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Theorem pjfo 20862

Description: A projection is a surjection onto 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
pjfo	⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝐾‘𝑇):𝑉–onto→𝑇)

Proof of Theorem pjfo Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pjf.k	. . 3 ⊢ 𝐾 = (proj‘𝑊)
2		pjf.v	. . 3 ⊢ 𝑉 = (Base‘𝑊)
3	1, 2	pjf2 20861	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝐾‘𝑇):𝑉⟶𝑇)
4	3	frnd 6524	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → ran (𝐾‘𝑇) ⊆ 𝑇)
5		eqid 2824	. . . . . . . 8 ⊢ (ocv‘𝑊) = (ocv‘𝑊)
6		eqid 2824	. . . . . . . 8 ⊢ (proj₁‘𝑊) = (proj₁‘𝑊)
7	5, 6, 1	pjval 20857	. . . . . . 7 ⊢ (𝑇 ∈ dom 𝐾 → (𝐾‘𝑇) = (𝑇(proj₁‘𝑊)((ocv‘𝑊)‘𝑇)))
8	7	ad2antlr 725	. . . . . 6 ⊢ (((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) ∧ 𝑥 ∈ 𝑇) → (𝐾‘𝑇) = (𝑇(proj₁‘𝑊)((ocv‘𝑊)‘𝑇)))
9	8	fveq1d 6675	. . . . 5 ⊢ (((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) ∧ 𝑥 ∈ 𝑇) → ((𝐾‘𝑇)‘𝑥) = ((𝑇(proj₁‘𝑊)((ocv‘𝑊)‘𝑇))‘𝑥))
10		eqid 2824	. . . . . 6 ⊢ (+_g‘𝑊) = (+_g‘𝑊)
11		eqid 2824	. . . . . 6 ⊢ (LSSum‘𝑊) = (LSSum‘𝑊)
12		eqid 2824	. . . . . 6 ⊢ (0_g‘𝑊) = (0_g‘𝑊)
13		eqid 2824	. . . . . 6 ⊢ (Cntz‘𝑊) = (Cntz‘𝑊)
14		phllmod 20777	. . . . . . . . 9 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod)
15	14	adantr 483	. . . . . . . 8 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑊 ∈ LMod)
16		eqid 2824	. . . . . . . . 9 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
17	16	lsssssubg 19733	. . . . . . . 8 ⊢ (𝑊 ∈ LMod → (LSubSp‘𝑊) ⊆ (SubGrp‘𝑊))
18	15, 17	syl 17	. . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (LSubSp‘𝑊) ⊆ (SubGrp‘𝑊))
19	2, 16, 5, 11, 1	pjdm2 20858	. . . . . . . 8 ⊢ (𝑊 ∈ PreHil → (𝑇 ∈ dom 𝐾 ↔ (𝑇 ∈ (LSubSp‘𝑊) ∧ (𝑇(LSSum‘𝑊)((ocv‘𝑊)‘𝑇)) = 𝑉)))
20	19	simprbda 501	. . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑇 ∈ (LSubSp‘𝑊))
21	18, 20	sseldd 3971	. . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑇 ∈ (SubGrp‘𝑊))
22	2, 16	lssss 19711	. . . . . . . . 9 ⊢ (𝑇 ∈ (LSubSp‘𝑊) → 𝑇 ⊆ 𝑉)
23	20, 22	syl 17	. . . . . . . 8 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑇 ⊆ 𝑉)
24	2, 5, 16	ocvlss 20819	. . . . . . . 8 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ⊆ 𝑉) → ((ocv‘𝑊)‘𝑇) ∈ (LSubSp‘𝑊))
25	23, 24	syldan 593	. . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → ((ocv‘𝑊)‘𝑇) ∈ (LSubSp‘𝑊))
26	18, 25	sseldd 3971	. . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → ((ocv‘𝑊)‘𝑇) ∈ (SubGrp‘𝑊))
27	5, 16, 12	ocvin 20821	. . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ (LSubSp‘𝑊)) → (𝑇 ∩ ((ocv‘𝑊)‘𝑇)) = {(0_g‘𝑊)})
28	20, 27	syldan 593	. . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝑇 ∩ ((ocv‘𝑊)‘𝑇)) = {(0_g‘𝑊)})
29		lmodabl 19684	. . . . . . . 8 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel)
30	15, 29	syl 17	. . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑊 ∈ Abel)
31	13, 30, 21, 26	ablcntzd 18980	. . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑇 ⊆ ((Cntz‘𝑊)‘((ocv‘𝑊)‘𝑇)))
32	10, 11, 12, 13, 21, 26, 28, 31, 6	pj1lid 18830	. . . . 5 ⊢ (((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) ∧ 𝑥 ∈ 𝑇) → ((𝑇(proj₁‘𝑊)((ocv‘𝑊)‘𝑇))‘𝑥) = 𝑥)
33	9, 32	eqtrd 2859	. . . 4 ⊢ (((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) ∧ 𝑥 ∈ 𝑇) → ((𝐾‘𝑇)‘𝑥) = 𝑥)
34	3	ffnd 6518	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝐾‘𝑇) Fn 𝑉)
35	23	sselda 3970	. . . . 5 ⊢ (((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) ∧ 𝑥 ∈ 𝑇) → 𝑥 ∈ 𝑉)
36		fnfvelrn 6851	. . . . 5 ⊢ (((𝐾‘𝑇) Fn 𝑉 ∧ 𝑥 ∈ 𝑉) → ((𝐾‘𝑇)‘𝑥) ∈ ran (𝐾‘𝑇))
37	34, 35, 36	syl2an2r 683	. . . 4 ⊢ (((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) ∧ 𝑥 ∈ 𝑇) → ((𝐾‘𝑇)‘𝑥) ∈ ran (𝐾‘𝑇))
38	33, 37	eqeltrrd 2917	. . 3 ⊢ (((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) ∧ 𝑥 ∈ 𝑇) → 𝑥 ∈ ran (𝐾‘𝑇))
39	4, 38	eqelssd 3991	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → ran (𝐾‘𝑇) = 𝑇)
40		dffo2 6597	. 2 ⊢ ((𝐾‘𝑇):𝑉–onto→𝑇 ↔ ((𝐾‘𝑇):𝑉⟶𝑇 ∧ ran (𝐾‘𝑇) = 𝑇))
41	3, 39, 40	sylanbrc 585	1 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝐾‘𝑇):𝑉–onto→𝑇)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∩ cin 3938 ⊆ wss 3939 {csn 4570 dom cdm 5558 ran crn 5559 Fn wfn 6353 ⟶wf 6354 –onto→wfo 6356 ‘cfv 6358 (class class class)co 7159 Basecbs 16486 +_gcplusg 16568 0_gc0g 16716 SubGrpcsubg 18276 Cntzccntz 18448 LSSumclsm 18762 proj₁cpj1 18763 Abelcabl 18910 LModclmod 19637 LSubSpclss 19706 PreHilcphl 20771 ocvcocv 20807 projcpj 20847

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-map 8411 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-sca 16584 df-vsca 16585 df-ip 16586 df-0g 16718 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-submnd 17960 df-grp 18109 df-minusg 18110 df-sbg 18111 df-subg 18279 df-ghm 18359 df-cntz 18450 df-lsm 18764 df-pj1 18765 df-cmn 18911 df-abl 18912 df-mgp 19243 df-ur 19255 df-ring 19302 df-lmod 19639 df-lss 19707 df-lmhm 19797 df-lvec 19878 df-sra 19947 df-rgmod 19948 df-phl 20773 df-ocv 20810 df-pj 20850

This theorem is referenced by: (None)

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