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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 3990 | . 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 40422 | . . . 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 40422 | . . . . 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 1123 | . . . 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 630 | . . 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 1194 | . . . . 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 38738 | . . . 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 1196 | . . . 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 1198 | . . . 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 18403 | . . . 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 1368 | . . 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 395 β§ w3a 1084 = wceq 1533 β wcel 2098 β wss 3941 class class class wbr 5139 βcfv 6534 Basecbs 17149 lecple 17209 Latclat 18392 HLchlt 38724 LHypclh 39359 DIsoAcdia 40403 |
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 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-riotaBAD 38327 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-iin 4991 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-1st 7969 df-2nd 7970 df-undef 8254 df-map 8819 df-proset 18256 df-poset 18274 df-plt 18291 df-lub 18307 df-glb 18308 df-join 18309 df-meet 18310 df-p0 18386 df-p1 18387 df-lat 18393 df-clat 18460 df-oposet 38550 df-ol 38552 df-oml 38553 df-covers 38640 df-ats 38641 df-atl 38672 df-cvlat 38696 df-hlat 38725 df-llines 38873 df-lplanes 38874 df-lvols 38875 df-lines 38876 df-psubsp 38878 df-pmap 38879 df-padd 39171 df-lhyp 39363 df-laut 39364 df-ldil 39479 df-ltrn 39480 df-trl 39534 df-disoa 40404 |
This theorem is referenced by: diaf11N 40424 |
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