![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > pmapjat2 | Structured version Visualization version GIF version |
Description: The projective map of the join of an atom with a lattice element. (Contributed by NM, 12-May-2012.) |
Ref | Expression |
---|---|
pmapjat.b | β’ π΅ = (BaseβπΎ) |
pmapjat.j | β’ β¨ = (joinβπΎ) |
pmapjat.a | β’ π΄ = (AtomsβπΎ) |
pmapjat.m | β’ π = (pmapβπΎ) |
pmapjat.p | β’ + = (+πβπΎ) |
Ref | Expression |
---|---|
pmapjat2 | β’ ((πΎ β HL β§ π β π΅ β§ π β π΄) β (πβ(π β¨ π)) = ((πβπ) + (πβπ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pmapjat.b | . . 3 β’ π΅ = (BaseβπΎ) | |
2 | pmapjat.j | . . 3 β’ β¨ = (joinβπΎ) | |
3 | pmapjat.a | . . 3 β’ π΄ = (AtomsβπΎ) | |
4 | pmapjat.m | . . 3 β’ π = (pmapβπΎ) | |
5 | pmapjat.p | . . 3 β’ + = (+πβπΎ) | |
6 | 1, 2, 3, 4, 5 | pmapjat1 39237 | . 2 β’ ((πΎ β HL β§ π β π΅ β§ π β π΄) β (πβ(π β¨ π)) = ((πβπ) + (πβπ))) |
7 | hllat 38746 | . . . . 5 β’ (πΎ β HL β πΎ β Lat) | |
8 | 7 | 3ad2ant1 1130 | . . . 4 β’ ((πΎ β HL β§ π β π΅ β§ π β π΄) β πΎ β Lat) |
9 | 1, 3 | atbase 38672 | . . . . 5 β’ (π β π΄ β π β π΅) |
10 | 9 | 3ad2ant3 1132 | . . . 4 β’ ((πΎ β HL β§ π β π΅ β§ π β π΄) β π β π΅) |
11 | simp2 1134 | . . . 4 β’ ((πΎ β HL β§ π β π΅ β§ π β π΄) β π β π΅) | |
12 | 1, 2 | latjcom 18412 | . . . 4 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β (π β¨ π) = (π β¨ π)) |
13 | 8, 10, 11, 12 | syl3anc 1368 | . . 3 β’ ((πΎ β HL β§ π β π΅ β§ π β π΄) β (π β¨ π) = (π β¨ π)) |
14 | 13 | fveq2d 6889 | . 2 β’ ((πΎ β HL β§ π β π΅ β§ π β π΄) β (πβ(π β¨ π)) = (πβ(π β¨ π))) |
15 | simp1 1133 | . . . 4 β’ ((πΎ β HL β§ π β π΅ β§ π β π΄) β πΎ β HL) | |
16 | 1, 3, 4 | pmapssat 39143 | . . . 4 β’ ((πΎ β HL β§ π β π΅) β (πβπ) β π΄) |
17 | 15, 10, 16 | syl2anc 583 | . . 3 β’ ((πΎ β HL β§ π β π΅ β§ π β π΄) β (πβπ) β π΄) |
18 | 1, 3, 4 | pmapssat 39143 | . . . 4 β’ ((πΎ β HL β§ π β π΅) β (πβπ) β π΄) |
19 | 18 | 3adant3 1129 | . . 3 β’ ((πΎ β HL β§ π β π΅ β§ π β π΄) β (πβπ) β π΄) |
20 | 3, 5 | paddcom 39197 | . . 3 β’ ((πΎ β Lat β§ (πβπ) β π΄ β§ (πβπ) β π΄) β ((πβπ) + (πβπ)) = ((πβπ) + (πβπ))) |
21 | 8, 17, 19, 20 | syl3anc 1368 | . 2 β’ ((πΎ β HL β§ π β π΅ β§ π β π΄) β ((πβπ) + (πβπ)) = ((πβπ) + (πβπ))) |
22 | 6, 14, 21 | 3eqtr4d 2776 | 1 β’ ((πΎ β HL β§ π β π΅ β§ π β π΄) β (πβ(π β¨ π)) = ((πβπ) + (πβπ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1084 = wceq 1533 β wcel 2098 β wss 3943 βcfv 6537 (class class class)co 7405 Basecbs 17153 joincjn 18276 Latclat 18396 Atomscatm 38646 HLchlt 38733 pmapcpmap 38881 +πcpadd 39179 |
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-mpo 7410 df-1st 7974 df-2nd 7975 df-proset 18260 df-poset 18278 df-plt 18295 df-lub 18311 df-glb 18312 df-join 18313 df-meet 18314 df-p0 18390 df-lat 18397 df-clat 18464 df-oposet 38559 df-ol 38561 df-oml 38562 df-covers 38649 df-ats 38650 df-atl 38681 df-cvlat 38705 df-hlat 38734 df-pmap 38888 df-padd 39180 |
This theorem is referenced by: atmod1i1 39241 |
Copyright terms: Public domain | W3C validator |