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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > diaf11N | Structured version Visualization version GIF version |
Description: The partial isomorphism A for a lattice πΎ is a one-to-one function. Part of Lemma M of [Crawley] p. 120 line 27. (Contributed by NM, 4-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dia1o.h | β’ π» = (LHypβπΎ) |
dia1o.i | β’ πΌ = ((DIsoAβπΎ)βπ) |
Ref | Expression |
---|---|
diaf11N | β’ ((πΎ β HL β§ π β π») β πΌ:dom πΌβ1-1-ontoβran πΌ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2726 | . . . 4 β’ (BaseβπΎ) = (BaseβπΎ) | |
2 | eqid 2726 | . . . 4 β’ (leβπΎ) = (leβπΎ) | |
3 | dia1o.h | . . . 4 β’ π» = (LHypβπΎ) | |
4 | dia1o.i | . . . 4 β’ πΌ = ((DIsoAβπΎ)βπ) | |
5 | 1, 2, 3, 4 | diafn 40417 | . . 3 β’ ((πΎ β HL β§ π β π») β πΌ Fn {π₯ β (BaseβπΎ) β£ π₯(leβπΎ)π}) |
6 | fnfun 6642 | . . . 4 β’ (πΌ Fn {π₯ β (BaseβπΎ) β£ π₯(leβπΎ)π} β Fun πΌ) | |
7 | funfn 6571 | . . . 4 β’ (Fun πΌ β πΌ Fn dom πΌ) | |
8 | 6, 7 | sylib 217 | . . 3 β’ (πΌ Fn {π₯ β (BaseβπΎ) β£ π₯(leβπΎ)π} β πΌ Fn dom πΌ) |
9 | 5, 8 | syl 17 | . 2 β’ ((πΎ β HL β§ π β π») β πΌ Fn dom πΌ) |
10 | eqidd 2727 | . 2 β’ ((πΎ β HL β§ π β π») β ran πΌ = ran πΌ) | |
11 | 1, 2, 3, 4 | diaeldm 40419 | . . . . 5 β’ ((πΎ β HL β§ π β π») β (π₯ β dom πΌ β (π₯ β (BaseβπΎ) β§ π₯(leβπΎ)π))) |
12 | 1, 2, 3, 4 | diaeldm 40419 | . . . . 5 β’ ((πΎ β HL β§ π β π») β (π¦ β dom πΌ β (π¦ β (BaseβπΎ) β§ π¦(leβπΎ)π))) |
13 | 11, 12 | anbi12d 630 | . . . 4 β’ ((πΎ β HL β§ π β π») β ((π₯ β dom πΌ β§ π¦ β dom πΌ) β ((π₯ β (BaseβπΎ) β§ π₯(leβπΎ)π) β§ (π¦ β (BaseβπΎ) β§ π¦(leβπΎ)π)))) |
14 | 1, 2, 3, 4 | dia11N 40431 | . . . . . 6 β’ (((πΎ β HL β§ π β π») β§ (π₯ β (BaseβπΎ) β§ π₯(leβπΎ)π) β§ (π¦ β (BaseβπΎ) β§ π¦(leβπΎ)π)) β ((πΌβπ₯) = (πΌβπ¦) β π₯ = π¦)) |
15 | 14 | biimpd 228 | . . . . 5 β’ (((πΎ β HL β§ π β π») β§ (π₯ β (BaseβπΎ) β§ π₯(leβπΎ)π) β§ (π¦ β (BaseβπΎ) β§ π¦(leβπΎ)π)) β ((πΌβπ₯) = (πΌβπ¦) β π₯ = π¦)) |
16 | 15 | 3expib 1119 | . . . 4 β’ ((πΎ β HL β§ π β π») β (((π₯ β (BaseβπΎ) β§ π₯(leβπΎ)π) β§ (π¦ β (BaseβπΎ) β§ π¦(leβπΎ)π)) β ((πΌβπ₯) = (πΌβπ¦) β π₯ = π¦))) |
17 | 13, 16 | sylbid 239 | . . 3 β’ ((πΎ β HL β§ π β π») β ((π₯ β dom πΌ β§ π¦ β dom πΌ) β ((πΌβπ₯) = (πΌβπ¦) β π₯ = π¦))) |
18 | 17 | ralrimivv 3192 | . 2 β’ ((πΎ β HL β§ π β π») β βπ₯ β dom πΌβπ¦ β dom πΌ((πΌβπ₯) = (πΌβπ¦) β π₯ = π¦)) |
19 | dff1o6 7268 | . 2 β’ (πΌ:dom πΌβ1-1-ontoβran πΌ β (πΌ Fn dom πΌ β§ ran πΌ = ran πΌ β§ βπ₯ β dom πΌβπ¦ β dom πΌ((πΌβπ₯) = (πΌβπ¦) β π₯ = π¦))) | |
20 | 9, 10, 18, 19 | syl3anbrc 1340 | 1 β’ ((πΎ β HL β§ π β π») β πΌ:dom πΌβ1-1-ontoβran πΌ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 βwral 3055 {crab 3426 class class class wbr 5141 dom cdm 5669 ran crn 5670 Fun wfun 6530 Fn wfn 6531 β1-1-ontoβwf1o 6535 βcfv 6536 Basecbs 17150 lecple 17210 HLchlt 38732 LHypclh 39367 DIsoAcdia 40411 |
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 7721 ax-riotaBAD 38335 |
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 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-iin 4993 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 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1st 7971 df-2nd 7972 df-undef 8256 df-map 8821 df-proset 18257 df-poset 18275 df-plt 18292 df-lub 18308 df-glb 18309 df-join 18310 df-meet 18311 df-p0 18387 df-p1 18388 df-lat 18394 df-clat 18461 df-oposet 38558 df-ol 38560 df-oml 38561 df-covers 38648 df-ats 38649 df-atl 38680 df-cvlat 38704 df-hlat 38733 df-llines 38881 df-lplanes 38882 df-lvols 38883 df-lines 38884 df-psubsp 38886 df-pmap 38887 df-padd 39179 df-lhyp 39371 df-laut 39372 df-ldil 39487 df-ltrn 39488 df-trl 39542 df-disoa 40412 |
This theorem is referenced by: diaclN 40433 diacnvclN 40434 dia1elN 40437 diainN 40440 diaintclN 40441 diasslssN 40442 docaclN 40507 diaocN 40508 doca3N 40510 diaf1oN 40513 |
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