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Theorem pjfo 21753

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 21752	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝐾‘𝑇):𝑉⟶𝑇)
4	3	frnd 6745	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → ran (𝐾‘𝑇) ⊆ 𝑇)
5		eqid 2735	. . . . . . . 8 ⊢ (ocv‘𝑊) = (ocv‘𝑊)
6		eqid 2735	. . . . . . . 8 ⊢ (proj₁‘𝑊) = (proj₁‘𝑊)
7	5, 6, 1	pjval 21748	. . . . . . 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 6909	. . . . 5 ⊢ (((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) ∧ 𝑥 ∈ 𝑇) → ((𝐾‘𝑇)‘𝑥) = ((𝑇(proj₁‘𝑊)((ocv‘𝑊)‘𝑇))‘𝑥))
10		eqid 2735	. . . . . 6 ⊢ (+_g‘𝑊) = (+_g‘𝑊)
11		eqid 2735	. . . . . 6 ⊢ (LSSum‘𝑊) = (LSSum‘𝑊)
12		eqid 2735	. . . . . 6 ⊢ (0_g‘𝑊) = (0_g‘𝑊)
13		eqid 2735	. . . . . 6 ⊢ (Cntz‘𝑊) = (Cntz‘𝑊)
14		phllmod 21666	. . . . . . . . 9 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod)
15	14	adantr 480	. . . . . . . 8 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑊 ∈ LMod)
16		eqid 2735	. . . . . . . . 9 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
17	16	lsssssubg 20974	. . . . . . . 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 21749	. . . . . . . 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 3996	. . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑇 ∈ (SubGrp‘𝑊))
22	2, 16	lssss 20952	. . . . . . . . 9 ⊢ (𝑇 ∈ (LSubSp‘𝑊) → 𝑇 ⊆ 𝑉)
23	20, 22	syl 17	. . . . . . . 8 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑇 ⊆ 𝑉)
24	2, 5, 16	ocvlss 21708	. . . . . . . 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 3996	. . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → ((ocv‘𝑊)‘𝑇) ∈ (SubGrp‘𝑊))
27	5, 16, 12	ocvin 21710	. . . . . . 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 20924	. . . . . . . 8 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel)
30	15, 29	syl 17	. . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑊 ∈ Abel)
31	13, 30, 21, 26	ablcntzd 19890	. . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑇 ⊆ ((Cntz‘𝑊)‘((ocv‘𝑊)‘𝑇)))
32	10, 11, 12, 13, 21, 26, 28, 31, 6	pj1lid 19734	. . . . 5 ⊢ (((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) ∧ 𝑥 ∈ 𝑇) → ((𝑇(proj₁‘𝑊)((ocv‘𝑊)‘𝑇))‘𝑥) = 𝑥)
33	9, 32	eqtrd 2775	. . . 4 ⊢ (((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) ∧ 𝑥 ∈ 𝑇) → ((𝐾‘𝑇)‘𝑥) = 𝑥)
34	3	ffnd 6738	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝐾‘𝑇) Fn 𝑉)
35	23	sselda 3995	. . . . 5 ⊢ (((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) ∧ 𝑥 ∈ 𝑇) → 𝑥 ∈ 𝑉)
36		fnfvelrn 7100	. . . . 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 2840	. . 3 ⊢ (((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) ∧ 𝑥 ∈ 𝑇) → 𝑥 ∈ ran (𝐾‘𝑇))
39	4, 38	eqelssd 4017	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → ran (𝐾‘𝑇) = 𝑇)
40		dffo2 6825	. 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 1537 ∈ wcel 2106 ∩ cin 3962 ⊆ wss 3963 {csn 4631 dom cdm 5689 ran crn 5690 Fn wfn 6558 ⟶wf 6559 –onto→wfo 6561 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +_gcplusg 17298 0_gc0g 17486 SubGrpcsubg 19151 Cntzccntz 19346 LSSumclsm 19667 proj₁cpj1 19668 Abelcabl 19814 LModclmod 20875 LSubSpclss 20947 PreHilcphl 21660 ocvcocv 21696 projcpj 21738

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-sca 17314 df-vsca 17315 df-ip 17316 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-ghm 19244 df-cntz 19348 df-lsm 19669 df-pj1 19670 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-lmod 20877 df-lss 20948 df-lmhm 21039 df-lvec 21120 df-sra 21190 df-rgmod 21191 df-phl 21662 df-ocv 21699 df-pj 21741

This theorem is referenced by: (None)

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