsoAβπΎ)βπ) |
Ref | Expression |
---|---|
dia11N | β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β€ π) β§ (π β π΅ β§ π β€ π)) β ((πΌβπ) = (πΌβπ) β π = π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqss 3963 | . 2 β’ ((πΌβπ) = (πΌβπ) β ((πΌβπ) β (πΌβπ) β§ (πΌβπ) β (πΌβπ))) | |
2 | dia11.b | . . . . 5 β’ π΅ = (BaseβπΎ) | |
3 | dia11.l | . . . . 5 β’ β€ = (leβπΎ) | |
4 | dia11.h | . . . . 5 β’ π» = (LHypβπΎ) | |
5 | dia11.i | . . . . 5 β’ πΌ = ((DIsoAβπΎ)βπ) | |
6 | 2, 3, 4, 5 | diaord 39560 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β€ π) β§ (π β π΅ β§ π β€ π)) β ((πΌβπ) β (πΌβπ) β π β€ π)) |
7 | 2, 3, 4, 5 | diaord 39560 | . . . . 5 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β€ π) β§ (π β π΅ β§ π β€ π)) β ((πΌβπ) β (πΌβπ) β π β€ π)) |
8 | 7 | 3com23 1127 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β€ π) β§ (π β π΅ β§ π β€ π)) β ((πΌβπ) β (πΌβπ) β π β€ π)) |
9 | 6, 8 | anbi12d 632 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β€ π) β§ (π β π΅ β§ π β€ π)) β (((πΌβπ) β (πΌβπ) β§ (πΌβπ) β (πΌβπ)) β (π β€ π β§ π β€ π))) |
10 | simp1l 1198 | . . . . 5 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β€ π) β§ (π β π΅ β§ π β€ π)) β πΎ β HL) | |
11 | 10 | hllatd 37876 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β€ π) β§ (π β π΅ β§ π β€ π)) β πΎ β Lat) |
12 | simp2l 1200 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β€ π) β§ (π β π΅ β§ π β€ π)) β π β π΅) | |
13 | simp3l 1202 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β€ π) β§ (π β π΅ β§ π β€ π)) β π β π΅) | |
14 | 2, 3 | latasymb 18339 | . . . 4 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β ((π β€ π β§ π β€ π) β π = π)) |
15 | 11, 12, 13, 14 | syl3anc 1372 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β€ π) β§ (π β π΅ β§ π β€ π)) β ((π β€ π β§ π β€ π) β π = π)) |
16 | 9, 15 | bitrd 279 | . 2 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β€ π) β§ (π β π΅ β§ π β€ π)) β (((πΌβπ) β (πΌβπ) β§ (πΌβπ) β (πΌβπ)) β π = π)) |
17 | 1, 16 | bitrid 283 | 1 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β€ π) β§ (π β π΅ β§ π β€ π)) β ((πΌβπ) = (πΌβπ) β π = π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 β wss 3914 class class class wbr 5109 βcfv 6500 Basecbs 17091 lecple 17148 Latclat 18328 HLchlt 37862 LHypclh 38497 DIsoAcdia 39541 |
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 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-riotaBAD 37465 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-iin 4961 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-1st 7925 df-2nd 7926 df-undef 8208 df-map 8773 df-proset 18192 df-poset 18210 df-plt 18227 df-lub 18243 df-glb 18244 df-join 18245 df-meet 18246 df-p0 18322 df-p1 18323 df-lat 18329 df-clat 18396 df-oposet 37688 df-ol 37690 df-oml 37691 df-covers 37778 df-ats 37779 df-atl 37810 df-cvlat 37834 df-hlat 37863 df-llines 38011 df-lplanes 38012 df-lvols 38013 df-lines 38014 df-psubsp 38016 df-pmap 38017 df-padd 38309 df-lhyp 38501 df-laut 38502 df-ldil 38617 df-ltrn 38618 df-trl 38672 df-disoa 39542 |
This theorem is referenced by: diaf11N 39562 |
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