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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 1137 | . . 3 β’ ((πΎ β HL β§ π β π β§ π β π) β πΎ β HL) | |
2 | eqid 2737 | . . . . 5 β’ (AtomsβπΎ) = (AtomsβπΎ) | |
3 | pclun2.s | . . . . 5 β’ π = (PSubSpβπΎ) | |
4 | 2, 3 | psubssat 38220 | . . . 4 β’ ((πΎ β HL β§ π β π) β π β (AtomsβπΎ)) |
5 | 4 | 3adant3 1133 | . . 3 β’ ((πΎ β HL β§ π β π β§ π β π) β π β (AtomsβπΎ)) |
6 | 2, 3 | psubssat 38220 | . . . 4 β’ ((πΎ β HL β§ π β π) β π β (AtomsβπΎ)) |
7 | 6 | 3adant2 1132 | . . 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 38364 | . . 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 1372 | . 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 38308 | . . 3 β’ ((πΎ β HL β§ π β π β§ π β π) β (π + π) β π) |
13 | 3, 9 | pclidN 38362 | . . 3 β’ ((πΎ β HL β§ (π + π) β π) β (πβ(π + π)) = (π + π)) |
14 | 1, 12, 13 | syl2anc 585 | . 2 β’ ((πΎ β HL β§ π β π β§ π β π) β (πβ(π + π)) = (π + π)) |
15 | 11, 14 | eqtrd 2777 | 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 1088 = wceq 1542 β wcel 2107 βͺ cun 3909 β wss 3911 βcfv 6497 (class class class)co 7358 Atomscatm 37728 HLchlt 37815 PSubSpcpsubsp 37962 +πcpadd 38261 PClcpclN 38353 |
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 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 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-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-int 4909 df-iun 4957 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-proset 18185 df-poset 18203 df-plt 18220 df-lub 18236 df-glb 18237 df-join 18238 df-meet 18239 df-p0 18315 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-psubsp 37969 df-padd 38262 df-pclN 38354 |
This theorem is referenced by: (None) |
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