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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 2737 | . . . 4 β’ (BaseβπΎ) = (BaseβπΎ) | |
2 | eqid 2737 | . . . 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 39500 | . . 3 β’ ((πΎ β HL β§ π β π») β πΌ Fn {π₯ β (BaseβπΎ) β£ π₯(leβπΎ)π}) |
6 | fnfun 6603 | . . . 4 β’ (πΌ Fn {π₯ β (BaseβπΎ) β£ π₯(leβπΎ)π} β Fun πΌ) | |
7 | funfn 6532 | . . . 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 2738 | . 2 β’ ((πΎ β HL β§ π β π») β ran πΌ = ran πΌ) | |
11 | 1, 2, 3, 4 | diaeldm 39502 | . . . . 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 39502 | . . . . 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 632 | . . . 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 39514 | . . . . . 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 1123 | . . . 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 7222 | . 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 1344 | 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 397 β§ w3a 1088 = wceq 1542 β wcel 2107 βwral 3065 {crab 3408 class class class wbr 5106 dom cdm 5634 ran crn 5635 Fun wfun 6491 Fn wfn 6492 β1-1-ontoβwf1o 6496 βcfv 6497 Basecbs 17084 lecple 17141 HLchlt 37815 LHypclh 38450 DIsoAcdia 39494 |
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 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-riotaBAD 37418 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-iin 4958 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-undef 8205 df-map 8768 df-proset 18185 df-poset 18203 df-plt 18220 df-lub 18236 df-glb 18237 df-join 18238 df-meet 18239 df-p0 18315 df-p1 18316 df-lat 18322 df-clat 18389 df-oposet 37641 df-ol 37643 df-oml 37644 df-covers 37731 df-ats 37732 df-atl 37763 df-cvlat 37787 df-hlat 37816 df-llines 37964 df-lplanes 37965 df-lvols 37966 df-lines 37967 df-psubsp 37969 df-pmap 37970 df-padd 38262 df-lhyp 38454 df-laut 38455 df-ldil 38570 df-ltrn 38571 df-trl 38625 df-disoa 39495 |
This theorem is referenced by: diaclN 39516 diacnvclN 39517 dia1elN 39520 diainN 39523 diaintclN 39524 diasslssN 39525 docaclN 39590 diaocN 39591 doca3N 39593 diaf1oN 39596 |
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