Theorem prjspval 42162

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 6909	. . . . 5 ⊢ (Base‘𝑣) ∈ V
2	1	difexi 5331	. . . 4 ⊢ ((Base‘𝑣) ∖ {(0g‘𝑣)}) ∈ V
3	2	a1i 11	. . 3 ⊢ (𝑣 = 𝑉 → ((Base‘𝑣) ∖ {(0g‘𝑣)}) ∈ V)
4		fveq2 6896	. . . . . . . . 9 ⊢ (𝑣 = 𝑉 → (Base‘𝑣) = (Base‘𝑉))
5		fveq2 6896	. . . . . . . . . 10 ⊢ (𝑣 = 𝑉 → (0g‘𝑣) = (0g‘𝑉))
6	5	sneqd 4642	. . . . . . . . 9 ⊢ (𝑣 = 𝑉 → {(0g‘𝑣)} = {(0g‘𝑉)})
7	4, 6	difeq12d 4119	. . . . . . . 8 ⊢ (𝑣 = 𝑉 → ((Base‘𝑣) ∖ {(0g‘𝑣)}) = ((Base‘𝑉) ∖ {(0g‘𝑉)}))
8		prjspval.b	. . . . . . . 8 ⊢ 𝐵 = ((Base‘𝑉) ∖ {(0g‘𝑉)})
9	7, 8	eqtr4di 2783	. . . . . . 7 ⊢ (𝑣 = 𝑉 → ((Base‘𝑣) ∖ {(0g‘𝑣)}) = 𝐵)
10	9	eqeq2d 2736	. . . . . 6 ⊢ (𝑣 = 𝑉 → (𝑏 = ((Base‘𝑣) ∖ {(0g‘𝑣)}) ↔ 𝑏 = 𝐵))
11	10	biimpd 228	. . . . 5 ⊢ (𝑣 = 𝑉 → (𝑏 = ((Base‘𝑣) ∖ {(0g‘𝑣)}) → 𝑏 = 𝐵))
12	11	imp 405	. . . 4 ⊢ ((𝑣 = 𝑉 ∧ 𝑏 = ((Base‘𝑣) ∖ {(0g‘𝑣)})) → 𝑏 = 𝐵)
13	11	imdistani 567	. . . . . 6 ⊢ ((𝑣 = 𝑉 ∧ 𝑏 = ((Base‘𝑣) ∖ {(0g‘𝑣)})) → (𝑣 = 𝑉 ∧ 𝑏 = 𝐵))
14		eleq2 2814	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → (𝑥 ∈ 𝑏 ↔ 𝑥 ∈ 𝐵))
15		eleq2 2814	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → (𝑦 ∈ 𝑏 ↔ 𝑦 ∈ 𝐵))
16	14, 15	anbi12d 630	. . . . . . 7 ⊢ (𝑏 = 𝐵 → ((𝑥 ∈ 𝑏 ∧ 𝑦 ∈ 𝑏) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)))
17		fveq2 6896	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑉 → (Scalar‘𝑣) = (Scalar‘𝑉))
18		prjspval.s	. . . . . . . . . . 11 ⊢ 𝑆 = (Scalar‘𝑉)
19	17, 18	eqtr4di 2783	. . . . . . . . . 10 ⊢ (𝑣 = 𝑉 → (Scalar‘𝑣) = 𝑆)
20	19	fveq2d 6900	. . . . . . . . 9 ⊢ (𝑣 = 𝑉 → (Base‘(Scalar‘𝑣)) = (Base‘𝑆))
21		prjspval.k	. . . . . . . . 9 ⊢ 𝐾 = (Base‘𝑆)
22	20, 21	eqtr4di 2783	. . . . . . . 8 ⊢ (𝑣 = 𝑉 → (Base‘(Scalar‘𝑣)) = 𝐾)
23		fveq2 6896	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑉 → ( ·𝑠 ‘𝑣) = ( ·𝑠 ‘𝑉))
24		prjspval.x	. . . . . . . . . . 11 ⊢ · = ( ·𝑠 ‘𝑉)
25	23, 24	eqtr4di 2783	. . . . . . . . . 10 ⊢ (𝑣 = 𝑉 → ( ·𝑠 ‘𝑣) = · )
26	25	oveqd 7436	. . . . . . . . 9 ⊢ (𝑣 = 𝑉 → (𝑙( ·𝑠 ‘𝑣)𝑦) = (𝑙 · 𝑦))
27	26	eqeq2d 2736	. . . . . . . 8 ⊢ (𝑣 = 𝑉 → (𝑥 = (𝑙( ·𝑠 ‘𝑣)𝑦) ↔ 𝑥 = (𝑙 · 𝑦)))
28	22, 27	rexeqbidv 3330	. . . . . . 7 ⊢ (𝑣 = 𝑉 → (∃𝑙 ∈ (Base‘(Scalar‘𝑣))𝑥 = (𝑙( ·𝑠 ‘𝑣)𝑦) ↔ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦)))
29	16, 28	bi2anan9r 637	. . . . . 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 5215	. . . 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 41863	. . 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 3927	. 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 42161	. 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 6909	. . . . 5 ⊢ (Base‘𝑉) ∈ V
36	35	difexi 5331	. . . 4 ⊢ ((Base‘𝑉) ∖ {(0g‘𝑉)}) ∈ V
37	8, 36	eqeltri 2821	. . 3 ⊢ 𝐵 ∈ V
38	37	qsex 8795	. 2 ⊢ (𝐵 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))}) ∈ V
39	33, 34, 38	fvmpt 7004	1 ⊢ (𝑉 ∈ LVec → (ℙ𝕣𝕠𝕛‘𝑉) = (𝐵 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))}))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ∃wrex 3059  Vcvv 3461  ⦋csb 3889   ∖ cdif 3941  {csn 4630  {copab 5211  ‘cfv 6549  (class class class)co 7419   / cqs 8724  Basecbs 17183  Scalarcsca 17239   ·_𝑠 cvsca 17240  0_gc0g 17424  LVecclvec 20999  ℙ𝕣𝕠𝕛cprjsp 42160

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fv 6557  df-ov 7422  df-ec 8727  df-qs 8731  df-prjsp 42161

This theorem is referenced by:  prjspval2  42172  prjspnval2  42177