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Theorem pjf2 20457

Description: A projection is a function from the base set to the subspace. (Contributed by Mario Carneiro, 16-Oct-2015.)

**Hypotheses**
Ref	Expression
pjf.k	⊢ 𝐾 = (proj‘𝑊)
pjf.v	⊢ 𝑉 = (Base‘𝑊)

**Assertion**
Ref	Expression
pjf2	⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝐾‘𝑇):𝑉⟶𝑇)

**Proof of Theorem pjf2**
Step	Hyp	Ref	Expression
1		eqid 2778	. . 3 ⊢ (+_g‘𝑊) = (+_g‘𝑊)
2		eqid 2778	. . 3 ⊢ (LSSum‘𝑊) = (LSSum‘𝑊)
3		eqid 2778	. . 3 ⊢ (0_g‘𝑊) = (0_g‘𝑊)
4		eqid 2778	. . 3 ⊢ (Cntz‘𝑊) = (Cntz‘𝑊)
5		phllmod 20373	. . . . . 6 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod)
6	5	adantr 474	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑊 ∈ LMod)
7		eqid 2778	. . . . . 6 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
8	7	lsssssubg 19353	. . . . 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 2778	. . . . . 6 ⊢ (ocv‘𝑊) = (ocv‘𝑊)
12		pjf.k	. . . . . 6 ⊢ 𝐾 = (proj‘𝑊)
13	10, 7, 11, 2, 12	pjdm2 20454	. . . . 5 ⊢ (𝑊 ∈ PreHil → (𝑇 ∈ dom 𝐾 ↔ (𝑇 ∈ (LSubSp‘𝑊) ∧ (𝑇(LSSum‘𝑊)((ocv‘𝑊)‘𝑇)) = 𝑉)))
14	13	simprbda 494	. . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑇 ∈ (LSubSp‘𝑊))
15	9, 14	sseldd 3822	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑇 ∈ (SubGrp‘𝑊))
16	10, 7	lssss 19329	. . . . . 6 ⊢ (𝑇 ∈ (LSubSp‘𝑊) → 𝑇 ⊆ 𝑉)
17	14, 16	syl 17	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑇 ⊆ 𝑉)
18	10, 11, 7	ocvlss 20415	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ⊆ 𝑉) → ((ocv‘𝑊)‘𝑇) ∈ (LSubSp‘𝑊))
19	17, 18	syldan 585	. . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → ((ocv‘𝑊)‘𝑇) ∈ (LSubSp‘𝑊))
20	9, 19	sseldd 3822	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → ((ocv‘𝑊)‘𝑇) ∈ (SubGrp‘𝑊))
21	11, 7, 3	ocvin 20417	. . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ (LSubSp‘𝑊)) → (𝑇 ∩ ((ocv‘𝑊)‘𝑇)) = {(0_g‘𝑊)})
22	14, 21	syldan 585	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝑇 ∩ ((ocv‘𝑊)‘𝑇)) = {(0_g‘𝑊)})
23		lmodabl 19302	. . . . 5 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel)
24	6, 23	syl 17	. . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑊 ∈ Abel)
25	4, 24, 15, 20	ablcntzd 18646	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑇 ⊆ ((Cntz‘𝑊)‘((ocv‘𝑊)‘𝑇)))
26		eqid 2778	. . 3 ⊢ (proj₁‘𝑊) = (proj₁‘𝑊)
27	1, 2, 3, 4, 15, 20, 22, 25, 26	pj1f 18494	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝑇(proj₁‘𝑊)((ocv‘𝑊)‘𝑇)):(𝑇(LSSum‘𝑊)((ocv‘𝑊)‘𝑇))⟶𝑇)
28	11, 26, 12	pjval 20453	. . . . 5 ⊢ (𝑇 ∈ dom 𝐾 → (𝐾‘𝑇) = (𝑇(proj₁‘𝑊)((ocv‘𝑊)‘𝑇)))
29	28	adantl 475	. . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝐾‘𝑇) = (𝑇(proj₁‘𝑊)((ocv‘𝑊)‘𝑇)))
30	29	eqcomd 2784	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝑇(proj₁‘𝑊)((ocv‘𝑊)‘𝑇)) = (𝐾‘𝑇))
31	13	simplbda 495	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝑇(LSSum‘𝑊)((ocv‘𝑊)‘𝑇)) = 𝑉)
32	30, 31	feq12d 6279	. 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 224	1 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝐾‘𝑇):𝑉⟶𝑇)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ∩ cin 3791 ⊆ wss 3792 {csn 4398 dom cdm 5355 ⟶wf 6131 ‘cfv 6135 (class class class)co 6922 Basecbs 16255 +_gcplusg 16338 0_gc0g 16486 SubGrpcsubg 17972 Cntzccntz 18131 LSSumclsm 18433 proj₁cpj1 18434 Abelcabl 18580 LModclmod 19255 LSubSpclss 19324 PreHilcphl 20367 ocvcocv 20403 projcpj 20443

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-sca 16354 df-vsca 16355 df-ip 16356 df-0g 16488 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-grp 17812 df-minusg 17813 df-sbg 17814 df-subg 17975 df-ghm 18042 df-cntz 18133 df-lsm 18435 df-pj1 18436 df-cmn 18581 df-abl 18582 df-mgp 18877 df-ur 18889 df-ring 18936 df-lmod 19257 df-lss 19325 df-lmhm 19417 df-lvec 19498 df-sra 19569 df-rgmod 19570 df-phl 20369 df-ocv 20406 df-pj 20446

This theorem is referenced by: pjfo 20458

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