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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 21117 | . 2 ⊢ (𝜑 → (𝑇𝑃𝑈) ∈ ((𝑊 ↾s (𝑇 ⊕ 𝑈)) LMHom 𝑊)) |
10 | eqid 2735 | . . . . 5 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
11 | eqid 2735 | . . . . 5 ⊢ (Cntz‘𝑊) = (Cntz‘𝑊) | |
12 | 1 | lsssssubg 20974 | . . . . . . 7 ⊢ (𝑊 ∈ LMod → 𝐿 ⊆ (SubGrp‘𝑊)) |
13 | 5, 12 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐿 ⊆ (SubGrp‘𝑊)) |
14 | 13, 6 | sseldd 3996 | . . . . 5 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝑊)) |
15 | 13, 7 | sseldd 3996 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊)) |
16 | lmodabl 20924 | . . . . . . 7 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel) | |
17 | 5, 16 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ Abel) |
18 | 11, 17, 14, 15 | ablcntzd 19890 | . . . . 5 ⊢ (𝜑 → 𝑇 ⊆ ((Cntz‘𝑊)‘𝑈)) |
19 | 10, 2, 3, 11, 14, 15, 8, 18, 4 | pj1f 19730 | . . . 4 ⊢ (𝜑 → (𝑇𝑃𝑈):(𝑇 ⊕ 𝑈)⟶𝑇) |
20 | 19 | frnd 6745 | . . 3 ⊢ (𝜑 → ran (𝑇𝑃𝑈) ⊆ 𝑇) |
21 | eqid 2735 | . . . 4 ⊢ (𝑊 ↾s 𝑇) = (𝑊 ↾s 𝑇) | |
22 | 21, 1 | reslmhm2b 21071 | . . 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 1370 | . 2 ⊢ (𝜑 → ((𝑇𝑃𝑈) ∈ ((𝑊 ↾s (𝑇 ⊕ 𝑈)) LMHom 𝑊) ↔ (𝑇𝑃𝑈) ∈ ((𝑊 ↾s (𝑇 ⊕ 𝑈)) LMHom (𝑊 ↾s 𝑇)))) |
24 | 9, 23 | mpbid 232 | 1 ⊢ (𝜑 → (𝑇𝑃𝑈) ∈ ((𝑊 ↾s (𝑇 ⊕ 𝑈)) LMHom (𝑊 ↾s 𝑇))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1537 ∈ wcel 2106 ∩ cin 3962 ⊆ wss 3963 {csn 4631 ran crn 5690 ‘cfv 6563 (class class class)co 7431 ↾s cress 17274 +gcplusg 17298 0gc0g 17486 SubGrpcsubg 19151 Cntzccntz 19346 LSSumclsm 19667 proj1cpj1 19668 Abelcabl 19814 LModclmod 20875 LSubSpclss 20947 LMHom clmhm 21036 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-sca 17314 df-vsca 17315 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-ghm 19244 df-cntz 19348 df-lsm 19669 df-pj1 19670 df-cmn 19815 df-abl 19816 df-mgp 20153 df-ur 20200 df-ring 20253 df-lmod 20877 df-lss 20948 df-lmhm 21039 |
This theorem is referenced by: (None) |
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