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Mirrors > Home > MPE Home > Th. List > Mathboxes > dalemrotps | Structured version Visualization version GIF version |
Description: Lemma for dath 39120. Rotate triangles π = πππ and π = πππ to allow reuse of analogous proofs. (Contributed by NM, 15-Aug-2012.) |
Ref | Expression |
---|---|
dalem.ph | β’ (π β (((πΎ β HL β§ πΆ β (BaseβπΎ)) 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§ ((Β¬ πΆ β€ (π β¨ π) β§ Β¬ πΆ β€ (π β¨ π ) β§ Β¬ πΆ β€ (π β¨ π)) β§ (Β¬ πΆ β€ (π β¨ π) β§ Β¬ πΆ β€ (π β¨ π) β§ Β¬ πΆ β€ (π β¨ π)) β§ (πΆ β€ (π β¨ π) β§ πΆ β€ (π β¨ π) β§ πΆ β€ (π β¨ π))))) |
dalem.l | β’ β€ = (leβπΎ) |
dalem.j | β’ β¨ = (joinβπΎ) |
dalem.a | β’ π΄ = (AtomsβπΎ) |
dalem.ps | β’ (π β ((π β π΄ β§ π β π΄) β§ Β¬ π β€ π β§ (π β π β§ Β¬ π β€ π β§ πΆ β€ (π β¨ π)))) |
dalemrotps.y | β’ π = ((π β¨ π) β¨ π ) |
Ref | Expression |
---|---|
dalemrotps | β’ ((π β§ π) β ((π β π΄ β§ π β π΄) β§ Β¬ π β€ ((π β¨ π ) β¨ π) β§ (π β π β§ Β¬ π β€ ((π β¨ π ) β¨ π) β§ πΆ β€ (π β¨ π)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dalem.ps | . . . . 5 β’ (π β ((π β π΄ β§ π β π΄) β§ Β¬ π β€ π β§ (π β π β§ Β¬ π β€ π β§ πΆ β€ (π β¨ π)))) | |
2 | 1 | dalemccea 39067 | . . . 4 β’ (π β π β π΄) |
3 | 1 | dalemddea 39068 | . . . 4 β’ (π β π β π΄) |
4 | 2, 3 | jca 511 | . . 3 β’ (π β (π β π΄ β§ π β π΄)) |
5 | 4 | adantl 481 | . 2 β’ ((π β§ π) β (π β π΄ β§ π β π΄)) |
6 | 1 | dalem-ccly 39069 | . . . 4 β’ (π β Β¬ π β€ π) |
7 | 6 | adantl 481 | . . 3 β’ ((π β§ π) β Β¬ π β€ π) |
8 | dalemrotps.y | . . . . . 6 β’ π = ((π β¨ π) β¨ π ) | |
9 | dalem.ph | . . . . . . 7 β’ (π β (((πΎ β HL β§ πΆ β (BaseβπΎ)) 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§ ((Β¬ πΆ β€ (π β¨ π) β§ Β¬ πΆ β€ (π β¨ π ) β§ Β¬ πΆ β€ (π β¨ π)) β§ (Β¬ πΆ β€ (π β¨ π) β§ Β¬ πΆ β€ (π β¨ π) β§ Β¬ πΆ β€ (π β¨ π)) β§ (πΆ β€ (π β¨ π) β§ πΆ β€ (π β¨ π) β§ πΆ β€ (π β¨ π))))) | |
10 | dalem.j | . . . . . . 7 β’ β¨ = (joinβπΎ) | |
11 | dalem.a | . . . . . . 7 β’ π΄ = (AtomsβπΎ) | |
12 | 9, 10, 11 | dalemqrprot 39032 | . . . . . 6 β’ (π β ((π β¨ π ) β¨ π) = ((π β¨ π) β¨ π )) |
13 | 8, 12 | eqtr4id 2785 | . . . . 5 β’ (π β π = ((π β¨ π ) β¨ π)) |
14 | 13 | breq2d 5153 | . . . 4 β’ (π β (π β€ π β π β€ ((π β¨ π ) β¨ π))) |
15 | 14 | adantr 480 | . . 3 β’ ((π β§ π) β (π β€ π β π β€ ((π β¨ π ) β¨ π))) |
16 | 7, 15 | mtbid 324 | . 2 β’ ((π β§ π) β Β¬ π β€ ((π β¨ π ) β¨ π)) |
17 | 1 | dalemccnedd 39071 | . . . . 5 β’ (π β π β π) |
18 | 17 | necomd 2990 | . . . 4 β’ (π β π β π) |
19 | 18 | adantl 481 | . . 3 β’ ((π β§ π) β π β π) |
20 | 1 | dalem-ddly 39070 | . . . . 5 β’ (π β Β¬ π β€ π) |
21 | 20 | adantl 481 | . . . 4 β’ ((π β§ π) β Β¬ π β€ π) |
22 | 13 | breq2d 5153 | . . . . 5 β’ (π β (π β€ π β π β€ ((π β¨ π ) β¨ π))) |
23 | 22 | adantr 480 | . . . 4 β’ ((π β§ π) β (π β€ π β π β€ ((π β¨ π ) β¨ π))) |
24 | 21, 23 | mtbid 324 | . . 3 β’ ((π β§ π) β Β¬ π β€ ((π β¨ π ) β¨ π)) |
25 | 1 | dalemclccjdd 39072 | . . . 4 β’ (π β πΆ β€ (π β¨ π)) |
26 | 25 | adantl 481 | . . 3 β’ ((π β§ π) β πΆ β€ (π β¨ π)) |
27 | 19, 24, 26 | 3jca 1125 | . 2 β’ ((π β§ π) β (π β π β§ Β¬ π β€ ((π β¨ π ) β¨ π) β§ πΆ β€ (π β¨ π))) |
28 | 5, 16, 27 | 3jca 1125 | 1 β’ ((π β§ π) β ((π β π΄ β§ π β π΄) β§ Β¬ π β€ ((π β¨ π ) β¨ π) β§ (π β π β§ Β¬ π β€ ((π β¨ π ) β¨ π) β§ πΆ β€ (π β¨ π)))) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2934 class class class wbr 5141 βcfv 6537 (class class class)co 7405 Basecbs 17153 lecple 17213 joincjn 18276 Atomscatm 38646 HLchlt 38733 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-proset 18260 df-poset 18278 df-lub 18311 df-glb 18312 df-join 18313 df-meet 18314 df-lat 18397 df-ats 38650 df-atl 38681 df-cvlat 38705 df-hlat 38734 |
This theorem is referenced by: dalem29 39085 dalem30 39086 dalem31N 39087 dalem32 39088 dalem33 39089 dalem34 39090 dalem35 39091 dalem36 39092 dalem37 39093 dalem40 39096 dalem46 39102 dalem47 39103 dalem49 39105 dalem50 39106 dalem58 39114 dalem59 39115 |
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