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Mirrors > Home > MPE Home > Th. List > Mathboxes > tendoplcl2 | Structured version Visualization version GIF version |
Description: Value of result of endomorphism sum operation. (Contributed by NM, 10-Jun-2013.) |
Ref | Expression |
---|---|
tendopl.h | β’ π» = (LHypβπΎ) |
tendopl.t | β’ π = ((LTrnβπΎ)βπ) |
tendopl.e | β’ πΈ = ((TEndoβπΎ)βπ) |
tendopl.p | β’ π = (π β πΈ, π‘ β πΈ β¦ (π β π β¦ ((π βπ) β (π‘βπ)))) |
Ref | Expression |
---|---|
tendoplcl2 | β’ (((πΎ β HL β§ π β π») β§ (π β πΈ β§ π β πΈ) β§ πΉ β π) β ((πππ)βπΉ) β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tendopl.p | . . . . 5 β’ π = (π β πΈ, π‘ β πΈ β¦ (π β π β¦ ((π βπ) β (π‘βπ)))) | |
2 | tendopl.t | . . . . 5 β’ π = ((LTrnβπΎ)βπ) | |
3 | 1, 2 | tendopl2 39951 | . . . 4 β’ ((π β πΈ β§ π β πΈ β§ πΉ β π) β ((πππ)βπΉ) = ((πβπΉ) β (πβπΉ))) |
4 | 3 | 3expa 1116 | . . 3 β’ (((π β πΈ β§ π β πΈ) β§ πΉ β π) β ((πππ)βπΉ) = ((πβπΉ) β (πβπΉ))) |
5 | 4 | 3adant1 1128 | . 2 β’ (((πΎ β HL β§ π β π») β§ (π β πΈ β§ π β πΈ) β§ πΉ β π) β ((πππ)βπΉ) = ((πβπΉ) β (πβπΉ))) |
6 | simp1 1134 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β πΈ β§ π β πΈ) β§ πΉ β π) β (πΎ β HL β§ π β π»)) | |
7 | tendopl.h | . . . . 5 β’ π» = (LHypβπΎ) | |
8 | tendopl.e | . . . . 5 β’ πΈ = ((TEndoβπΎ)βπ) | |
9 | 7, 2, 8 | tendocl 39941 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ π β πΈ β§ πΉ β π) β (πβπΉ) β π) |
10 | 9 | 3adant2r 1177 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β πΈ β§ π β πΈ) β§ πΉ β π) β (πβπΉ) β π) |
11 | 7, 2, 8 | tendocl 39941 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ π β πΈ β§ πΉ β π) β (πβπΉ) β π) |
12 | 11 | 3adant2l 1176 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β πΈ β§ π β πΈ) β§ πΉ β π) β (πβπΉ) β π) |
13 | 7, 2 | ltrnco 39893 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (πβπΉ) β π β§ (πβπΉ) β π) β ((πβπΉ) β (πβπΉ)) β π) |
14 | 6, 10, 12, 13 | syl3anc 1369 | . 2 β’ (((πΎ β HL β§ π β π») β§ (π β πΈ β§ π β πΈ) β§ πΉ β π) β ((πβπΉ) β (πβπΉ)) β π) |
15 | 5, 14 | eqeltrd 2831 | 1 β’ (((πΎ β HL β§ π β π») β§ (π β πΈ β§ π β πΈ) β§ πΉ β π) β ((πππ)βπΉ) β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 β§ w3a 1085 = wceq 1539 β wcel 2104 β¦ cmpt 5230 β ccom 5679 βcfv 6542 (class class class)co 7411 β cmpo 7413 HLchlt 38523 LHypclh 39158 LTrncltrn 39275 TEndoctendo 39926 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-riotaBAD 38126 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 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 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-1st 7977 df-2nd 7978 df-undef 8260 df-map 8824 df-proset 18252 df-poset 18270 df-plt 18287 df-lub 18303 df-glb 18304 df-join 18305 df-meet 18306 df-p0 18382 df-p1 18383 df-lat 18389 df-clat 18456 df-oposet 38349 df-ol 38351 df-oml 38352 df-covers 38439 df-ats 38440 df-atl 38471 df-cvlat 38495 df-hlat 38524 df-llines 38672 df-lplanes 38673 df-lvols 38674 df-lines 38675 df-psubsp 38677 df-pmap 38678 df-padd 38970 df-lhyp 39162 df-laut 39163 df-ldil 39278 df-ltrn 39279 df-trl 39333 df-tendo 39929 |
This theorem is referenced by: tendopltp 39954 |
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