Theorem prjspval 43060

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 6847	. . . . 5 ⊢ (Base‘𝑣) ∈ V
2	1	difexi 5265	. . . 4 ⊢ ((Base‘𝑣) ∖ {(0g‘𝑣)}) ∈ V
3	2	a1i 11	. . 3 ⊢ (𝑣 = 𝑉 → ((Base‘𝑣) ∖ {(0g‘𝑣)}) ∈ V)
4		fveq2 6834	. . . . . . . . 9 ⊢ (𝑣 = 𝑉 → (Base‘𝑣) = (Base‘𝑉))
5		fveq2 6834	. . . . . . . . . 10 ⊢ (𝑣 = 𝑉 → (0g‘𝑣) = (0g‘𝑉))
6	5	sneqd 4574	. . . . . . . . 9 ⊢ (𝑣 = 𝑉 → {(0g‘𝑣)} = {(0g‘𝑉)})
7	4, 6	difeq12d 4065	. . . . . . . 8 ⊢ (𝑣 = 𝑉 → ((Base‘𝑣) ∖ {(0g‘𝑣)}) = ((Base‘𝑉) ∖ {(0g‘𝑉)}))
8		prjspval.b	. . . . . . . 8 ⊢ 𝐵 = ((Base‘𝑉) ∖ {(0g‘𝑉)})
9	7, 8	eqtr4di 2793	. . . . . . 7 ⊢ (𝑣 = 𝑉 → ((Base‘𝑣) ∖ {(0g‘𝑣)}) = 𝐵)
10	9	eqeq2d 2751	. . . . . 6 ⊢ (𝑣 = 𝑉 → (𝑏 = ((Base‘𝑣) ∖ {(0g‘𝑣)}) ↔ 𝑏 = 𝐵))
11	10	biimpd 230	. . . . 5 ⊢ (𝑣 = 𝑉 → (𝑏 = ((Base‘𝑣) ∖ {(0g‘𝑣)}) → 𝑏 = 𝐵))
12	11	imp 407	. . . 4 ⊢ ((𝑣 = 𝑉 ∧ 𝑏 = ((Base‘𝑣) ∖ {(0g‘𝑣)})) → 𝑏 = 𝐵)
13	11	imdistani 573	. . . . . 6 ⊢ ((𝑣 = 𝑉 ∧ 𝑏 = ((Base‘𝑣) ∖ {(0g‘𝑣)})) → (𝑣 = 𝑉 ∧ 𝑏 = 𝐵))
14		eleq2 2829	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → (𝑥 ∈ 𝑏 ↔ 𝑥 ∈ 𝐵))
15		eleq2 2829	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → (𝑦 ∈ 𝑏 ↔ 𝑦 ∈ 𝐵))
16	14, 15	anbi12d 638	. . . . . . 7 ⊢ (𝑏 = 𝐵 → ((𝑥 ∈ 𝑏 ∧ 𝑦 ∈ 𝑏) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)))
17		fveq2 6834	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑉 → (Scalar‘𝑣) = (Scalar‘𝑉))
18		prjspval.s	. . . . . . . . . . 11 ⊢ 𝑆 = (Scalar‘𝑉)
19	17, 18	eqtr4di 2793	. . . . . . . . . 10 ⊢ (𝑣 = 𝑉 → (Scalar‘𝑣) = 𝑆)
20	19	fveq2d 6838	. . . . . . . . 9 ⊢ (𝑣 = 𝑉 → (Base‘(Scalar‘𝑣)) = (Base‘𝑆))
21		prjspval.k	. . . . . . . . 9 ⊢ 𝐾 = (Base‘𝑆)
22	20, 21	eqtr4di 2793	. . . . . . . 8 ⊢ (𝑣 = 𝑉 → (Base‘(Scalar‘𝑣)) = 𝐾)
23		fveq2 6834	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑉 → ( ·𝑠 ‘𝑣) = ( ·𝑠 ‘𝑉))
24		prjspval.x	. . . . . . . . . . 11 ⊢ · = ( ·𝑠 ‘𝑉)
25	23, 24	eqtr4di 2793	. . . . . . . . . 10 ⊢ (𝑣 = 𝑉 → ( ·𝑠 ‘𝑣) = · )
26	25	oveqd 7380	. . . . . . . . 9 ⊢ (𝑣 = 𝑉 → (𝑙( ·𝑠 ‘𝑣)𝑦) = (𝑙 · 𝑦))
27	26	eqeq2d 2751	. . . . . . . 8 ⊢ (𝑣 = 𝑉 → (𝑥 = (𝑙( ·𝑠 ‘𝑣)𝑦) ↔ 𝑥 = (𝑙 · 𝑦)))
28	22, 27	rexeqbidv 3315	. . . . . . 7 ⊢ (𝑣 = 𝑉 → (∃𝑙 ∈ (Base‘(Scalar‘𝑣))𝑥 = (𝑙( ·𝑠 ‘𝑣)𝑦) ↔ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦)))
29	16, 28	bi2anan9r 645	. . . . . 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 5145	. . . 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 42731	. . 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 3874	. 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 43059	. 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 6847	. . . . 5 ⊢ (Base‘𝑉) ∈ V
36	35	difexi 5265	. . . 4 ⊢ ((Base‘𝑉) ∖ {(0g‘𝑉)}) ∈ V
37	8, 36	eqeltri 2836	. . 3 ⊢ 𝐵 ∈ V
38	37	qsex 8716	. 2 ⊢ (𝐵 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))}) ∈ V
39	33, 34, 38	fvmpt 6942	1 ⊢ (𝑉 ∈ LVec → (ℙ𝕣𝕠𝕛‘𝑉) = (𝐵 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))}))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∃wrex 3064  Vcvv 3432  ⦋csb 3838   ∖ cdif 3887  {csn 4562  {copab 5141  ‘cfv 6492  (class class class)co 7363   / cqs 8639  Basecbs 17177  Scalarcsca 17221   ·_𝑠 cvsca 17222  0_gc0g 17400  LVecclvec 21099  ℙ𝕣𝕠𝕛cprjsp 43058

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7366  df-ec 8642  df-qs 8646  df-prjsp 43059

This theorem is referenced by:  prjspval2  43070  prjspnval2  43075