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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 20948 | . 2 โข (๐ โ (๐๐๐) โ ((๐ โพs (๐ โ ๐)) LMHom ๐)) |
10 | eqid 2726 | . . . . 5 โข (+gโ๐) = (+gโ๐) | |
11 | eqid 2726 | . . . . 5 โข (Cntzโ๐) = (Cntzโ๐) | |
12 | 1 | lsssssubg 20805 | . . . . . . 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 3978 | . . . . 5 โข (๐ โ ๐ โ (SubGrpโ๐)) |
15 | 13, 7 | sseldd 3978 | . . . . 5 โข (๐ โ ๐ โ (SubGrpโ๐)) |
16 | lmodabl 20755 | . . . . . . 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 19777 | . . . . 5 โข (๐ โ ๐ โ ((Cntzโ๐)โ๐)) |
19 | 10, 2, 3, 11, 14, 15, 8, 18, 4 | pj1f 19617 | . . . 4 โข (๐ โ (๐๐๐):(๐ โ ๐)โถ๐) |
20 | 19 | frnd 6719 | . . 3 โข (๐ โ ran (๐๐๐) โ ๐) |
21 | eqid 2726 | . . . 4 โข (๐ โพs ๐) = (๐ โพs ๐) | |
22 | 21, 1 | reslmhm2b 20902 | . . 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 1368 | . 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 1533 โ wcel 2098 โฉ cin 3942 โ wss 3943 {csn 4623 ran crn 5670 โcfv 6537 (class class class)co 7405 โพs cress 17182 +gcplusg 17206 0gc0g 17394 SubGrpcsubg 19047 Cntzccntz 19231 LSSumclsm 19554 proj1cpj1 19555 Abelcabl 19701 LModclmod 20706 LSubSpclss 20778 LMHom clmhm 20867 |
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 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 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-pss 3962 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-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 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-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 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-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-sca 17222 df-vsca 17223 df-0g 17396 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-mhm 18713 df-submnd 18714 df-grp 18866 df-minusg 18867 df-sbg 18868 df-subg 19050 df-ghm 19139 df-cntz 19233 df-lsm 19556 df-pj1 19557 df-cmn 19702 df-abl 19703 df-mgp 20040 df-ur 20087 df-ring 20140 df-lmod 20708 df-lss 20779 df-lmhm 20870 |
This theorem is referenced by: (None) |
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