0prjspn - Mathbox for Steven Nguyen

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Theorem 0prjspn 41057

Description: A zero-dimensional projective space has only 1 point. (Contributed by Steven Nguyen, 9-Jun-2023.)

**Hypotheses**
Ref	Expression
0prjspn.w	⊢ 𝑊 = (𝐾 freeLMod (0...0))
0prjspn.b	⊢ 𝐵 = ((Base‘𝑊) ∖ {(0_g‘𝑊)})

**Assertion**
Ref	Expression
0prjspn	⊢ (𝐾 ∈ DivRing → (0ℙ𝕣𝕠𝕛_n𝐾) = {𝐵})

Proof of Theorem 0prjspn Dummy variables 𝑎 𝑏 𝑙 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0nn0 12452	. . 3 ⊢ 0 ∈ ℕ₀
2		eqid 2731	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘𝐾)𝑥 = (𝑙( ·_𝑠 ‘𝑊)𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘𝐾)𝑥 = (𝑙( ·_𝑠 ‘𝑊)𝑦))}
3		0prjspn.w	. . . 4 ⊢ 𝑊 = (𝐾 freeLMod (0...0))
4		0prjspn.b	. . . 4 ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0_g‘𝑊)})
5		eqid 2731	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
6		eqid 2731	. . . 4 ⊢ ( ·_𝑠 ‘𝑊) = ( ·_𝑠 ‘𝑊)
7	2, 3, 4, 5, 6	prjspnval2 41047	. . 3 ⊢ ((0 ∈ ℕ₀ ∧ 𝐾 ∈ DivRing) → (0ℙ𝕣𝕠𝕛_n𝐾) = (𝐵 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘𝐾)𝑥 = (𝑙( ·_𝑠 ‘𝑊)𝑦))}))
8	1, 7	mpan 688	. 2 ⊢ (𝐾 ∈ DivRing → (0ℙ𝕣𝕠𝕛_n𝐾) = (𝐵 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘𝐾)𝑥 = (𝑙( ·_𝑠 ‘𝑊)𝑦))}))
9		ovex 7410	. . . . . . . 8 ⊢ (0...0) ∈ V
10	3	frlmsca 21211	. . . . . . . 8 ⊢ ((𝐾 ∈ DivRing ∧ (0...0) ∈ V) → 𝐾 = (Scalar‘𝑊))
11	9, 10	mpan2 689	. . . . . . 7 ⊢ (𝐾 ∈ DivRing → 𝐾 = (Scalar‘𝑊))
12	11	fveq2d 6866	. . . . . 6 ⊢ (𝐾 ∈ DivRing → (Base‘𝐾) = (Base‘(Scalar‘𝑊)))
13	12	rexeqdv 3325	. . . . 5 ⊢ (𝐾 ∈ DivRing → (∃𝑙 ∈ (Base‘𝐾)𝑥 = (𝑙( ·_𝑠 ‘𝑊)𝑦) ↔ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙( ·_𝑠 ‘𝑊)𝑦)))
14	13	anbi2d 629	. . . 4 ⊢ (𝐾 ∈ DivRing → (((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘𝐾)𝑥 = (𝑙( ·_𝑠 ‘𝑊)𝑦)) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙( ·_𝑠 ‘𝑊)𝑦))))
15	14	opabbidv 5191	. . 3 ⊢ (𝐾 ∈ DivRing → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘𝐾)𝑥 = (𝑙( ·_𝑠 ‘𝑊)𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙( ·_𝑠 ‘𝑊)𝑦))})
16	15	qseq2d 8727	. 2 ⊢ (𝐾 ∈ DivRing → (𝐵 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘𝐾)𝑥 = (𝑙( ·_𝑠 ‘𝑊)𝑦))}) = (𝐵 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙( ·_𝑠 ‘𝑊)𝑦))}))
17	3	frlmlvec 21219	. . . . . . 7 ⊢ ((𝐾 ∈ DivRing ∧ (0...0) ∈ V) → 𝑊 ∈ LVec)
18	9, 17	mpan2 689	. . . . . 6 ⊢ (𝐾 ∈ DivRing → 𝑊 ∈ LVec)
19		lveclmod 20639	. . . . . 6 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
20	18, 19	syl 17	. . . . 5 ⊢ (𝐾 ∈ DivRing → 𝑊 ∈ LMod)
21	20	adantr 481	. . . 4 ⊢ ((𝐾 ∈ DivRing ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑊 ∈ LMod)
22	15	adantr 481	. . . . . 6 ⊢ ((𝐾 ∈ DivRing ∧ 𝑎 ∈ 𝐵) → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘𝐾)𝑥 = (𝑙( ·_𝑠 ‘𝑊)𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙( ·_𝑠 ‘𝑊)𝑦))})
23		eqid 2731	. . . . . . 7 ⊢ ((𝐾 unitVec (0...0))‘0) = ((𝐾 unitVec (0...0))‘0)
24	2, 4, 6, 5, 3, 23	0prjspnrel 41056	. . . . . 6 ⊢ ((𝐾 ∈ DivRing ∧ 𝑎 ∈ 𝐵) → 𝑎{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘𝐾)𝑥 = (𝑙( ·_𝑠 ‘𝑊)𝑦))} ((𝐾 unitVec (0...0))‘0))
25	22, 24	breqdi 5140	. . . . 5 ⊢ ((𝐾 ∈ DivRing ∧ 𝑎 ∈ 𝐵) → 𝑎{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙( ·_𝑠 ‘𝑊)𝑦))} ((𝐾 unitVec (0...0))‘0))
26	25	adantrr 715	. . . 4 ⊢ ((𝐾 ∈ DivRing ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑎{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙( ·_𝑠 ‘𝑊)𝑦))} ((𝐾 unitVec (0...0))‘0))
27	15	adantr 481	. . . . . . 7 ⊢ ((𝐾 ∈ DivRing ∧ 𝑏 ∈ 𝐵) → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘𝐾)𝑥 = (𝑙( ·_𝑠 ‘𝑊)𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙( ·_𝑠 ‘𝑊)𝑦))})
28	2, 4, 6, 5, 3, 23	0prjspnrel 41056	. . . . . . 7 ⊢ ((𝐾 ∈ DivRing ∧ 𝑏 ∈ 𝐵) → 𝑏{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘𝐾)𝑥 = (𝑙( ·_𝑠 ‘𝑊)𝑦))} ((𝐾 unitVec (0...0))‘0))
29	27, 28	breqdi 5140	. . . . . 6 ⊢ ((𝐾 ∈ DivRing ∧ 𝑏 ∈ 𝐵) → 𝑏{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙( ·_𝑠 ‘𝑊)𝑦))} ((𝐾 unitVec (0...0))‘0))
30		eqid 2731	. . . . . . 7 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙( ·_𝑠 ‘𝑊)𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙( ·_𝑠 ‘𝑊)𝑦))}
31		eqid 2731	. . . . . . 7 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
32		eqid 2731	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
33	30, 4, 31, 6, 32	prjspersym 41036	. . . . . 6 ⊢ ((𝑊 ∈ LVec ∧ 𝑏{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙( ·_𝑠 ‘𝑊)𝑦))} ((𝐾 unitVec (0...0))‘0)) → ((𝐾 unitVec (0...0))‘0){⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙( ·_𝑠 ‘𝑊)𝑦))}𝑏)
34	18, 29, 33	syl2an2r 683	. . . . 5 ⊢ ((𝐾 ∈ DivRing ∧ 𝑏 ∈ 𝐵) → ((𝐾 unitVec (0...0))‘0){⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙( ·_𝑠 ‘𝑊)𝑦))}𝑏)
35	34	adantrl 714	. . . 4 ⊢ ((𝐾 ∈ DivRing ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ((𝐾 unitVec (0...0))‘0){⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙( ·_𝑠 ‘𝑊)𝑦))}𝑏)
36	30, 4, 31, 6, 32	prjspertr 41034	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ (𝑎{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙( ·_𝑠 ‘𝑊)𝑦))} ((𝐾 unitVec (0...0))‘0) ∧ ((𝐾 unitVec (0...0))‘0){⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙( ·_𝑠 ‘𝑊)𝑦))}𝑏)) → 𝑎{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙( ·_𝑠 ‘𝑊)𝑦))}𝑏)
37	21, 26, 35, 36	syl12anc 835	. . 3 ⊢ ((𝐾 ∈ DivRing ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑎{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙( ·_𝑠 ‘𝑊)𝑦))}𝑏)
38	30, 4, 31, 6, 32	prjsper 41037	. . . 4 ⊢ (𝑊 ∈ LVec → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙( ·_𝑠 ‘𝑊)𝑦))} Er 𝐵)
39	18, 38	syl 17	. . 3 ⊢ (𝐾 ∈ DivRing → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙( ·_𝑠 ‘𝑊)𝑦))} Er 𝐵)
40	4, 3, 23	0prjspnlem 41052	. . 3 ⊢ (𝐾 ∈ DivRing → ((𝐾 unitVec (0...0))‘0) ∈ 𝐵)
41	37, 39, 40	qsalrel 40768	. 2 ⊢ (𝐾 ∈ DivRing → (𝐵 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙( ·_𝑠 ‘𝑊)𝑦))}) = {𝐵})
42	8, 16, 41	3eqtrd 2775	1 ⊢ (𝐾 ∈ DivRing → (0ℙ𝕣𝕠𝕛_n𝐾) = {𝐵})

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∃wrex 3069 Vcvv 3459 ∖ cdif 3925 {csn 4606 class class class wbr 5125 {copab 5187 ‘cfv 6516 (class class class)co 7377 Er wer 8667 / cqs 8669 0cc0 11075 ℕ₀cn0 12437 ...cfz 13449 Basecbs 17109 Scalarcsca 17165 ·_𝑠 cvsca 17166 0_gc0g 17350 DivRingcdr 20240 LModclmod 20393 LVecclvec 20635 freeLMod cfrlm 21204 unitVec cuvc 21240 ℙ𝕣𝕠𝕛_ncprjspn 41043

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5262 ax-sep 5276 ax-nul 5283 ax-pow 5340 ax-pr 5404 ax-un 7692 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3364 df-reu 3365 df-rab 3419 df-v 3461 df-sbc 3758 df-csb 3874 df-dif 3931 df-un 3933 df-in 3935 df-ss 3945 df-pss 3947 df-nul 4303 df-if 4507 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4886 df-iun 4976 df-br 5126 df-opab 5188 df-mpt 5209 df-tr 5243 df-id 5551 df-eprel 5557 df-po 5565 df-so 5566 df-fr 5608 df-we 5610 df-xp 5659 df-rel 5660 df-cnv 5661 df-co 5662 df-dm 5663 df-rn 5664 df-res 5665 df-ima 5666 df-pred 6273 df-ord 6340 df-on 6341 df-lim 6342 df-suc 6343 df-iota 6468 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-riota 7333 df-ov 7380 df-oprab 7381 df-mpo 7382 df-of 7637 df-om 7823 df-1st 7941 df-2nd 7942 df-supp 8113 df-tpos 8177 df-frecs 8232 df-wrecs 8263 df-recs 8337 df-rdg 8376 df-1o 8432 df-er 8670 df-ec 8672 df-qs 8676 df-map 8789 df-ixp 8858 df-en 8906 df-dom 8907 df-sdom 8908 df-fin 8909 df-fsupp 9328 df-sup 9402 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11411 df-neg 11412 df-nn 12178 df-2 12240 df-3 12241 df-4 12242 df-5 12243 df-6 12244 df-7 12245 df-8 12246 df-9 12247 df-n0 12438 df-z 12524 df-dec 12643 df-uz 12788 df-fz 13450 df-struct 17045 df-sets 17062 df-slot 17080 df-ndx 17092 df-base 17110 df-ress 17139 df-plusg 17175 df-mulr 17176 df-sca 17178 df-vsca 17179 df-ip 17180 df-tset 17181 df-ple 17182 df-ds 17184 df-hom 17186 df-cco 17187 df-0g 17352 df-prds 17358 df-pws 17360 df-mgm 18526 df-sgrp 18575 df-mnd 18586 df-grp 18780 df-minusg 18781 df-sbg 18782 df-subg 18954 df-mgp 19926 df-ur 19943 df-ring 19995 df-oppr 20078 df-dvdsr 20099 df-unit 20100 df-invr 20130 df-nzr 20217 df-drng 20242 df-subrg 20283 df-lmod 20395 df-lss 20465 df-lvec 20636 df-sra 20707 df-rgmod 20708 df-dsmm 21190 df-frlm 21205 df-uvc 21241 df-prjsp 41031 df-prjspn 41044

This theorem is referenced by: (None)

W3C validator