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Mirrors > Home > MPE Home > Th. List > Mathboxes > paddasslem6 | Structured version Visualization version GIF version |
Description: Lemma for paddass 39235. (Contributed by NM, 8-Jan-2012.) |
Ref | Expression |
---|---|
paddasslem.l | β’ β€ = (leβπΎ) |
paddasslem.j | β’ β¨ = (joinβπΎ) |
paddasslem.a | β’ π΄ = (AtomsβπΎ) |
Ref | Expression |
---|---|
paddasslem6 | β’ (((πΎ β HL β§ (π β π΄ β§ π β π΄) β§ π§ β π΄) β§ (π β π§ β§ π β€ (π β¨ π§))) β π β€ (π β¨ π§)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 1189 | . . 3 β’ (((πΎ β HL β§ (π β π΄ β§ π β π΄) β§ π§ β π΄) β§ (π β π§ β§ π β€ (π β¨ π§))) β πΎ β HL) | |
2 | simpl2r 1225 | . . . 4 β’ (((πΎ β HL β§ (π β π΄ β§ π β π΄) β§ π§ β π΄) β§ (π β π§ β§ π β€ (π β¨ π§))) β π β π΄) | |
3 | simpl2l 1224 | . . . 4 β’ (((πΎ β HL β§ (π β π΄ β§ π β π΄) β§ π§ β π΄) β§ (π β π§ β§ π β€ (π β¨ π§))) β π β π΄) | |
4 | simpl3 1191 | . . . 4 β’ (((πΎ β HL β§ (π β π΄ β§ π β π΄) β§ π§ β π΄) β§ (π β π§ β§ π β€ (π β¨ π§))) β π§ β π΄) | |
5 | 2, 3, 4 | 3jca 1126 | . . 3 β’ (((πΎ β HL β§ (π β π΄ β§ π β π΄) β§ π§ β π΄) β§ (π β π§ β§ π β€ (π β¨ π§))) β (π β π΄ β§ π β π΄ β§ π§ β π΄)) |
6 | simprl 770 | . . 3 β’ (((πΎ β HL β§ (π β π΄ β§ π β π΄) β§ π§ β π΄) β§ (π β π§ β§ π β€ (π β¨ π§))) β π β π§) | |
7 | 1, 5, 6 | 3jca 1126 | . 2 β’ (((πΎ β HL β§ (π β π΄ β§ π β π΄) β§ π§ β π΄) β§ (π β π§ β§ π β€ (π β¨ π§))) β (πΎ β HL β§ (π β π΄ β§ π β π΄ β§ π§ β π΄) β§ π β π§)) |
8 | simprr 772 | . 2 β’ (((πΎ β HL β§ (π β π΄ β§ π β π΄) β§ π§ β π΄) β§ (π β π§ β§ π β€ (π β¨ π§))) β π β€ (π β¨ π§)) | |
9 | paddasslem.l | . . 3 β’ β€ = (leβπΎ) | |
10 | paddasslem.j | . . 3 β’ β¨ = (joinβπΎ) | |
11 | paddasslem.a | . . 3 β’ π΄ = (AtomsβπΎ) | |
12 | 9, 10, 11 | hlatexch2 38793 | . 2 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄ β§ π§ β π΄) β§ π β π§) β (π β€ (π β¨ π§) β π β€ (π β¨ π§))) |
13 | 7, 8, 12 | sylc 65 | 1 β’ (((πΎ β HL β§ (π β π΄ β§ π β π΄) β§ π§ β π΄) β§ (π β π§ β§ π β€ (π β¨ π§))) β π β€ (π β¨ π§)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 β§ w3a 1085 = wceq 1534 β wcel 2099 β wne 2935 class class class wbr 5142 βcfv 6542 (class class class)co 7414 lecple 17225 joincjn 18288 Atomscatm 38659 HLchlt 38746 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-proset 18272 df-poset 18290 df-plt 18307 df-lub 18323 df-glb 18324 df-join 18325 df-meet 18326 df-p0 18402 df-lat 18409 df-covers 38662 df-ats 38663 df-atl 38694 df-cvlat 38718 df-hlat 38747 |
This theorem is referenced by: paddasslem7 39223 |
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