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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 20999 | . 2 โข (๐ โ (๐๐๐) โ ((๐ โพs (๐ โ ๐)) LMHom ๐)) |
10 | eqid 2728 | . . . . 5 โข (+gโ๐) = (+gโ๐) | |
11 | eqid 2728 | . . . . 5 โข (Cntzโ๐) = (Cntzโ๐) | |
12 | 1 | lsssssubg 20856 | . . . . . . 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 3983 | . . . . 5 โข (๐ โ ๐ โ (SubGrpโ๐)) |
15 | 13, 7 | sseldd 3983 | . . . . 5 โข (๐ โ ๐ โ (SubGrpโ๐)) |
16 | lmodabl 20806 | . . . . . . 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 19826 | . . . . 5 โข (๐ โ ๐ โ ((Cntzโ๐)โ๐)) |
19 | 10, 2, 3, 11, 14, 15, 8, 18, 4 | pj1f 19666 | . . . 4 โข (๐ โ (๐๐๐):(๐ โ ๐)โถ๐) |
20 | 19 | frnd 6735 | . . 3 โข (๐ โ ran (๐๐๐) โ ๐) |
21 | eqid 2728 | . . . 4 โข (๐ โพs ๐) = (๐ โพs ๐) | |
22 | 21, 1 | reslmhm2b 20953 | . . 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 3948 โ wss 3949 {csn 4632 ran crn 5683 โcfv 6553 (class class class)co 7426 โพs cress 17218 +gcplusg 17242 0gc0g 17430 SubGrpcsubg 19089 Cntzccntz 19280 LSSumclsm 19603 proj1cpj1 19604 Abelcabl 19750 LModclmod 20757 LSubSpclss 20829 LMHom clmhm 20918 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8001 df-2nd 8002 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-er 8733 df-map 8855 df-en 8973 df-dom 8974 df-sdom 8975 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-nn 12253 df-2 12315 df-3 12316 df-4 12317 df-5 12318 df-6 12319 df-sets 17142 df-slot 17160 df-ndx 17172 df-base 17190 df-ress 17219 df-plusg 17255 df-sca 17258 df-vsca 17259 df-0g 17432 df-mgm 18609 df-sgrp 18688 df-mnd 18704 df-mhm 18749 df-submnd 18750 df-grp 18907 df-minusg 18908 df-sbg 18909 df-subg 19092 df-ghm 19182 df-cntz 19282 df-lsm 19605 df-pj1 19606 df-cmn 19751 df-abl 19752 df-mgp 20089 df-ur 20136 df-ring 20189 df-lmod 20759 df-lss 20830 df-lmhm 20921 |
This theorem is referenced by: (None) |
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