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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dia1elN | Structured version Visualization version GIF version |
Description: The largest subspace in the range of partial isomorphism A. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dia1.h | β’ π» = (LHypβπΎ) |
dia1.t | β’ π = ((LTrnβπΎ)βπ) |
dia1.i | β’ πΌ = ((DIsoAβπΎ)βπ) |
Ref | Expression |
---|---|
dia1elN | β’ ((πΎ β HL β§ π β π») β π β ran πΌ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dia1.h | . . 3 β’ π» = (LHypβπΎ) | |
2 | dia1.t | . . 3 β’ π = ((LTrnβπΎ)βπ) | |
3 | dia1.i | . . 3 β’ πΌ = ((DIsoAβπΎ)βπ) | |
4 | 1, 2, 3 | dia1N 40435 | . 2 β’ ((πΎ β HL β§ π β π») β (πΌβπ) = π) |
5 | 1, 3 | diaf11N 40431 | . . . 4 β’ ((πΎ β HL β§ π β π») β πΌ:dom πΌβ1-1-ontoβran πΌ) |
6 | f1ofun 6828 | . . . 4 β’ (πΌ:dom πΌβ1-1-ontoβran πΌ β Fun πΌ) | |
7 | 5, 6 | syl 17 | . . 3 β’ ((πΎ β HL β§ π β π») β Fun πΌ) |
8 | 1, 3 | dia1eldmN 40423 | . . 3 β’ ((πΎ β HL β§ π β π») β π β dom πΌ) |
9 | fvelrn 7071 | . . 3 β’ ((Fun πΌ β§ π β dom πΌ) β (πΌβπ) β ran πΌ) | |
10 | 7, 8, 9 | syl2anc 583 | . 2 β’ ((πΎ β HL β§ π β π») β (πΌβπ) β ran πΌ) |
11 | 4, 10 | eqeltrrd 2828 | 1 β’ ((πΎ β HL β§ π β π») β π β ran πΌ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 dom cdm 5669 ran crn 5670 Fun wfun 6530 β1-1-ontoβwf1o 6535 βcfv 6536 HLchlt 38731 LHypclh 39366 LTrncltrn 39483 DIsoAcdia 40410 |
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 38334 |
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 18258 df-poset 18276 df-plt 18293 df-lub 18309 df-glb 18310 df-join 18311 df-meet 18312 df-p0 18388 df-p1 18389 df-lat 18395 df-clat 18462 df-oposet 38557 df-ol 38559 df-oml 38560 df-covers 38647 df-ats 38648 df-atl 38679 df-cvlat 38703 df-hlat 38732 df-llines 38880 df-lplanes 38881 df-lvols 38882 df-lines 38883 df-psubsp 38885 df-pmap 38886 df-padd 39178 df-lhyp 39370 df-laut 39371 df-ldil 39486 df-ltrn 39487 df-trl 39541 df-disoa 40411 |
This theorem is referenced by: docaclN 40506 |
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