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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 19597 | . 2 ⊢ (𝜑 → (𝑇𝑃𝑈) ∈ ((𝑊 ↾s (𝑇 ⊕ 𝑈)) LMHom 𝑊)) |
10 | eqid 2778 | . . . . 5 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
11 | eqid 2778 | . . . . 5 ⊢ (Cntz‘𝑊) = (Cntz‘𝑊) | |
12 | 1 | lsssssubg 19455 | . . . . . . 7 ⊢ (𝑊 ∈ LMod → 𝐿 ⊆ (SubGrp‘𝑊)) |
13 | 5, 12 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐿 ⊆ (SubGrp‘𝑊)) |
14 | 13, 6 | sseldd 3861 | . . . . 5 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝑊)) |
15 | 13, 7 | sseldd 3861 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊)) |
16 | lmodabl 19406 | . . . . . . 7 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel) | |
17 | 5, 16 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ Abel) |
18 | 11, 17, 14, 15 | ablcntzd 18736 | . . . . 5 ⊢ (𝜑 → 𝑇 ⊆ ((Cntz‘𝑊)‘𝑈)) |
19 | 10, 2, 3, 11, 14, 15, 8, 18, 4 | pj1f 18584 | . . . 4 ⊢ (𝜑 → (𝑇𝑃𝑈):(𝑇 ⊕ 𝑈)⟶𝑇) |
20 | 19 | frnd 6353 | . . 3 ⊢ (𝜑 → ran (𝑇𝑃𝑈) ⊆ 𝑇) |
21 | eqid 2778 | . . . 4 ⊢ (𝑊 ↾s 𝑇) = (𝑊 ↾s 𝑇) | |
22 | 21, 1 | reslmhm2b 19551 | . . 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 1351 | . 2 ⊢ (𝜑 → ((𝑇𝑃𝑈) ∈ ((𝑊 ↾s (𝑇 ⊕ 𝑈)) LMHom 𝑊) ↔ (𝑇𝑃𝑈) ∈ ((𝑊 ↾s (𝑇 ⊕ 𝑈)) LMHom (𝑊 ↾s 𝑇)))) |
24 | 9, 23 | mpbid 224 | 1 ⊢ (𝜑 → (𝑇𝑃𝑈) ∈ ((𝑊 ↾s (𝑇 ⊕ 𝑈)) LMHom (𝑊 ↾s 𝑇))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1507 ∈ wcel 2050 ∩ cin 3830 ⊆ wss 3831 {csn 4442 ran crn 5409 ‘cfv 6190 (class class class)co 6978 ↾s cress 16343 +gcplusg 16424 0gc0g 16572 SubGrpcsubg 18060 Cntzccntz 18219 LSSumclsm 18523 proj1cpj1 18524 Abelcabl 18670 LModclmod 19359 LSubSpclss 19428 LMHom clmhm 19516 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5050 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 ax-cnex 10393 ax-resscn 10394 ax-1cn 10395 ax-icn 10396 ax-addcl 10397 ax-addrcl 10398 ax-mulcl 10399 ax-mulrcl 10400 ax-mulcom 10401 ax-addass 10402 ax-mulass 10403 ax-distr 10404 ax-i2m1 10405 ax-1ne0 10406 ax-1rid 10407 ax-rnegex 10408 ax-rrecex 10409 ax-cnre 10410 ax-pre-lttri 10411 ax-pre-lttrn 10412 ax-pre-ltadd 10413 ax-pre-mulgt0 10414 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-tp 4447 df-op 4449 df-uni 4714 df-iun 4795 df-br 4931 df-opab 4993 df-mpt 5010 df-tr 5032 df-id 5313 df-eprel 5318 df-po 5327 df-so 5328 df-fr 5367 df-we 5369 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-pred 5988 df-ord 6034 df-on 6035 df-lim 6036 df-suc 6037 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-riota 6939 df-ov 6981 df-oprab 6982 df-mpo 6983 df-om 7399 df-1st 7503 df-2nd 7504 df-wrecs 7752 df-recs 7814 df-rdg 7852 df-er 8091 df-map 8210 df-en 8309 df-dom 8310 df-sdom 8311 df-pnf 10478 df-mnf 10479 df-xr 10480 df-ltxr 10481 df-le 10482 df-sub 10674 df-neg 10675 df-nn 11442 df-2 11506 df-3 11507 df-4 11508 df-5 11509 df-6 11510 df-ndx 16345 df-slot 16346 df-base 16348 df-sets 16349 df-ress 16350 df-plusg 16437 df-sca 16440 df-vsca 16441 df-0g 16574 df-mgm 17713 df-sgrp 17755 df-mnd 17766 df-mhm 17806 df-submnd 17807 df-grp 17897 df-minusg 17898 df-sbg 17899 df-subg 18063 df-ghm 18130 df-cntz 18221 df-lsm 18525 df-pj1 18526 df-cmn 18671 df-abl 18672 df-mgp 18966 df-ur 18978 df-ring 19025 df-lmod 19361 df-lss 19429 df-lmhm 19519 |
This theorem is referenced by: (None) |
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