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Mirrors > Home > MPE Home > Th. List > Mathboxes > atexchltN | Structured version Visualization version GIF version |
Description: Atom exchange property. Version of hlatexch2 37905 with less-than ordering. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
atexchlt.s | β’ < = (ltβπΎ) |
atexchlt.j | β’ β¨ = (joinβπΎ) |
atexchlt.a | β’ π΄ = (AtomsβπΎ) |
Ref | Expression |
---|---|
atexchltN | β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄ β§ π β π΄) β§ π β π ) β (π < (π β¨ π ) β π < (π β¨ π ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | atexchlt.j | . . 3 β’ β¨ = (joinβπΎ) | |
2 | atexchlt.a | . . 3 β’ π΄ = (AtomsβπΎ) | |
3 | eqid 2733 | . . 3 β’ ( β βπΎ) = ( β βπΎ) | |
4 | 1, 2, 3 | atexchcvrN 37949 | . 2 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄ β§ π β π΄) β§ π β π ) β (π( β βπΎ)(π β¨ π ) β π( β βπΎ)(π β¨ π ))) |
5 | atexchlt.s | . . . 4 β’ < = (ltβπΎ) | |
6 | 5, 1, 2, 3 | atltcvr 37944 | . . 3 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄ β§ π β π΄)) β (π < (π β¨ π ) β π( β βπΎ)(π β¨ π ))) |
7 | 6 | 3adant3 1133 | . 2 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄ β§ π β π΄) β§ π β π ) β (π < (π β¨ π ) β π( β βπΎ)(π β¨ π ))) |
8 | simpl 484 | . . . 4 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄ β§ π β π΄)) β πΎ β HL) | |
9 | simpr2 1196 | . . . 4 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄ β§ π β π΄)) β π β π΄) | |
10 | simpr1 1195 | . . . 4 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄ β§ π β π΄)) β π β π΄) | |
11 | simpr3 1197 | . . . 4 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄ β§ π β π΄)) β π β π΄) | |
12 | 5, 1, 2, 3 | atltcvr 37944 | . . . 4 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄ β§ π β π΄)) β (π < (π β¨ π ) β π( β βπΎ)(π β¨ π ))) |
13 | 8, 9, 10, 11, 12 | syl13anc 1373 | . . 3 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄ β§ π β π΄)) β (π < (π β¨ π ) β π( β βπΎ)(π β¨ π ))) |
14 | 13 | 3adant3 1133 | . 2 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄ β§ π β π΄) β§ π β π ) β (π < (π β¨ π ) β π( β βπΎ)(π β¨ π ))) |
15 | 4, 7, 14 | 3imtr4d 294 | 1 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄ β§ π β π΄) β§ π β π ) β (π < (π β¨ π ) β π < (π β¨ π ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 β wne 2940 class class class wbr 5106 βcfv 6497 (class class class)co 7358 ltcplt 18202 joincjn 18205 β ccvr 37770 Atomscatm 37771 HLchlt 37858 |
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-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
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 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-proset 18189 df-poset 18207 df-plt 18224 df-lub 18240 df-glb 18241 df-join 18242 df-meet 18243 df-p0 18319 df-lat 18326 df-clat 18393 df-oposet 37684 df-ol 37686 df-oml 37687 df-covers 37774 df-ats 37775 df-atl 37806 df-cvlat 37830 df-hlat 37859 |
This theorem is referenced by: (None) |
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