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Mirrors > Home > MPE Home > Th. List > Mathboxes > paddvaln0N | Structured version Visualization version GIF version |
Description: Projective subspace sum operation value for nonempty sets. (Contributed by NM, 27-Jan-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
paddfval.l | β’ β€ = (leβπΎ) |
paddfval.j | β’ β¨ = (joinβπΎ) |
paddfval.a | β’ π΄ = (AtomsβπΎ) |
paddfval.p | β’ + = (+πβπΎ) |
Ref | Expression |
---|---|
paddvaln0N | β’ (((πΎ β Lat β§ π β π΄ β§ π β π΄) β§ (π β β β§ π β β )) β (π + π) = {π β π΄ β£ βπ β π βπ β π π β€ (π β¨ π)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | paddfval.l | . . . 4 β’ β€ = (leβπΎ) | |
2 | paddfval.j | . . . 4 β’ β¨ = (joinβπΎ) | |
3 | paddfval.a | . . . 4 β’ π΄ = (AtomsβπΎ) | |
4 | paddfval.p | . . . 4 β’ + = (+πβπΎ) | |
5 | 1, 2, 3, 4 | elpaddn0 38076 | . . 3 β’ (((πΎ β Lat β§ π β π΄ β§ π β π΄) β§ (π β β β§ π β β )) β (π β (π + π) β (π β π΄ β§ βπ β π βπ β π π β€ (π β¨ π)))) |
6 | breq1 5095 | . . . . 5 β’ (π = π β (π β€ (π β¨ π) β π β€ (π β¨ π))) | |
7 | 6 | 2rexbidv 3209 | . . . 4 β’ (π = π β (βπ β π βπ β π π β€ (π β¨ π) β βπ β π βπ β π π β€ (π β¨ π))) |
8 | 7 | elrab 3634 | . . 3 β’ (π β {π β π΄ β£ βπ β π βπ β π π β€ (π β¨ π)} β (π β π΄ β§ βπ β π βπ β π π β€ (π β¨ π))) |
9 | 5, 8 | bitr4di 288 | . 2 β’ (((πΎ β Lat β§ π β π΄ β§ π β π΄) β§ (π β β β§ π β β )) β (π β (π + π) β π β {π β π΄ β£ βπ β π βπ β π π β€ (π β¨ π)})) |
10 | 9 | eqrdv 2734 | 1 β’ (((πΎ β Lat β§ π β π΄ β§ π β π΄) β§ (π β β β§ π β β )) β (π + π) = {π β π΄ β£ βπ β π βπ β π π β€ (π β¨ π)}) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 β§ w3a 1086 = wceq 1540 β wcel 2105 β wne 2940 βwrex 3070 {crab 3403 β wss 3898 β c0 4269 class class class wbr 5092 βcfv 6479 (class class class)co 7337 lecple 17066 joincjn 18126 Latclat 18246 Atomscatm 37538 +πcpadd 38071 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-id 5518 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-riota 7293 df-ov 7340 df-oprab 7341 df-mpo 7342 df-1st 7899 df-2nd 7900 df-lub 18161 df-join 18163 df-lat 18247 df-ats 37542 df-padd 38072 |
This theorem is referenced by: (None) |
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