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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemg1fvawlemN | Structured version Visualization version GIF version |
Description: Lemma for ltrniotafvawN 39939. (Contributed by NM, 18-Apr-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cdlemg1.b | β’ π΅ = (BaseβπΎ) |
cdlemg1.l | β’ β€ = (leβπΎ) |
cdlemg1.j | β’ β¨ = (joinβπΎ) |
cdlemg1.m | β’ β§ = (meetβπΎ) |
cdlemg1.a | β’ π΄ = (AtomsβπΎ) |
cdlemg1.h | β’ π» = (LHypβπΎ) |
cdlemg1.u | β’ π = ((π β¨ π) β§ π) |
cdlemg1.d | β’ π· = ((π‘ β¨ π) β§ (π β¨ ((π β¨ π‘) β§ π))) |
cdlemg1.e | β’ πΈ = ((π β¨ π) β§ (π· β¨ ((π β¨ π‘) β§ π))) |
cdlemg1.g | β’ πΊ = (π₯ β π΅ β¦ if((π β π β§ Β¬ π₯ β€ π), (β©π§ β π΅ βπ β π΄ ((Β¬ π β€ π β§ (π β¨ (π₯ β§ π)) = π₯) β π§ = (if(π β€ (π β¨ π), (β©π¦ β π΅ βπ‘ β π΄ ((Β¬ π‘ β€ π β§ Β¬ π‘ β€ (π β¨ π)) β π¦ = πΈ)), β¦π / π‘β¦π·) β¨ (π₯ β§ π)))), π₯)) |
cdlemg1.t | β’ π = ((LTrnβπΎ)βπ) |
cdlemg1.f | β’ πΉ = (β©π β π (πβπ) = π) |
Ref | Expression |
---|---|
cdlemg1fvawlemN | β’ ((((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β π΄ β§ Β¬ π β€ π)) β ((πΉβπ ) β π΄ β§ Β¬ (πΉβπ ) β€ π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdlemg1.b | . . 3 β’ π΅ = (BaseβπΎ) | |
2 | cdlemg1.l | . . 3 β’ β€ = (leβπΎ) | |
3 | cdlemg1.j | . . 3 β’ β¨ = (joinβπΎ) | |
4 | cdlemg1.m | . . 3 β’ β§ = (meetβπΎ) | |
5 | cdlemg1.a | . . 3 β’ π΄ = (AtomsβπΎ) | |
6 | cdlemg1.h | . . 3 β’ π» = (LHypβπΎ) | |
7 | cdlemg1.u | . . 3 β’ π = ((π β¨ π) β§ π) | |
8 | cdlemg1.d | . . 3 β’ π· = ((π‘ β¨ π) β§ (π β¨ ((π β¨ π‘) β§ π))) | |
9 | cdlemg1.e | . . 3 β’ πΈ = ((π β¨ π) β§ (π· β¨ ((π β¨ π‘) β§ π))) | |
10 | cdlemg1.g | . . 3 β’ πΊ = (π₯ β π΅ β¦ if((π β π β§ Β¬ π₯ β€ π), (β©π§ β π΅ βπ β π΄ ((Β¬ π β€ π β§ (π β¨ (π₯ β§ π)) = π₯) β π§ = (if(π β€ (π β¨ π), (β©π¦ β π΅ βπ‘ β π΄ ((Β¬ π‘ β€ π β§ Β¬ π‘ β€ (π β¨ π)) β π¦ = πΈ)), β¦π / π‘β¦π·) β¨ (π₯ β§ π)))), π₯)) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | cdleme46fvaw 39862 | . 2 β’ ((((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β π΄ β§ Β¬ π β€ π)) β ((πΊβπ ) β π΄ β§ Β¬ (πΊβπ ) β€ π)) |
12 | cdlemg1.t | . . . . . . 7 β’ π = ((LTrnβπΎ)βπ) | |
13 | cdlemg1.f | . . . . . . 7 β’ πΉ = (β©π β π (πβπ) = π) | |
14 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13 | cdlemg1b2 39932 | . . . . . 6 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β πΉ = πΊ) |
15 | 14 | adantr 480 | . . . . 5 β’ ((((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β π΄ β§ Β¬ π β€ π)) β πΉ = πΊ) |
16 | 15 | fveq1d 6883 | . . . 4 β’ ((((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β π΄ β§ Β¬ π β€ π)) β (πΉβπ ) = (πΊβπ )) |
17 | 16 | eleq1d 2810 | . . 3 β’ ((((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β π΄ β§ Β¬ π β€ π)) β ((πΉβπ ) β π΄ β (πΊβπ ) β π΄)) |
18 | 16 | breq1d 5148 | . . . 4 β’ ((((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β π΄ β§ Β¬ π β€ π)) β ((πΉβπ ) β€ π β (πΊβπ ) β€ π)) |
19 | 18 | notbid 318 | . . 3 β’ ((((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β π΄ β§ Β¬ π β€ π)) β (Β¬ (πΉβπ ) β€ π β Β¬ (πΊβπ ) β€ π)) |
20 | 17, 19 | anbi12d 630 | . 2 β’ ((((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β π΄ β§ Β¬ π β€ π)) β (((πΉβπ ) β π΄ β§ Β¬ (πΉβπ ) β€ π) β ((πΊβπ ) β π΄ β§ Β¬ (πΊβπ ) β€ π))) |
21 | 11, 20 | mpbird 257 | 1 β’ ((((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β π΄ β§ Β¬ π β€ π)) β ((πΉβπ ) β π΄ β§ Β¬ (πΉβπ ) β€ π)) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2932 βwral 3053 β¦csb 3885 ifcif 4520 class class class wbr 5138 β¦ cmpt 5221 βcfv 6533 β©crio 7356 (class class class)co 7401 Basecbs 17143 lecple 17203 joincjn 18266 meetcmee 18267 Atomscatm 38623 HLchlt 38710 LHypclh 39345 LTrncltrn 39462 |
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 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-riotaBAD 38313 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-1st 7968 df-2nd 7969 df-undef 8253 df-map 8818 df-proset 18250 df-poset 18268 df-plt 18285 df-lub 18301 df-glb 18302 df-join 18303 df-meet 18304 df-p0 18380 df-p1 18381 df-lat 18387 df-clat 18454 df-oposet 38536 df-ol 38538 df-oml 38539 df-covers 38626 df-ats 38627 df-atl 38658 df-cvlat 38682 df-hlat 38711 df-llines 38859 df-lplanes 38860 df-lvols 38861 df-lines 38862 df-psubsp 38864 df-pmap 38865 df-padd 39157 df-lhyp 39349 df-laut 39350 df-ldil 39465 df-ltrn 39466 df-trl 39520 |
This theorem is referenced by: ltrniotafvawN 39939 |
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