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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 | ⊢ 𝐿 = (LSubSp‘𝑊) |
pj1lmhm.s | ⊢ ⊕ = (LSSum‘𝑊) |
pj1lmhm.z | ⊢ 0 = (0g‘𝑊) |
pj1lmhm.p | ⊢ 𝑃 = (proj1‘𝑊) |
pj1lmhm.1 | ⊢ (𝜑 → 𝑊 ∈ LMod) |
pj1lmhm.2 | ⊢ (𝜑 → 𝑇 ∈ 𝐿) |
pj1lmhm.3 | ⊢ (𝜑 → 𝑈 ∈ 𝐿) |
pj1lmhm.4 | ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 }) |
Ref | Expression |
---|---|
pj1lmhm2 | ⊢ (𝜑 → (𝑇𝑃𝑈) ∈ ((𝑊 ↾s (𝑇 ⊕ 𝑈)) LMHom (𝑊 ↾s 𝑇))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pj1lmhm.l | . . 3 ⊢ 𝐿 = (LSubSp‘𝑊) | |
2 | pj1lmhm.s | . . 3 ⊢ ⊕ = (LSSum‘𝑊) | |
3 | pj1lmhm.z | . . 3 ⊢ 0 = (0g‘𝑊) | |
4 | pj1lmhm.p | . . 3 ⊢ 𝑃 = (proj1‘𝑊) | |
5 | pj1lmhm.1 | . . 3 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
6 | pj1lmhm.2 | . . 3 ⊢ (𝜑 → 𝑇 ∈ 𝐿) | |
7 | pj1lmhm.3 | . . 3 ⊢ (𝜑 → 𝑈 ∈ 𝐿) | |
8 | pj1lmhm.4 | . . 3 ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 }) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | pj1lmhm 19100 | . 2 ⊢ (𝜑 → (𝑇𝑃𝑈) ∈ ((𝑊 ↾s (𝑇 ⊕ 𝑈)) LMHom 𝑊)) |
10 | eqid 2622 | . . . . 5 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
11 | eqid 2622 | . . . . 5 ⊢ (Cntz‘𝑊) = (Cntz‘𝑊) | |
12 | 1 | lsssssubg 18958 | . . . . . . 7 ⊢ (𝑊 ∈ LMod → 𝐿 ⊆ (SubGrp‘𝑊)) |
13 | 5, 12 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐿 ⊆ (SubGrp‘𝑊)) |
14 | 13, 6 | sseldd 3604 | . . . . 5 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝑊)) |
15 | 13, 7 | sseldd 3604 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊)) |
16 | lmodabl 18910 | . . . . . . 7 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel) | |
17 | 5, 16 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ Abel) |
18 | 11, 17, 14, 15 | ablcntzd 18260 | . . . . 5 ⊢ (𝜑 → 𝑇 ⊆ ((Cntz‘𝑊)‘𝑈)) |
19 | 10, 2, 3, 11, 14, 15, 8, 18, 4 | pj1f 18110 | . . . 4 ⊢ (𝜑 → (𝑇𝑃𝑈):(𝑇 ⊕ 𝑈)⟶𝑇) |
20 | frn 6053 | . . . 4 ⊢ ((𝑇𝑃𝑈):(𝑇 ⊕ 𝑈)⟶𝑇 → ran (𝑇𝑃𝑈) ⊆ 𝑇) | |
21 | 19, 20 | syl 17 | . . 3 ⊢ (𝜑 → ran (𝑇𝑃𝑈) ⊆ 𝑇) |
22 | eqid 2622 | . . . 4 ⊢ (𝑊 ↾s 𝑇) = (𝑊 ↾s 𝑇) | |
23 | 22, 1 | reslmhm2b 19054 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝐿 ∧ ran (𝑇𝑃𝑈) ⊆ 𝑇) → ((𝑇𝑃𝑈) ∈ ((𝑊 ↾s (𝑇 ⊕ 𝑈)) LMHom 𝑊) ↔ (𝑇𝑃𝑈) ∈ ((𝑊 ↾s (𝑇 ⊕ 𝑈)) LMHom (𝑊 ↾s 𝑇)))) |
24 | 5, 6, 21, 23 | syl3anc 1326 | . 2 ⊢ (𝜑 → ((𝑇𝑃𝑈) ∈ ((𝑊 ↾s (𝑇 ⊕ 𝑈)) LMHom 𝑊) ↔ (𝑇𝑃𝑈) ∈ ((𝑊 ↾s (𝑇 ⊕ 𝑈)) LMHom (𝑊 ↾s 𝑇)))) |
25 | 9, 24 | mpbid 222 | 1 ⊢ (𝜑 → (𝑇𝑃𝑈) ∈ ((𝑊 ↾s (𝑇 ⊕ 𝑈)) LMHom (𝑊 ↾s 𝑇))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1483 ∈ wcel 1990 ∩ cin 3573 ⊆ wss 3574 {csn 4177 ran crn 5115 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↾s cress 15858 +gcplusg 15941 0gc0g 16100 SubGrpcsubg 17588 Cntzccntz 17748 LSSumclsm 18049 proj1cpj1 18050 Abelcabl 18194 LModclmod 18863 LSubSpclss 18932 LMHom clmhm 19019 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-sca 15957 df-vsca 15958 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-subg 17591 df-ghm 17658 df-cntz 17750 df-lsm 18051 df-pj1 18052 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-lmod 18865 df-lss 18933 df-lmhm 19022 |
This theorem is referenced by: (None) |
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