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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pclun2N | Structured version Visualization version GIF version |
Description: The projective subspace closure of the union of two subspaces equals their projective sum. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pclun2.s | β’ π = (PSubSpβπΎ) |
pclun2.p | β’ + = (+πβπΎ) |
pclun2.c | β’ π = (PClβπΎ) |
Ref | Expression |
---|---|
pclun2N | β’ ((πΎ β HL β§ π β π β§ π β π) β (πβ(π βͺ π)) = (π + π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1133 | . . 3 β’ ((πΎ β HL β§ π β π β§ π β π) β πΎ β HL) | |
2 | eqid 2726 | . . . . 5 β’ (AtomsβπΎ) = (AtomsβπΎ) | |
3 | pclun2.s | . . . . 5 β’ π = (PSubSpβπΎ) | |
4 | 2, 3 | psubssat 39137 | . . . 4 β’ ((πΎ β HL β§ π β π) β π β (AtomsβπΎ)) |
5 | 4 | 3adant3 1129 | . . 3 β’ ((πΎ β HL β§ π β π β§ π β π) β π β (AtomsβπΎ)) |
6 | 2, 3 | psubssat 39137 | . . . 4 β’ ((πΎ β HL β§ π β π) β π β (AtomsβπΎ)) |
7 | 6 | 3adant2 1128 | . . 3 β’ ((πΎ β HL β§ π β π β§ π β π) β π β (AtomsβπΎ)) |
8 | pclun2.p | . . . 4 β’ + = (+πβπΎ) | |
9 | pclun2.c | . . . 4 β’ π = (PClβπΎ) | |
10 | 2, 8, 9 | pclunN 39281 | . . 3 β’ ((πΎ β HL β§ π β (AtomsβπΎ) β§ π β (AtomsβπΎ)) β (πβ(π βͺ π)) = (πβ(π + π))) |
11 | 1, 5, 7, 10 | syl3anc 1368 | . 2 β’ ((πΎ β HL β§ π β π β§ π β π) β (πβ(π βͺ π)) = (πβ(π + π))) |
12 | 3, 8 | paddclN 39225 | . . 3 β’ ((πΎ β HL β§ π β π β§ π β π) β (π + π) β π) |
13 | 3, 9 | pclidN 39279 | . . 3 β’ ((πΎ β HL β§ (π + π) β π) β (πβ(π + π)) = (π + π)) |
14 | 1, 12, 13 | syl2anc 583 | . 2 β’ ((πΎ β HL β§ π β π β§ π β π) β (πβ(π + π)) = (π + π)) |
15 | 11, 14 | eqtrd 2766 | 1 β’ ((πΎ β HL β§ π β π β§ π β π) β (πβ(π βͺ π)) = (π + π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1084 = wceq 1533 β wcel 2098 βͺ cun 3941 β wss 3943 βcfv 6536 (class class class)co 7404 Atomscatm 38645 HLchlt 38732 PSubSpcpsubsp 38879 +πcpadd 39178 PClcpclN 39270 |
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 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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-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-proset 18257 df-poset 18275 df-plt 18292 df-lub 18308 df-glb 18309 df-join 18310 df-meet 18311 df-p0 18387 df-lat 18394 df-clat 18461 df-oposet 38558 df-ol 38560 df-oml 38561 df-covers 38648 df-ats 38649 df-atl 38680 df-cvlat 38704 df-hlat 38733 df-psubsp 38886 df-padd 39179 df-pclN 39271 |
This theorem is referenced by: (None) |
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