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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 2728 | . . . 4 β’ (BaseβπΎ) = (BaseβπΎ) | |
2 | eqid 2728 | . . . 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 40539 | . . 3 β’ ((πΎ β HL β§ π β π») β πΌ Fn {π₯ β (BaseβπΎ) β£ π₯(leβπΎ)π}) |
6 | fnfun 6659 | . . . 4 β’ (πΌ Fn {π₯ β (BaseβπΎ) β£ π₯(leβπΎ)π} β Fun πΌ) | |
7 | funfn 6588 | . . . 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 2729 | . 2 β’ ((πΎ β HL β§ π β π») β ran πΌ = ran πΌ) | |
11 | 1, 2, 3, 4 | diaeldm 40541 | . . . . 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 40541 | . . . . 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 40553 | . . . . . 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 3196 | . 2 β’ ((πΎ β HL β§ π β π») β βπ₯ β dom πΌβπ¦ β dom πΌ((πΌβπ₯) = (πΌβπ¦) β π₯ = π¦)) |
19 | dff1o6 7290 | . 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 394 β§ w3a 1084 = wceq 1533 β wcel 2098 βwral 3058 {crab 3430 class class class wbr 5152 dom cdm 5682 ran crn 5683 Fun wfun 6547 Fn wfn 6548 β1-1-ontoβwf1o 6552 βcfv 6553 Basecbs 17187 lecple 17247 HLchlt 38854 LHypclh 39489 DIsoAcdia 40533 |
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 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-riotaBAD 38457 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 7999 df-2nd 8000 df-undef 8285 df-map 8853 df-proset 18294 df-poset 18312 df-plt 18329 df-lub 18345 df-glb 18346 df-join 18347 df-meet 18348 df-p0 18424 df-p1 18425 df-lat 18431 df-clat 18498 df-oposet 38680 df-ol 38682 df-oml 38683 df-covers 38770 df-ats 38771 df-atl 38802 df-cvlat 38826 df-hlat 38855 df-llines 39003 df-lplanes 39004 df-lvols 39005 df-lines 39006 df-psubsp 39008 df-pmap 39009 df-padd 39301 df-lhyp 39493 df-laut 39494 df-ldil 39609 df-ltrn 39610 df-trl 39664 df-disoa 40534 |
This theorem is referenced by: diaclN 40555 diacnvclN 40556 dia1elN 40559 diainN 40562 diaintclN 40563 diasslssN 40564 docaclN 40629 diaocN 40630 doca3N 40632 diaf1oN 40635 |
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