Theorem prjspval 38666

Description: Value of the projective space function, which is also known as the projectivization of 𝑉. (Contributed by Steven Nguyen, 29-Apr-2023.)

Hypotheses

Ref	Expression
prjspval.b	⊢ 𝐵 = ((Base‘𝑉) ∖ {(0g‘𝑉)})
prjspval.x	⊢ · = ( ·𝑠 ‘𝑉)
prjspval.s	⊢ 𝑆 = (Scalar‘𝑉)
prjspval.k	⊢ 𝐾 = (Base‘𝑆)

Assertion

Ref	Expression
prjspval	⊢ (𝑉 ∈ LVec → (ℙ𝕣𝕠𝕛‘𝑉) = (𝐵 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))}))

Distinct variable group:   𝑥,𝑙,𝑦,𝑉

Allowed substitution hints:   𝐵(𝑥,𝑦,𝑙)   𝑆(𝑥,𝑦,𝑙)   · (𝑥,𝑦,𝑙)   𝐾(𝑥,𝑦,𝑙)

Proof of Theorem prjspval

Dummy variables 𝑏 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6512	. . . . 5 ⊢ (Base‘𝑣) ∈ V
2	1	difexi 5088	. . . 4 ⊢ ((Base‘𝑣) ∖ {(0g‘𝑣)}) ∈ V
3	2	a1i 11	. . 3 ⊢ (𝑣 = 𝑉 → ((Base‘𝑣) ∖ {(0g‘𝑣)}) ∈ V)
4		fveq2 6499	. . . . . . . . 9 ⊢ (𝑣 = 𝑉 → (Base‘𝑣) = (Base‘𝑉))
5		fveq2 6499	. . . . . . . . . 10 ⊢ (𝑣 = 𝑉 → (0g‘𝑣) = (0g‘𝑉))
6	5	sneqd 4453	. . . . . . . . 9 ⊢ (𝑣 = 𝑉 → {(0g‘𝑣)} = {(0g‘𝑉)})
7	4, 6	difeq12d 3990	. . . . . . . 8 ⊢ (𝑣 = 𝑉 → ((Base‘𝑣) ∖ {(0g‘𝑣)}) = ((Base‘𝑉) ∖ {(0g‘𝑉)}))
8		prjspval.b	. . . . . . . 8 ⊢ 𝐵 = ((Base‘𝑉) ∖ {(0g‘𝑉)})
9	7, 8	syl6eqr 2832	. . . . . . 7 ⊢ (𝑣 = 𝑉 → ((Base‘𝑣) ∖ {(0g‘𝑣)}) = 𝐵)
10	9	eqeq2d 2788	. . . . . 6 ⊢ (𝑣 = 𝑉 → (𝑏 = ((Base‘𝑣) ∖ {(0g‘𝑣)}) ↔ 𝑏 = 𝐵))
11	10	biimpd 221	. . . . 5 ⊢ (𝑣 = 𝑉 → (𝑏 = ((Base‘𝑣) ∖ {(0g‘𝑣)}) → 𝑏 = 𝐵))
12	11	imp 398	. . . 4 ⊢ ((𝑣 = 𝑉 ∧ 𝑏 = ((Base‘𝑣) ∖ {(0g‘𝑣)})) → 𝑏 = 𝐵)
13	11	imdistani 561	. . . . . 6 ⊢ ((𝑣 = 𝑉 ∧ 𝑏 = ((Base‘𝑣) ∖ {(0g‘𝑣)})) → (𝑣 = 𝑉 ∧ 𝑏 = 𝐵))
14		eleq2 2854	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → (𝑥 ∈ 𝑏 ↔ 𝑥 ∈ 𝐵))
15		eleq2 2854	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → (𝑦 ∈ 𝑏 ↔ 𝑦 ∈ 𝐵))
16	14, 15	anbi12d 621	. . . . . . 7 ⊢ (𝑏 = 𝐵 → ((𝑥 ∈ 𝑏 ∧ 𝑦 ∈ 𝑏) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)))
17		fveq2 6499	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑉 → (Scalar‘𝑣) = (Scalar‘𝑉))
18		prjspval.s	. . . . . . . . . . 11 ⊢ 𝑆 = (Scalar‘𝑉)
19	17, 18	syl6eqr 2832	. . . . . . . . . 10 ⊢ (𝑣 = 𝑉 → (Scalar‘𝑣) = 𝑆)
20	19	fveq2d 6503	. . . . . . . . 9 ⊢ (𝑣 = 𝑉 → (Base‘(Scalar‘𝑣)) = (Base‘𝑆))
21		prjspval.k	. . . . . . . . 9 ⊢ 𝐾 = (Base‘𝑆)
22	20, 21	syl6eqr 2832	. . . . . . . 8 ⊢ (𝑣 = 𝑉 → (Base‘(Scalar‘𝑣)) = 𝐾)
23		fveq2 6499	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑉 → ( ·𝑠 ‘𝑣) = ( ·𝑠 ‘𝑉))
24		prjspval.x	. . . . . . . . . . 11 ⊢ · = ( ·𝑠 ‘𝑉)
25	23, 24	syl6eqr 2832	. . . . . . . . . 10 ⊢ (𝑣 = 𝑉 → ( ·𝑠 ‘𝑣) = · )
26	25	oveqd 6993	. . . . . . . . 9 ⊢ (𝑣 = 𝑉 → (𝑙( ·𝑠 ‘𝑣)𝑦) = (𝑙 · 𝑦))
27	26	eqeq2d 2788	. . . . . . . 8 ⊢ (𝑣 = 𝑉 → (𝑥 = (𝑙( ·𝑠 ‘𝑣)𝑦) ↔ 𝑥 = (𝑙 · 𝑦)))
28	22, 27	rexeqbidv 3342	. . . . . . 7 ⊢ (𝑣 = 𝑉 → (∃𝑙 ∈ (Base‘(Scalar‘𝑣))𝑥 = (𝑙( ·𝑠 ‘𝑣)𝑦) ↔ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦)))
29	16, 28	bi2anan9r 627	. . . . . 6 ⊢ ((𝑣 = 𝑉 ∧ 𝑏 = 𝐵) → (((𝑥 ∈ 𝑏 ∧ 𝑦 ∈ 𝑏) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑣))𝑥 = (𝑙( ·𝑠 ‘𝑣)𝑦)) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))))
30	13, 29	syl 17	. . . . 5 ⊢ ((𝑣 = 𝑉 ∧ 𝑏 = ((Base‘𝑣) ∖ {(0g‘𝑣)})) → (((𝑥 ∈ 𝑏 ∧ 𝑦 ∈ 𝑏) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑣))𝑥 = (𝑙( ·𝑠 ‘𝑣)𝑦)) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))))
31	30	opabbidv 4995	. . . 4 ⊢ ((𝑣 = 𝑉 ∧ 𝑏 = ((Base‘𝑣) ∖ {(0g‘𝑣)})) → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑦 ∈ 𝑏) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑣))𝑥 = (𝑙( ·𝑠 ‘𝑣)𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))})
32	12, 31	qseq12d 38575	. . 3 ⊢ ((𝑣 = 𝑉 ∧ 𝑏 = ((Base‘𝑣) ∖ {(0g‘𝑣)})) → (𝑏 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑦 ∈ 𝑏) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑣))𝑥 = (𝑙( ·𝑠 ‘𝑣)𝑦))}) = (𝐵 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))}))
33	3, 32	csbied 3815	. 2 ⊢ (𝑣 = 𝑉 → ⦋((Base‘𝑣) ∖ {(0g‘𝑣)}) / 𝑏⦌(𝑏 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑦 ∈ 𝑏) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑣))𝑥 = (𝑙( ·𝑠 ‘𝑣)𝑦))}) = (𝐵 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))}))
34		df-prjsp 38665	. 2 ⊢ ℙ𝕣𝕠𝕛 = (𝑣 ∈ LVec ↦ ⦋((Base‘𝑣) ∖ {(0g‘𝑣)}) / 𝑏⦌(𝑏 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑦 ∈ 𝑏) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑣))𝑥 = (𝑙( ·𝑠 ‘𝑣)𝑦))}))
35		fvex 6512	. . . . 5 ⊢ (Base‘𝑉) ∈ V
36	35	difexi 5088	. . . 4 ⊢ ((Base‘𝑉) ∖ {(0g‘𝑉)}) ∈ V
37	8, 36	eqeltri 2862	. . 3 ⊢ 𝐵 ∈ V
38	37	qsex 8156	. 2 ⊢ (𝐵 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))}) ∈ V
39	33, 34, 38	fvmpt 6595	1 ⊢ (𝑉 ∈ LVec → (ℙ𝕣𝕠𝕛‘𝑉) = (𝐵 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))}))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   = wceq 1507   ∈ wcel 2050  ∃wrex 3089  Vcvv 3415  ⦋csb 3786   ∖ cdif 3826  {csn 4441  {copab 4991  ‘cfv 6188  (class class class)co 6976   / cqs 8088  Basecbs 16339  Scalarcsca 16424   ·_𝑠 cvsca 16425  0_gc0g 16569  LVecclvec 19596  ℙ𝕣𝕠𝕛cprjsp 38664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pr 5186  ax-un 7279

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-ov 6979  df-ec 8091  df-qs 8095  df-prjsp 38665

This theorem is referenced by:  prjspval2  38676  prjspnval2  38680