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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 2731 | . . 3 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
2 | eqid 2731 | . . 3 ⊢ (LSSum‘𝑊) = (LSSum‘𝑊) | |
3 | eqid 2731 | . . 3 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
4 | eqid 2731 | . . 3 ⊢ (Cntz‘𝑊) = (Cntz‘𝑊) | |
5 | phllmod 21493 | . . . . . 6 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod) | |
6 | 5 | adantr 480 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑊 ∈ LMod) |
7 | eqid 2731 | . . . . . 6 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
8 | 7 | lsssssubg 20801 | . . . . 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 2731 | . . . . . 6 ⊢ (ocv‘𝑊) = (ocv‘𝑊) | |
12 | pjf.k | . . . . . 6 ⊢ 𝐾 = (proj‘𝑊) | |
13 | 10, 7, 11, 2, 12 | pjdm2 21576 | . . . . 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 3983 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑇 ∈ (SubGrp‘𝑊)) |
16 | 10, 7 | lssss 20779 | . . . . . 6 ⊢ (𝑇 ∈ (LSubSp‘𝑊) → 𝑇 ⊆ 𝑉) |
17 | 14, 16 | syl 17 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑇 ⊆ 𝑉) |
18 | 10, 11, 7 | ocvlss 21535 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ⊆ 𝑉) → ((ocv‘𝑊)‘𝑇) ∈ (LSubSp‘𝑊)) |
19 | 17, 18 | syldan 590 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → ((ocv‘𝑊)‘𝑇) ∈ (LSubSp‘𝑊)) |
20 | 9, 19 | sseldd 3983 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → ((ocv‘𝑊)‘𝑇) ∈ (SubGrp‘𝑊)) |
21 | 11, 7, 3 | ocvin 21537 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ (LSubSp‘𝑊)) → (𝑇 ∩ ((ocv‘𝑊)‘𝑇)) = {(0g‘𝑊)}) |
22 | 14, 21 | syldan 590 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝑇 ∩ ((ocv‘𝑊)‘𝑇)) = {(0g‘𝑊)}) |
23 | lmodabl 20751 | . . . . 5 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel) | |
24 | 6, 23 | syl 17 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑊 ∈ Abel) |
25 | 4, 24, 15, 20 | ablcntzd 19773 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑇 ⊆ ((Cntz‘𝑊)‘((ocv‘𝑊)‘𝑇))) |
26 | eqid 2731 | . . 3 ⊢ (proj1‘𝑊) = (proj1‘𝑊) | |
27 | 1, 2, 3, 4, 15, 20, 22, 25, 26 | pj1f 19613 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝑇(proj1‘𝑊)((ocv‘𝑊)‘𝑇)):(𝑇(LSSum‘𝑊)((ocv‘𝑊)‘𝑇))⟶𝑇) |
28 | 11, 26, 12 | pjval 21575 | . . . . 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 2737 | . . 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 6705 | . 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 231 | 1 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝐾‘𝑇):𝑉⟶𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ∩ cin 3947 ⊆ wss 3948 {csn 4628 dom cdm 5676 ⟶wf 6539 ‘cfv 6543 (class class class)co 7412 Basecbs 17151 +gcplusg 17204 0gc0g 17392 SubGrpcsubg 19043 Cntzccntz 19227 LSSumclsm 19550 proj1cpj1 19551 Abelcabl 19697 LModclmod 20702 LSubSpclss 20774 PreHilcphl 21487 ocvcocv 21523 projcpj 21565 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-er 8709 df-map 8828 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-sca 17220 df-vsca 17221 df-ip 17222 df-0g 17394 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-grp 18864 df-minusg 18865 df-sbg 18866 df-subg 19046 df-ghm 19135 df-cntz 19229 df-lsm 19552 df-pj1 19553 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-lmod 20704 df-lss 20775 df-lmhm 20866 df-lvec 20947 df-sra 21019 df-rgmod 21020 df-phl 21489 df-ocv 21526 df-pj 21568 |
This theorem is referenced by: pjfo 21580 |
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