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Theorem pjfo 21654

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 21653	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝐾‘𝑇):𝑉⟶𝑇)
4	3	frnd 6664	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → ran (𝐾‘𝑇) ⊆ 𝑇)
5		eqid 2733	. . . . . . . 8 ⊢ (ocv‘𝑊) = (ocv‘𝑊)
6		eqid 2733	. . . . . . . 8 ⊢ (proj₁‘𝑊) = (proj₁‘𝑊)
7	5, 6, 1	pjval 21649	. . . . . . 7 ⊢ (𝑇 ∈ dom 𝐾 → (𝐾‘𝑇) = (𝑇(proj₁‘𝑊)((ocv‘𝑊)‘𝑇)))
8	7	ad2antlr 727	. . . . . 6 ⊢ (((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) ∧ 𝑥 ∈ 𝑇) → (𝐾‘𝑇) = (𝑇(proj₁‘𝑊)((ocv‘𝑊)‘𝑇)))
9	8	fveq1d 6830	. . . . 5 ⊢ (((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) ∧ 𝑥 ∈ 𝑇) → ((𝐾‘𝑇)‘𝑥) = ((𝑇(proj₁‘𝑊)((ocv‘𝑊)‘𝑇))‘𝑥))
10		eqid 2733	. . . . . 6 ⊢ (+_g‘𝑊) = (+_g‘𝑊)
11		eqid 2733	. . . . . 6 ⊢ (LSSum‘𝑊) = (LSSum‘𝑊)
12		eqid 2733	. . . . . 6 ⊢ (0_g‘𝑊) = (0_g‘𝑊)
13		eqid 2733	. . . . . 6 ⊢ (Cntz‘𝑊) = (Cntz‘𝑊)
14		phllmod 21569	. . . . . . . . 9 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod)
15	14	adantr 480	. . . . . . . 8 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑊 ∈ LMod)
16		eqid 2733	. . . . . . . . 9 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
17	16	lsssssubg 20893	. . . . . . . 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 21650	. . . . . . . 8 ⊢ (𝑊 ∈ PreHil → (𝑇 ∈ dom 𝐾 ↔ (𝑇 ∈ (LSubSp‘𝑊) ∧ (𝑇(LSSum‘𝑊)((ocv‘𝑊)‘𝑇)) = 𝑉)))
20	19	simprbda 498	. . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑇 ∈ (LSubSp‘𝑊))
21	18, 20	sseldd 3931	. . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑇 ∈ (SubGrp‘𝑊))
22	2, 16	lssss 20871	. . . . . . . . 9 ⊢ (𝑇 ∈ (LSubSp‘𝑊) → 𝑇 ⊆ 𝑉)
23	20, 22	syl 17	. . . . . . . 8 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑇 ⊆ 𝑉)
24	2, 5, 16	ocvlss 21611	. . . . . . . 8 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ⊆ 𝑉) → ((ocv‘𝑊)‘𝑇) ∈ (LSubSp‘𝑊))
25	23, 24	syldan 591	. . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → ((ocv‘𝑊)‘𝑇) ∈ (LSubSp‘𝑊))
26	18, 25	sseldd 3931	. . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → ((ocv‘𝑊)‘𝑇) ∈ (SubGrp‘𝑊))
27	5, 16, 12	ocvin 21613	. . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ (LSubSp‘𝑊)) → (𝑇 ∩ ((ocv‘𝑊)‘𝑇)) = {(0_g‘𝑊)})
28	20, 27	syldan 591	. . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝑇 ∩ ((ocv‘𝑊)‘𝑇)) = {(0_g‘𝑊)})
29		lmodabl 20844	. . . . . . . 8 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel)
30	15, 29	syl 17	. . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑊 ∈ Abel)
31	13, 30, 21, 26	ablcntzd 19771	. . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑇 ⊆ ((Cntz‘𝑊)‘((ocv‘𝑊)‘𝑇)))
32	10, 11, 12, 13, 21, 26, 28, 31, 6	pj1lid 19615	. . . . 5 ⊢ (((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) ∧ 𝑥 ∈ 𝑇) → ((𝑇(proj₁‘𝑊)((ocv‘𝑊)‘𝑇))‘𝑥) = 𝑥)
33	9, 32	eqtrd 2768	. . . 4 ⊢ (((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) ∧ 𝑥 ∈ 𝑇) → ((𝐾‘𝑇)‘𝑥) = 𝑥)
34	3	ffnd 6657	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝐾‘𝑇) Fn 𝑉)
35	23	sselda 3930	. . . . 5 ⊢ (((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) ∧ 𝑥 ∈ 𝑇) → 𝑥 ∈ 𝑉)
36		fnfvelrn 7019	. . . . 5 ⊢ (((𝐾‘𝑇) Fn 𝑉 ∧ 𝑥 ∈ 𝑉) → ((𝐾‘𝑇)‘𝑥) ∈ ran (𝐾‘𝑇))
37	34, 35, 36	syl2an2r 685	. . . 4 ⊢ (((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) ∧ 𝑥 ∈ 𝑇) → ((𝐾‘𝑇)‘𝑥) ∈ ran (𝐾‘𝑇))
38	33, 37	eqeltrrd 2834	. . 3 ⊢ (((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) ∧ 𝑥 ∈ 𝑇) → 𝑥 ∈ ran (𝐾‘𝑇))
39	4, 38	eqelssd 3952	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → ran (𝐾‘𝑇) = 𝑇)
40		dffo2 6744	. 2 ⊢ ((𝐾‘𝑇):𝑉–onto→𝑇 ↔ ((𝐾‘𝑇):𝑉⟶𝑇 ∧ ran (𝐾‘𝑇) = 𝑇))
41	3, 39, 40	sylanbrc 583	1 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝐾‘𝑇):𝑉–onto→𝑇)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ∩ cin 3897 ⊆ wss 3898 {csn 4575 dom cdm 5619 ran crn 5620 Fn wfn 6481 ⟶wf 6482 –onto→wfo 6484 ‘cfv 6486 (class class class)co 7352 Basecbs 17122 +_gcplusg 17163 0_gc0g 17345 SubGrpcsubg 19035 Cntzccntz 19229 LSSumclsm 19548 proj₁cpj1 19549 Abelcabl 19695 LModclmod 20795 LSubSpclss 20866 PreHilcphl 21563 ocvcocv 21599 projcpj 21639

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 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-cnex 11069 ax-resscn 11070 ax-1cn 11071 ax-icn 11072 ax-addcl 11073 ax-addrcl 11074 ax-mulcl 11075 ax-mulrcl 11076 ax-mulcom 11077 ax-addass 11078 ax-mulass 11079 ax-distr 11080 ax-i2m1 11081 ax-1ne0 11082 ax-1rid 11083 ax-rnegex 11084 ax-rrecex 11085 ax-cnre 11086 ax-pre-lttri 11087 ax-pre-lttrn 11088 ax-pre-ltadd 11089 ax-pre-mulgt0 11090

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-op 4582 df-uni 4859 df-int 4898 df-iun 4943 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-om 7803 df-1st 7927 df-2nd 7928 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-er 8628 df-map 8758 df-en 8876 df-dom 8877 df-sdom 8878 df-pnf 11155 df-mnf 11156 df-xr 11157 df-ltxr 11158 df-le 11159 df-sub 11353 df-neg 11354 df-nn 12133 df-2 12195 df-3 12196 df-4 12197 df-5 12198 df-6 12199 df-7 12200 df-8 12201 df-sets 17077 df-slot 17095 df-ndx 17107 df-base 17123 df-ress 17144 df-plusg 17176 df-sca 17179 df-vsca 17180 df-ip 17181 df-0g 17347 df-mgm 18550 df-sgrp 18629 df-mnd 18645 df-submnd 18694 df-grp 18851 df-minusg 18852 df-sbg 18853 df-subg 19038 df-ghm 19127 df-cntz 19231 df-lsm 19550 df-pj1 19551 df-cmn 19696 df-abl 19697 df-mgp 20061 df-rng 20073 df-ur 20102 df-ring 20155 df-lmod 20797 df-lss 20867 df-lmhm 20958 df-lvec 21039 df-sra 21109 df-rgmod 21110 df-phl 21565 df-ocv 21602 df-pj 21642

This theorem is referenced by: (None)

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