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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 20137 | . 2 ⊢ (𝜑 → (𝑇𝑃𝑈) ∈ ((𝑊 ↾s (𝑇 ⊕ 𝑈)) LMHom 𝑊)) |
10 | eqid 2737 | . . . . 5 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
11 | eqid 2737 | . . . . 5 ⊢ (Cntz‘𝑊) = (Cntz‘𝑊) | |
12 | 1 | lsssssubg 19995 | . . . . . . 7 ⊢ (𝑊 ∈ LMod → 𝐿 ⊆ (SubGrp‘𝑊)) |
13 | 5, 12 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐿 ⊆ (SubGrp‘𝑊)) |
14 | 13, 6 | sseldd 3902 | . . . . 5 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝑊)) |
15 | 13, 7 | sseldd 3902 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊)) |
16 | lmodabl 19946 | . . . . . . 7 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel) | |
17 | 5, 16 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ Abel) |
18 | 11, 17, 14, 15 | ablcntzd 19242 | . . . . 5 ⊢ (𝜑 → 𝑇 ⊆ ((Cntz‘𝑊)‘𝑈)) |
19 | 10, 2, 3, 11, 14, 15, 8, 18, 4 | pj1f 19087 | . . . 4 ⊢ (𝜑 → (𝑇𝑃𝑈):(𝑇 ⊕ 𝑈)⟶𝑇) |
20 | 19 | frnd 6553 | . . 3 ⊢ (𝜑 → ran (𝑇𝑃𝑈) ⊆ 𝑇) |
21 | eqid 2737 | . . . 4 ⊢ (𝑊 ↾s 𝑇) = (𝑊 ↾s 𝑇) | |
22 | 21, 1 | reslmhm2b 20091 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝐿 ∧ ran (𝑇𝑃𝑈) ⊆ 𝑇) → ((𝑇𝑃𝑈) ∈ ((𝑊 ↾s (𝑇 ⊕ 𝑈)) LMHom 𝑊) ↔ (𝑇𝑃𝑈) ∈ ((𝑊 ↾s (𝑇 ⊕ 𝑈)) LMHom (𝑊 ↾s 𝑇)))) |
23 | 5, 6, 20, 22 | syl3anc 1373 | . 2 ⊢ (𝜑 → ((𝑇𝑃𝑈) ∈ ((𝑊 ↾s (𝑇 ⊕ 𝑈)) LMHom 𝑊) ↔ (𝑇𝑃𝑈) ∈ ((𝑊 ↾s (𝑇 ⊕ 𝑈)) LMHom (𝑊 ↾s 𝑇)))) |
24 | 9, 23 | mpbid 235 | 1 ⊢ (𝜑 → (𝑇𝑃𝑈) ∈ ((𝑊 ↾s (𝑇 ⊕ 𝑈)) LMHom (𝑊 ↾s 𝑇))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1543 ∈ wcel 2110 ∩ cin 3865 ⊆ wss 3866 {csn 4541 ran crn 5552 ‘cfv 6380 (class class class)co 7213 ↾s cress 16784 +gcplusg 16802 0gc0g 16944 SubGrpcsubg 18537 Cntzccntz 18709 LSSumclsm 19023 proj1cpj1 19024 Abelcabl 19171 LModclmod 19899 LSubSpclss 19968 LMHom clmhm 20056 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-er 8391 df-map 8510 df-en 8627 df-dom 8628 df-sdom 8629 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-sca 16818 df-vsca 16819 df-0g 16946 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-mhm 18218 df-submnd 18219 df-grp 18368 df-minusg 18369 df-sbg 18370 df-subg 18540 df-ghm 18620 df-cntz 18711 df-lsm 19025 df-pj1 19026 df-cmn 19172 df-abl 19173 df-mgp 19505 df-ur 19517 df-ring 19564 df-lmod 19901 df-lss 19969 df-lmhm 20059 |
This theorem is referenced by: (None) |
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