![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > pltval | Structured version Visualization version GIF version |
Description: Less-than relation. (df-pss 3968 analog.) (Contributed by NM, 12-Oct-2011.) |
Ref | Expression |
---|---|
pltval.l | β’ β€ = (leβπΎ) |
pltval.s | β’ < = (ltβπΎ) |
Ref | Expression |
---|---|
pltval | β’ ((πΎ β π΄ β§ π β π΅ β§ π β πΆ) β (π < π β (π β€ π β§ π β π))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pltval.l | . . . . 5 β’ β€ = (leβπΎ) | |
2 | pltval.s | . . . . 5 β’ < = (ltβπΎ) | |
3 | 1, 2 | pltfval 18284 | . . . 4 β’ (πΎ β π΄ β < = ( β€ β I )) |
4 | 3 | breqd 5160 | . . 3 β’ (πΎ β π΄ β (π < π β π( β€ β I )π)) |
5 | brdif 5202 | . . . 4 β’ (π( β€ β I )π β (π β€ π β§ Β¬ π I π)) | |
6 | ideqg 5852 | . . . . . . 7 β’ (π β πΆ β (π I π β π = π)) | |
7 | 6 | necon3bbid 2979 | . . . . . 6 β’ (π β πΆ β (Β¬ π I π β π β π)) |
8 | 7 | adantl 483 | . . . . 5 β’ ((π β π΅ β§ π β πΆ) β (Β¬ π I π β π β π)) |
9 | 8 | anbi2d 630 | . . . 4 β’ ((π β π΅ β§ π β πΆ) β ((π β€ π β§ Β¬ π I π) β (π β€ π β§ π β π))) |
10 | 5, 9 | bitrid 283 | . . 3 β’ ((π β π΅ β§ π β πΆ) β (π( β€ β I )π β (π β€ π β§ π β π))) |
11 | 4, 10 | sylan9bb 511 | . 2 β’ ((πΎ β π΄ β§ (π β π΅ β§ π β πΆ)) β (π < π β (π β€ π β§ π β π))) |
12 | 11 | 3impb 1116 | 1 β’ ((πΎ β π΄ β§ π β π΅ β§ π β πΆ) β (π < π β (π β€ π β§ π β π))) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 β wne 2941 β cdif 3946 class class class wbr 5149 I cid 5574 βcfv 6544 lecple 17204 ltcplt 18261 |
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 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-iota 6496 df-fun 6546 df-fv 6552 df-plt 18283 |
This theorem is referenced by: pltle 18286 pltne 18287 pleval2i 18289 pltnle 18291 pltval3 18292 plttr 18295 latnlemlt 18425 latnle 18426 ipolt 18488 ogrpaddlt 32235 ogrpsublt 32239 ornglmullt 32425 orngrmullt 32426 orngmullt 32427 ofldlt1 32431 opltn0 38060 cvrval2 38144 cvrnbtwn2 38145 cvrnbtwn3 38146 cvrle 38148 cvrnbtwn4 38149 cvrne 38151 atlltn0 38176 hlrelat5N 38272 llnle 38389 lplnle 38411 llncvrlpln2 38428 lplncvrlvol2 38486 lhp2lt 38872 lautlt 38962 |
Copyright terms: Public domain | W3C validator |