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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 19925 | . 2 ⊢ (𝜑 → (𝑇𝑃𝑈) ∈ ((𝑊 ↾s (𝑇 ⊕ 𝑈)) LMHom 𝑊)) |
10 | eqid 2759 | . . . . 5 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
11 | eqid 2759 | . . . . 5 ⊢ (Cntz‘𝑊) = (Cntz‘𝑊) | |
12 | 1 | lsssssubg 19783 | . . . . . . 7 ⊢ (𝑊 ∈ LMod → 𝐿 ⊆ (SubGrp‘𝑊)) |
13 | 5, 12 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐿 ⊆ (SubGrp‘𝑊)) |
14 | 13, 6 | sseldd 3889 | . . . . 5 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝑊)) |
15 | 13, 7 | sseldd 3889 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊)) |
16 | lmodabl 19734 | . . . . . . 7 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel) | |
17 | 5, 16 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ Abel) |
18 | 11, 17, 14, 15 | ablcntzd 19030 | . . . . 5 ⊢ (𝜑 → 𝑇 ⊆ ((Cntz‘𝑊)‘𝑈)) |
19 | 10, 2, 3, 11, 14, 15, 8, 18, 4 | pj1f 18875 | . . . 4 ⊢ (𝜑 → (𝑇𝑃𝑈):(𝑇 ⊕ 𝑈)⟶𝑇) |
20 | 19 | frnd 6498 | . . 3 ⊢ (𝜑 → ran (𝑇𝑃𝑈) ⊆ 𝑇) |
21 | eqid 2759 | . . . 4 ⊢ (𝑊 ↾s 𝑇) = (𝑊 ↾s 𝑇) | |
22 | 21, 1 | reslmhm2b 19879 | . . 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 1369 | . 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 1539 ∈ wcel 2112 ∩ cin 3853 ⊆ wss 3854 {csn 4515 ran crn 5518 ‘cfv 6328 (class class class)co 7143 ↾s cress 16527 +gcplusg 16608 0gc0g 16756 SubGrpcsubg 18325 Cntzccntz 18497 LSSumclsm 18811 proj1cpj1 18812 Abelcabl 18959 LModclmod 19687 LSubSpclss 19756 LMHom clmhm 19844 |
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 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-rep 5149 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 ax-cnex 10616 ax-resscn 10617 ax-1cn 10618 ax-icn 10619 ax-addcl 10620 ax-addrcl 10621 ax-mulcl 10622 ax-mulrcl 10623 ax-mulcom 10624 ax-addass 10625 ax-mulass 10626 ax-distr 10627 ax-i2m1 10628 ax-1ne0 10629 ax-1rid 10630 ax-rnegex 10631 ax-rrecex 10632 ax-cnre 10633 ax-pre-lttri 10634 ax-pre-lttrn 10635 ax-pre-ltadd 10636 ax-pre-mulgt0 10637 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-nel 3054 df-ral 3073 df-rex 3074 df-reu 3075 df-rmo 3076 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-pss 3873 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-tp 4520 df-op 4522 df-uni 4792 df-iun 4878 df-br 5026 df-opab 5088 df-mpt 5106 df-tr 5132 df-id 5423 df-eprel 5428 df-po 5436 df-so 5437 df-fr 5476 df-we 5478 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-pred 6119 df-ord 6165 df-on 6166 df-lim 6167 df-suc 6168 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-riota 7101 df-ov 7146 df-oprab 7147 df-mpo 7148 df-om 7573 df-1st 7686 df-2nd 7687 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-map 8411 df-en 8521 df-dom 8522 df-sdom 8523 df-pnf 10700 df-mnf 10701 df-xr 10702 df-ltxr 10703 df-le 10704 df-sub 10895 df-neg 10896 df-nn 11660 df-2 11722 df-3 11723 df-4 11724 df-5 11725 df-6 11726 df-ndx 16529 df-slot 16530 df-base 16532 df-sets 16533 df-ress 16534 df-plusg 16621 df-sca 16624 df-vsca 16625 df-0g 16758 df-mgm 17903 df-sgrp 17952 df-mnd 17963 df-mhm 18007 df-submnd 18008 df-grp 18157 df-minusg 18158 df-sbg 18159 df-subg 18328 df-ghm 18408 df-cntz 18499 df-lsm 18813 df-pj1 18814 df-cmn 18960 df-abl 18961 df-mgp 19293 df-ur 19305 df-ring 19352 df-lmod 19689 df-lss 19757 df-lmhm 19847 |
This theorem is referenced by: (None) |
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