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Theorem pjf2 19877

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 2610	. . 3 ⊢ (+_g‘𝑊) = (+_g‘𝑊)
2		eqid 2610	. . 3 ⊢ (LSSum‘𝑊) = (LSSum‘𝑊)
3		eqid 2610	. . 3 ⊢ (0_g‘𝑊) = (0_g‘𝑊)
4		eqid 2610	. . 3 ⊢ (Cntz‘𝑊) = (Cntz‘𝑊)
5		phllmod 19794	. . . . . 6 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod)
6	5	adantr 480	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑊 ∈ LMod)
7		eqid 2610	. . . . . 6 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
8	7	lsssssubg 18779	. . . . 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 2610	. . . . . 6 ⊢ (ocv‘𝑊) = (ocv‘𝑊)
12		pjf.k	. . . . . 6 ⊢ 𝐾 = (proj‘𝑊)
13	10, 7, 11, 2, 12	pjdm2 19874	. . . . 5 ⊢ (𝑊 ∈ PreHil → (𝑇 ∈ dom 𝐾 ↔ (𝑇 ∈ (LSubSp‘𝑊) ∧ (𝑇(LSSum‘𝑊)((ocv‘𝑊)‘𝑇)) = 𝑉)))
14	13	simprbda 651	. . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑇 ∈ (LSubSp‘𝑊))
15	9, 14	sseldd 3569	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑇 ∈ (SubGrp‘𝑊))
16	10, 7	lssss 18758	. . . . . 6 ⊢ (𝑇 ∈ (LSubSp‘𝑊) → 𝑇 ⊆ 𝑉)
17	14, 16	syl 17	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑇 ⊆ 𝑉)
18	10, 11, 7	ocvlss 19835	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ⊆ 𝑉) → ((ocv‘𝑊)‘𝑇) ∈ (LSubSp‘𝑊))
19	17, 18	syldan 486	. . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → ((ocv‘𝑊)‘𝑇) ∈ (LSubSp‘𝑊))
20	9, 19	sseldd 3569	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → ((ocv‘𝑊)‘𝑇) ∈ (SubGrp‘𝑊))
21	11, 7, 3	ocvin 19837	. . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ (LSubSp‘𝑊)) → (𝑇 ∩ ((ocv‘𝑊)‘𝑇)) = {(0_g‘𝑊)})
22	14, 21	syldan 486	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝑇 ∩ ((ocv‘𝑊)‘𝑇)) = {(0_g‘𝑊)})
23		lmodabl 18733	. . . . 5 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel)
24	6, 23	syl 17	. . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑊 ∈ Abel)
25	4, 24, 15, 20	ablcntzd 18083	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑇 ⊆ ((Cntz‘𝑊)‘((ocv‘𝑊)‘𝑇)))
26		eqid 2610	. . 3 ⊢ (proj₁‘𝑊) = (proj₁‘𝑊)
27	1, 2, 3, 4, 15, 20, 22, 25, 26	pj1f 17933	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝑇(proj₁‘𝑊)((ocv‘𝑊)‘𝑇)):(𝑇(LSSum‘𝑊)((ocv‘𝑊)‘𝑇))⟶𝑇)
28	11, 26, 12	pjval 19873	. . . . 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 2616	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝑇(proj₁‘𝑊)((ocv‘𝑊)‘𝑇)) = (𝐾‘𝑇))
31	13	simplbda 652	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝑇(LSSum‘𝑊)((ocv‘𝑊)‘𝑇)) = 𝑉)
32	30, 31	feq12d 5946	. 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 221	1 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝐾‘𝑇):𝑉⟶𝑇)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∩ cin 3539 ⊆ wss 3540 {csn 4125 dom cdm 5038 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +_gcplusg 15768 0_gc0g 15923 SubGrpcsubg 17411 Cntzccntz 17571 LSSumclsm 17872 proj₁cpj1 17873 Abelcabl 18017 LModclmod 18686 LSubSpclss 18753 PreHilcphl 19788 ocvcocv 19823 projcpj 19863

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-sca 15784 df-vsca 15785 df-ip 15786 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-sbg 17250 df-subg 17414 df-ghm 17481 df-cntz 17573 df-lsm 17874 df-pj1 17875 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-lmod 18688 df-lss 18754 df-lmhm 18843 df-lvec 18924 df-sra 18993 df-rgmod 18994 df-phl 19790 df-ocv 19826 df-pj 19866

This theorem is referenced by: pjfo 19878

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