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Mirrors > Home > MPE Home > Th. List > Mathboxes > paddssat | Structured version Visualization version GIF version |
Description: A projective subspace sum is a set of atoms. (Contributed by NM, 3-Jan-2012.) |
Ref | Expression |
---|---|
padd0.a | β’ π΄ = (AtomsβπΎ) |
padd0.p | β’ + = (+πβπΎ) |
Ref | Expression |
---|---|
paddssat | β’ ((πΎ β π΅ β§ π β π΄ β§ π β π΄) β (π + π) β π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2732 | . . 3 β’ (leβπΎ) = (leβπΎ) | |
2 | eqid 2732 | . . 3 β’ (joinβπΎ) = (joinβπΎ) | |
3 | padd0.a | . . 3 β’ π΄ = (AtomsβπΎ) | |
4 | padd0.p | . . 3 β’ + = (+πβπΎ) | |
5 | 1, 2, 3, 4 | paddval 38755 | . 2 β’ ((πΎ β π΅ β§ π β π΄ β§ π β π΄) β (π + π) = ((π βͺ π) βͺ {π β π΄ β£ βπ β π βπ β π π(leβπΎ)(π(joinβπΎ)π)})) |
6 | unss 4184 | . . . . . 6 β’ ((π β π΄ β§ π β π΄) β (π βͺ π) β π΄) | |
7 | 6 | biimpi 215 | . . . . 5 β’ ((π β π΄ β§ π β π΄) β (π βͺ π) β π΄) |
8 | ssrab2 4077 | . . . . 5 β’ {π β π΄ β£ βπ β π βπ β π π(leβπΎ)(π(joinβπΎ)π)} β π΄ | |
9 | 7, 8 | jctir 521 | . . . 4 β’ ((π β π΄ β§ π β π΄) β ((π βͺ π) β π΄ β§ {π β π΄ β£ βπ β π βπ β π π(leβπΎ)(π(joinβπΎ)π)} β π΄)) |
10 | unss 4184 | . . . 4 β’ (((π βͺ π) β π΄ β§ {π β π΄ β£ βπ β π βπ β π π(leβπΎ)(π(joinβπΎ)π)} β π΄) β ((π βͺ π) βͺ {π β π΄ β£ βπ β π βπ β π π(leβπΎ)(π(joinβπΎ)π)}) β π΄) | |
11 | 9, 10 | sylib 217 | . . 3 β’ ((π β π΄ β§ π β π΄) β ((π βͺ π) βͺ {π β π΄ β£ βπ β π βπ β π π(leβπΎ)(π(joinβπΎ)π)}) β π΄) |
12 | 11 | 3adant1 1130 | . 2 β’ ((πΎ β π΅ β§ π β π΄ β§ π β π΄) β ((π βͺ π) βͺ {π β π΄ β£ βπ β π βπ β π π(leβπΎ)(π(joinβπΎ)π)}) β π΄) |
13 | 5, 12 | eqsstrd 4020 | 1 β’ ((πΎ β π΅ β§ π β π΄ β§ π β π΄) β (π + π) β π΄) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 β§ w3a 1087 = wceq 1541 β wcel 2106 βwrex 3070 {crab 3432 βͺ cun 3946 β wss 3948 class class class wbr 5148 βcfv 6543 (class class class)co 7411 lecple 17206 joincjn 18266 Atomscatm 38219 +πcpadd 38752 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7414 df-oprab 7415 df-mpo 7416 df-1st 7977 df-2nd 7978 df-padd 38753 |
This theorem is referenced by: paddasslem8 38784 paddasslem11 38787 paddasslem12 38788 paddasslem13 38789 paddasslem16 38792 paddasslem17 38793 paddass 38795 padd4N 38797 paddclN 38799 pmodl42N 38808 pclunN 38855 paddunN 38884 pmapocjN 38887 pclfinclN 38907 osumcllem1N 38913 osumcllem2N 38914 osumcllem9N 38921 osumcllem11N 38923 osumclN 38924 pexmidlem6N 38932 pexmidlem8N 38934 pl42lem3N 38938 |
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