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Mirrors > Home > MPE Home > Th. List > pjf2 | Structured version Visualization version GIF version |
Description: A projection is a function from the base set to the subspace. (Contributed by Mario Carneiro, 16-Oct-2015.) |
Ref | Expression |
---|---|
pjf.k | ⊢ 𝐾 = (proj‘𝑊) |
pjf.v | ⊢ 𝑉 = (Base‘𝑊) |
Ref | Expression |
---|---|
pjf2 | ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝐾‘𝑇):𝑉⟶𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2735 | . . 3 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
2 | eqid 2735 | . . 3 ⊢ (LSSum‘𝑊) = (LSSum‘𝑊) | |
3 | eqid 2735 | . . 3 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
4 | eqid 2735 | . . 3 ⊢ (Cntz‘𝑊) = (Cntz‘𝑊) | |
5 | phllmod 21666 | . . . . . 6 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod) | |
6 | 5 | adantr 480 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑊 ∈ LMod) |
7 | eqid 2735 | . . . . . 6 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
8 | 7 | lsssssubg 20974 | . . . . 5 ⊢ (𝑊 ∈ LMod → (LSubSp‘𝑊) ⊆ (SubGrp‘𝑊)) |
9 | 6, 8 | syl 17 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (LSubSp‘𝑊) ⊆ (SubGrp‘𝑊)) |
10 | pjf.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑊) | |
11 | eqid 2735 | . . . . . 6 ⊢ (ocv‘𝑊) = (ocv‘𝑊) | |
12 | pjf.k | . . . . . 6 ⊢ 𝐾 = (proj‘𝑊) | |
13 | 10, 7, 11, 2, 12 | pjdm2 21749 | . . . . 5 ⊢ (𝑊 ∈ PreHil → (𝑇 ∈ dom 𝐾 ↔ (𝑇 ∈ (LSubSp‘𝑊) ∧ (𝑇(LSSum‘𝑊)((ocv‘𝑊)‘𝑇)) = 𝑉))) |
14 | 13 | simprbda 498 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑇 ∈ (LSubSp‘𝑊)) |
15 | 9, 14 | sseldd 3996 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑇 ∈ (SubGrp‘𝑊)) |
16 | 10, 7 | lssss 20952 | . . . . . 6 ⊢ (𝑇 ∈ (LSubSp‘𝑊) → 𝑇 ⊆ 𝑉) |
17 | 14, 16 | syl 17 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑇 ⊆ 𝑉) |
18 | 10, 11, 7 | ocvlss 21708 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ⊆ 𝑉) → ((ocv‘𝑊)‘𝑇) ∈ (LSubSp‘𝑊)) |
19 | 17, 18 | syldan 591 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → ((ocv‘𝑊)‘𝑇) ∈ (LSubSp‘𝑊)) |
20 | 9, 19 | sseldd 3996 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → ((ocv‘𝑊)‘𝑇) ∈ (SubGrp‘𝑊)) |
21 | 11, 7, 3 | ocvin 21710 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ (LSubSp‘𝑊)) → (𝑇 ∩ ((ocv‘𝑊)‘𝑇)) = {(0g‘𝑊)}) |
22 | 14, 21 | syldan 591 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝑇 ∩ ((ocv‘𝑊)‘𝑇)) = {(0g‘𝑊)}) |
23 | lmodabl 20924 | . . . . 5 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel) | |
24 | 6, 23 | syl 17 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑊 ∈ Abel) |
25 | 4, 24, 15, 20 | ablcntzd 19890 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑇 ⊆ ((Cntz‘𝑊)‘((ocv‘𝑊)‘𝑇))) |
26 | eqid 2735 | . . 3 ⊢ (proj1‘𝑊) = (proj1‘𝑊) | |
27 | 1, 2, 3, 4, 15, 20, 22, 25, 26 | pj1f 19730 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝑇(proj1‘𝑊)((ocv‘𝑊)‘𝑇)):(𝑇(LSSum‘𝑊)((ocv‘𝑊)‘𝑇))⟶𝑇) |
28 | 11, 26, 12 | pjval 21748 | . . . . 5 ⊢ (𝑇 ∈ dom 𝐾 → (𝐾‘𝑇) = (𝑇(proj1‘𝑊)((ocv‘𝑊)‘𝑇))) |
29 | 28 | adantl 481 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝐾‘𝑇) = (𝑇(proj1‘𝑊)((ocv‘𝑊)‘𝑇))) |
30 | 29 | eqcomd 2741 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝑇(proj1‘𝑊)((ocv‘𝑊)‘𝑇)) = (𝐾‘𝑇)) |
31 | 13 | simplbda 499 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝑇(LSSum‘𝑊)((ocv‘𝑊)‘𝑇)) = 𝑉) |
32 | 30, 31 | feq12d 6725 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → ((𝑇(proj1‘𝑊)((ocv‘𝑊)‘𝑇)):(𝑇(LSSum‘𝑊)((ocv‘𝑊)‘𝑇))⟶𝑇 ↔ (𝐾‘𝑇):𝑉⟶𝑇)) |
33 | 27, 32 | mpbid 232 | 1 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝐾‘𝑇):𝑉⟶𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∩ cin 3962 ⊆ wss 3963 {csn 4631 dom cdm 5689 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +gcplusg 17298 0gc0g 17486 SubGrpcsubg 19151 Cntzccntz 19346 LSSumclsm 19667 proj1cpj1 19668 Abelcabl 19814 LModclmod 20875 LSubSpclss 20947 PreHilcphl 21660 ocvcocv 21696 projcpj 21738 |
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-int 4952 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-7 12332 df-8 12333 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-ip 17316 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 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-rng 20171 df-ur 20200 df-ring 20253 df-lmod 20877 df-lss 20948 df-lmhm 21039 df-lvec 21120 df-sra 21190 df-rgmod 21191 df-phl 21662 df-ocv 21699 df-pj 21741 |
This theorem is referenced by: pjfo 21753 |
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