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Mirrors > Home > MPE Home > Th. List > Mathboxes > sspadd1 | Structured version Visualization version GIF version |
Description: A projective subspace sum is a superset of its first summand. (ssun1 4112 analog.) (Contributed by NM, 3-Jan-2012.) |
Ref | Expression |
---|---|
padd0.a | β’ π΄ = (AtomsβπΎ) |
padd0.p | β’ + = (+πβπΎ) |
Ref | Expression |
---|---|
sspadd1 | β’ ((πΎ β π΅ β§ π β π΄ β§ π β π΄) β π β (π + π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssun1 4112 | . . 3 β’ π β (π βͺ π) | |
2 | ssun1 4112 | . . 3 β’ (π βͺ π) β ((π βͺ π) βͺ {π β π΄ β£ βπ β π βπ β π π(leβπΎ)(π(joinβπΎ)π)}) | |
3 | 1, 2 | sstri 3935 | . 2 β’ π β ((π βͺ π) βͺ {π β π΄ β£ βπ β π βπ β π π(leβπΎ)(π(joinβπΎ)π)}) |
4 | eqid 2736 | . . 3 β’ (leβπΎ) = (leβπΎ) | |
5 | eqid 2736 | . . 3 β’ (joinβπΎ) = (joinβπΎ) | |
6 | padd0.a | . . 3 β’ π΄ = (AtomsβπΎ) | |
7 | padd0.p | . . 3 β’ + = (+πβπΎ) | |
8 | 4, 5, 6, 7 | paddval 37854 | . 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)})) |
9 | 3, 8 | sseqtrrid 3979 | 1 β’ ((πΎ β π΅ β§ π β π΄ β§ π β π΄) β π β (π + π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1087 = wceq 1539 β wcel 2104 βwrex 3071 {crab 3284 βͺ cun 3890 β wss 3892 class class class wbr 5081 βcfv 6458 (class class class)co 7307 lecple 17014 joincjn 18074 Atomscatm 37319 +πcpadd 37851 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-iun 4933 df-br 5082 df-opab 5144 df-mpt 5165 df-id 5500 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-ov 7310 df-oprab 7311 df-mpo 7312 df-1st 7863 df-2nd 7864 df-padd 37852 |
This theorem is referenced by: paddasslem13 37888 paddasslem17 37892 paddidm 37897 paddssw2 37900 pmodlem1 37902 pmodlem2 37903 pmodl42N 37907 osumcllem1N 38012 osumcllem2N 38013 osumcllem10N 38021 pexmidlem6N 38031 pexmidlem7N 38032 |
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