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Mirrors > Home > MPE Home > Th. List > pltle | Structured version Visualization version GIF version |
Description: "Less than" implies "less than or equal to". (pssss 4096 analog.) (Contributed by NM, 4-Dec-2011.) |
Ref | Expression |
---|---|
pltval.l | β’ β€ = (leβπΎ) |
pltval.s | β’ < = (ltβπΎ) |
Ref | Expression |
---|---|
pltle | β’ ((πΎ β π΄ β§ π β π΅ β§ π β πΆ) β (π < π β π β€ π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pltval.l | . . . 4 β’ β€ = (leβπΎ) | |
2 | pltval.s | . . . 4 β’ < = (ltβπΎ) | |
3 | 1, 2 | pltval 18285 | . . 3 β’ ((πΎ β π΄ β§ π β π΅ β§ π β πΆ) β (π < π β (π β€ π β§ π β π))) |
4 | 3 | simprbda 500 | . 2 β’ (((πΎ β π΄ β§ π β π΅ β§ π β πΆ) β§ π < π) β π β€ π) |
5 | 4 | ex 414 | 1 β’ ((πΎ β π΄ β§ π β π΅ β§ π β πΆ) β (π < π β π β€ π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1088 = wceq 1542 β wcel 2107 β wne 2941 class class class wbr 5149 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: pleval2 18290 pltnlt 18293 pltn2lp 18294 plttr 18295 pospo 18298 ogrpaddlt 32235 isarchi3 32333 archirngz 32335 archiabllem2a 32340 orngsqr 32422 ornglmullt 32425 orngrmullt 32426 atnlt 38183 cvlcvr1 38209 hlrelat 38273 hlrelat3 38283 cvratlem 38292 atltcvr 38306 atlelt 38309 llnnlt 38394 lplnnle2at 38412 lplnnlt 38436 lvolnle3at 38453 lvolnltN 38489 cdlemblem 38664 cdlemb 38665 lhpexle1 38879 |
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