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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 39053 | . . . 4 β’ ((π β πΈ β§ π β πΈ β§ πΉ β π) β ((πππ)βπΉ) = ((πβπΉ) β (πβπΉ))) |
4 | 3 | 3expa 1117 | . . 3 β’ (((π β πΈ β§ π β πΈ) β§ πΉ β π) β ((πππ)βπΉ) = ((πβπΉ) β (πβπΉ))) |
5 | 4 | 3adant1 1129 | . 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 1135 | . . 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 39043 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ π β πΈ β§ πΉ β π) β (πβπΉ) β π) |
10 | 9 | 3adant2r 1178 | . . 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 39043 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ π β πΈ β§ πΉ β π) β (πβπΉ) β π) |
12 | 11 | 3adant2l 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) |
13 | 7, 2 | ltrnco 38995 | . . 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 1370 | . 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 2837 | 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 396 β§ w3a 1086 = wceq 1540 β wcel 2105 β¦ cmpt 5175 β ccom 5624 βcfv 6479 (class class class)co 7337 β cmpo 7339 HLchlt 37625 LHypclh 38260 LTrncltrn 38377 TEndoctendo 39028 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-riotaBAD 37228 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-iun 4943 df-iin 4944 df-br 5093 df-opab 5155 df-mpt 5176 df-id 5518 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-riota 7293 df-ov 7340 df-oprab 7341 df-mpo 7342 df-1st 7899 df-2nd 7900 df-undef 8159 df-map 8688 df-proset 18110 df-poset 18128 df-plt 18145 df-lub 18161 df-glb 18162 df-join 18163 df-meet 18164 df-p0 18240 df-p1 18241 df-lat 18247 df-clat 18314 df-oposet 37451 df-ol 37453 df-oml 37454 df-covers 37541 df-ats 37542 df-atl 37573 df-cvlat 37597 df-hlat 37626 df-llines 37774 df-lplanes 37775 df-lvols 37776 df-lines 37777 df-psubsp 37779 df-pmap 37780 df-padd 38072 df-lhyp 38264 df-laut 38265 df-ldil 38380 df-ltrn 38381 df-trl 38435 df-tendo 39031 |
This theorem is referenced by: tendopltp 39056 |
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