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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 40524 | . . 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 40599 | . . 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 40522 | . . . . 5 β’ ((πΎ β HL β§ π β π») β πΌ:dom πΌβ1-1-ontoβran πΌ) |
8 | f1ocnvfv2 7286 | . . . . 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 6901 | . . 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 6901 | . 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 2776 | 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 1534 β wcel 2099 β‘ccnv 5677 dom cdm 5678 ran crn 5679 β1-1-ontoβwf1o 6547 βcfv 6548 HLchlt 38822 LHypclh 39457 DIsoAcdia 40501 ocAcocaN 40592 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-riotaBAD 38425 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 7993 df-2nd 7994 df-undef 8279 df-map 8847 df-proset 18287 df-poset 18305 df-plt 18322 df-lub 18338 df-glb 18339 df-join 18340 df-meet 18341 df-p0 18417 df-p1 18418 df-lat 18424 df-clat 18491 df-oposet 38648 df-cmtN 38649 df-ol 38650 df-oml 38651 df-covers 38738 df-ats 38739 df-atl 38770 df-cvlat 38794 df-hlat 38823 df-llines 38971 df-lplanes 38972 df-lvols 38973 df-lines 38974 df-psubsp 38976 df-pmap 38977 df-padd 39269 df-lhyp 39461 df-laut 39462 df-ldil 39577 df-ltrn 39578 df-trl 39632 df-disoa 40502 df-docaN 40593 |
This theorem is referenced by: diarnN 40602 |
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