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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > doca3N | Structured version Visualization version GIF version |
Description: Double orthocomplement of partial isomorphism A. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
doca2.h | β’ π» = (LHypβπΎ) |
doca2.i | β’ πΌ = ((DIsoAβπΎ)βπ) |
doca2.n | β’ β₯ = ((ocAβπΎ)βπ) |
Ref | Expression |
---|---|
doca3N | β’ (((πΎ β HL β§ π β π») β§ π β ran πΌ) β ( β₯ β( β₯ βπ)) = π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | doca2.h | . . . 4 β’ π» = (LHypβπΎ) | |
2 | doca2.i | . . . 4 β’ πΌ = ((DIsoAβπΎ)βπ) | |
3 | 1, 2 | diacnvclN 40433 | . . 3 β’ (((πΎ β HL β§ π β π») β§ π β ran πΌ) β (β‘πΌβπ) β dom πΌ) |
4 | doca2.n | . . . 4 β’ β₯ = ((ocAβπΎ)βπ) | |
5 | 1, 2, 4 | doca2N 40508 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (β‘πΌβπ) β dom πΌ) β ( β₯ β( β₯ β(πΌβ(β‘πΌβπ)))) = (πΌβ(β‘πΌβπ))) |
6 | 3, 5 | syldan 590 | . 2 β’ (((πΎ β HL β§ π β π») β§ π β ran πΌ) β ( β₯ β( β₯ β(πΌβ(β‘πΌβπ)))) = (πΌβ(β‘πΌβπ))) |
7 | 1, 2 | diaf11N 40431 | . . . . 5 β’ ((πΎ β HL β§ π β π») β πΌ:dom πΌβ1-1-ontoβran πΌ) |
8 | f1ocnvfv2 7270 | . . . . 5 β’ ((πΌ:dom πΌβ1-1-ontoβran πΌ β§ π β ran πΌ) β (πΌβ(β‘πΌβπ)) = π) | |
9 | 7, 8 | sylan 579 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ π β ran πΌ) β (πΌβ(β‘πΌβπ)) = π) |
10 | 9 | fveq2d 6888 | . . 3 β’ (((πΎ β HL β§ π β π») β§ π β ran πΌ) β ( β₯ β(πΌβ(β‘πΌβπ))) = ( β₯ βπ)) |
11 | 10 | fveq2d 6888 | . 2 β’ (((πΎ β HL β§ π β π») β§ π β ran πΌ) β ( β₯ β( β₯ β(πΌβ(β‘πΌβπ)))) = ( β₯ β( β₯ βπ))) |
12 | 6, 11, 9 | 3eqtr3d 2774 | 1 β’ (((πΎ β HL β§ π β π») β§ π β ran πΌ) β ( β₯ β( β₯ βπ)) = π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 β‘ccnv 5668 dom cdm 5669 ran crn 5670 β1-1-ontoβwf1o 6535 βcfv 6536 HLchlt 38731 LHypclh 39366 DIsoAcdia 40410 ocAcocaN 40501 |
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-int 4944 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-cmtN 38558 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 df-docaN 40502 |
This theorem is referenced by: diarnN 40511 |
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