pjf2 - Metamath Proof Explorer

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Theorem pjf2 21916

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 2765	. . 3 ⊢ (+_g‘𝑊) = (+_g‘𝑊)
2		eqid 2765	. . 3 ⊢ (LSSum‘𝑊) = (LSSum‘𝑊)
3		eqid 2765	. . 3 ⊢ (0_g‘𝑊) = (0_g‘𝑊)
4		eqid 2765	. . 3 ⊢ (Cntz‘𝑊) = (Cntz‘𝑊)
5		phllmod 21832	. . . . . 6 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod)
6	5	adantr 486	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑊 ∈ LMod)
7		eqid 2765	. . . . . 6 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
8	7	lsssssubg 21131	. . . . 5 ⊢ (𝑊 ∈ LMod → (LSubSp‘𝑊) ⊆ (SubGrp‘𝑊))
9	6, 8	syl 18	. . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (LSubSp‘𝑊) ⊆ (SubGrp‘𝑊))
10		pjf.v	. . . . . 6 ⊢ 𝑉 = (Base‘𝑊)
11		eqid 2765	. . . . . 6 ⊢ (ocv‘𝑊) = (ocv‘𝑊)
12		pjf.k	. . . . . 6 ⊢ 𝐾 = (proj‘𝑊)
13	10, 7, 11, 2, 12	pjdm2 21913	. . . . 5 ⊢ (𝑊 ∈ PreHil → (𝑇 ∈ dom 𝐾 ↔ (𝑇 ∈ (LSubSp‘𝑊) ∧ (𝑇(LSSum‘𝑊)((ocv‘𝑊)‘𝑇)) = 𝑉)))
14	13	simprbda 504	. . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑇 ∈ (LSubSp‘𝑊))
15	9, 14	sseldd 3939	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑇 ∈ (SubGrp‘𝑊))
16	10, 7	lssss 21109	. . . . . 6 ⊢ (𝑇 ∈ (LSubSp‘𝑊) → 𝑇 ⊆ 𝑉)
17	14, 16	syl 18	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑇 ⊆ 𝑉)
18	10, 11, 7	ocvlss 21874	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ⊆ 𝑉) → ((ocv‘𝑊)‘𝑇) ∈ (LSubSp‘𝑊))
19	17, 18	syldan 603	. . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → ((ocv‘𝑊)‘𝑇) ∈ (LSubSp‘𝑊))
20	9, 19	sseldd 3939	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → ((ocv‘𝑊)‘𝑇) ∈ (SubGrp‘𝑊))
21	11, 7, 3	ocvin 21876	. . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ (LSubSp‘𝑊)) → (𝑇 ∩ ((ocv‘𝑊)‘𝑇)) = {(0_g‘𝑊)})
22	14, 21	syldan 603	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝑇 ∩ ((ocv‘𝑊)‘𝑇)) = {(0_g‘𝑊)})
23		lmodabl 21082	. . . . 5 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel)
24	6, 23	syl 18	. . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑊 ∈ Abel)
25	4, 24, 15, 20	ablcntzd 19973	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑇 ⊆ ((Cntz‘𝑊)‘((ocv‘𝑊)‘𝑇)))
26		eqid 2765	. . 3 ⊢ (proj₁‘𝑊) = (proj₁‘𝑊)
27	1, 2, 3, 4, 15, 20, 22, 25, 26	pj1f 19813	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝑇(proj₁‘𝑊)((ocv‘𝑊)‘𝑇)):(𝑇(LSSum‘𝑊)((ocv‘𝑊)‘𝑇))⟶𝑇)
28	11, 26, 12	pjval 21912	. . . . 5 ⊢ (𝑇 ∈ dom 𝐾 → (𝐾‘𝑇) = (𝑇(proj₁‘𝑊)((ocv‘𝑊)‘𝑇)))
29	28	adantl 487	. . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝐾‘𝑇) = (𝑇(proj₁‘𝑊)((ocv‘𝑊)‘𝑇)))
30	29	eqcomd 2771	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝑇(proj₁‘𝑊)((ocv‘𝑊)‘𝑇)) = (𝐾‘𝑇))
31	13	simplbda 505	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝑇(LSSum‘𝑊)((ocv‘𝑊)‘𝑇)) = 𝑉)
32	30, 31	feq12d 6697	. 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 235	1 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝐾‘𝑇):𝑉⟶𝑇)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2146 ∩ cin 3905 ⊆ wss 3906 {csn 4591 dom cdm 5663 ⟶wf 6536 ‘cfv 6540 (class class class)co 7419 Basecbs 17293 +_gcplusg 17334 0_gc0g 17516 SubGrpcsubg 19232 Cntzccntz 19431 LSSumclsm 19750 proj₁cpj1 19751 Abelcabl 19897 LModclmod 21033 LSubSpclss 21104 PreHilcphl 21826 ocvcocv 21862 projcpj 21902

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11173 ax-resscn 11174 ax-1cn 11175 ax-icn 11176 ax-addcl 11177 ax-addrcl 11178 ax-mulcl 11179 ax-mulrcl 11180 ax-mulcom 11181 ax-addass 11182 ax-mulass 11183 ax-distr 11184 ax-i2m1 11185 ax-1ne0 11186 ax-1rid 11187 ax-rnegex 11188 ax-rrecex 11189 ax-cnre 11190 ax-pre-lttri 11191 ax-pre-lttrn 11192 ax-pre-ltadd 11193 ax-pre-mulgt0 11194

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7869 df-1st 7992 df-2nd 7993 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-er 8700 df-map 8832 df-en 8950 df-dom 8951 df-sdom 8952 df-pnf 11262 df-mnf 11263 df-xr 11264 df-ltxr 11265 df-le 11266 df-sub 11460 df-neg 11461 df-nn 12251 df-2 12320 df-3 12321 df-4 12322 df-5 12323 df-6 12324 df-7 12325 df-8 12326 df-sets 17248 df-slot 17266 df-ndx 17278 df-base 17294 df-ress 17315 df-plusg 17347 df-sca 17350 df-vsca 17351 df-ip 17352 df-0g 17518 df-mgm 18722 df-sgrp 18811 df-mnd 18827 df-grp 19049 df-minusg 19050 df-sbg 19051 df-subg 19235 df-ghm 19330 df-cntz 19433 df-lsm 19752 df-pj1 19753 df-cmn 19898 df-abl 19899 df-mgp 20263 df-rng 20277 df-ur 20310 df-ring 20363 df-lmod 21035 df-lss 21105 df-lmhm 21195 df-lvec 21276 df-sra 21346 df-rgmod 21347 df-phl 21828 df-ocv 21865 df-pj 21905

This theorem is used by: pjfo 21917

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