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Mirrors > Home > MPE Home > Th. List > pj1lmhm2 | Structured version Visualization version GIF version |
Description: The left projection function is a linear operator. (Contributed by Mario Carneiro, 15-Oct-2015.) (Revised by Mario Carneiro, 21-Apr-2016.) |
Ref | Expression |
---|---|
pj1lmhm.l | โข ๐ฟ = (LSubSpโ๐) |
pj1lmhm.s | โข โ = (LSSumโ๐) |
pj1lmhm.z | โข 0 = (0gโ๐) |
pj1lmhm.p | โข ๐ = (proj1โ๐) |
pj1lmhm.1 | โข (๐ โ ๐ โ LMod) |
pj1lmhm.2 | โข (๐ โ ๐ โ ๐ฟ) |
pj1lmhm.3 | โข (๐ โ ๐ โ ๐ฟ) |
pj1lmhm.4 | โข (๐ โ (๐ โฉ ๐) = { 0 }) |
Ref | Expression |
---|---|
pj1lmhm2 | โข (๐ โ (๐๐๐) โ ((๐ โพs (๐ โ ๐)) LMHom (๐ โพs ๐))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pj1lmhm.l | . . 3 โข ๐ฟ = (LSubSpโ๐) | |
2 | pj1lmhm.s | . . 3 โข โ = (LSSumโ๐) | |
3 | pj1lmhm.z | . . 3 โข 0 = (0gโ๐) | |
4 | pj1lmhm.p | . . 3 โข ๐ = (proj1โ๐) | |
5 | pj1lmhm.1 | . . 3 โข (๐ โ ๐ โ LMod) | |
6 | pj1lmhm.2 | . . 3 โข (๐ โ ๐ โ ๐ฟ) | |
7 | pj1lmhm.3 | . . 3 โข (๐ โ ๐ โ ๐ฟ) | |
8 | pj1lmhm.4 | . . 3 โข (๐ โ (๐ โฉ ๐) = { 0 }) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | pj1lmhm 20576 | . 2 โข (๐ โ (๐๐๐) โ ((๐ โพs (๐ โ ๐)) LMHom ๐)) |
10 | eqid 2733 | . . . . 5 โข (+gโ๐) = (+gโ๐) | |
11 | eqid 2733 | . . . . 5 โข (Cntzโ๐) = (Cntzโ๐) | |
12 | 1 | lsssssubg 20434 | . . . . . . 7 โข (๐ โ LMod โ ๐ฟ โ (SubGrpโ๐)) |
13 | 5, 12 | syl 17 | . . . . . 6 โข (๐ โ ๐ฟ โ (SubGrpโ๐)) |
14 | 13, 6 | sseldd 3946 | . . . . 5 โข (๐ โ ๐ โ (SubGrpโ๐)) |
15 | 13, 7 | sseldd 3946 | . . . . 5 โข (๐ โ ๐ โ (SubGrpโ๐)) |
16 | lmodabl 20384 | . . . . . . 7 โข (๐ โ LMod โ ๐ โ Abel) | |
17 | 5, 16 | syl 17 | . . . . . 6 โข (๐ โ ๐ โ Abel) |
18 | 11, 17, 14, 15 | ablcntzd 19640 | . . . . 5 โข (๐ โ ๐ โ ((Cntzโ๐)โ๐)) |
19 | 10, 2, 3, 11, 14, 15, 8, 18, 4 | pj1f 19484 | . . . 4 โข (๐ โ (๐๐๐):(๐ โ ๐)โถ๐) |
20 | 19 | frnd 6677 | . . 3 โข (๐ โ ran (๐๐๐) โ ๐) |
21 | eqid 2733 | . . . 4 โข (๐ โพs ๐) = (๐ โพs ๐) | |
22 | 21, 1 | reslmhm2b 20530 | . . 3 โข ((๐ โ LMod โง ๐ โ ๐ฟ โง ran (๐๐๐) โ ๐) โ ((๐๐๐) โ ((๐ โพs (๐ โ ๐)) LMHom ๐) โ (๐๐๐) โ ((๐ โพs (๐ โ ๐)) LMHom (๐ โพs ๐)))) |
23 | 5, 6, 20, 22 | syl3anc 1372 | . 2 โข (๐ โ ((๐๐๐) โ ((๐ โพs (๐ โ ๐)) LMHom ๐) โ (๐๐๐) โ ((๐ โพs (๐ โ ๐)) LMHom (๐ โพs ๐)))) |
24 | 9, 23 | mpbid 231 | 1 โข (๐ โ (๐๐๐) โ ((๐ โพs (๐ โ ๐)) LMHom (๐ โพs ๐))) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โ wb 205 = wceq 1542 โ wcel 2107 โฉ cin 3910 โ wss 3911 {csn 4587 ran crn 5635 โcfv 6497 (class class class)co 7358 โพs cress 17117 +gcplusg 17138 0gc0g 17326 SubGrpcsubg 18927 Cntzccntz 19100 LSSumclsm 19421 proj1cpj1 19422 Abelcabl 19568 LModclmod 20336 LSubSpclss 20407 LMHom clmhm 20495 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-map 8770 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ress 17118 df-plusg 17151 df-sca 17154 df-vsca 17155 df-0g 17328 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-mhm 18606 df-submnd 18607 df-grp 18756 df-minusg 18757 df-sbg 18758 df-subg 18930 df-ghm 19011 df-cntz 19102 df-lsm 19423 df-pj1 19424 df-cmn 19569 df-abl 19570 df-mgp 19902 df-ur 19919 df-ring 19971 df-lmod 20338 df-lss 20408 df-lmhm 20498 |
This theorem is referenced by: (None) |
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