prjspnval2 - Mathbox for Steven Nguyen

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Theorem prjspnval2 43241

Description: Value of the n-dimensional projective space function, expanded. (Contributed by Steven Nguyen, 15-Jul-2023.)

**Hypotheses**
Ref	Expression
prjspnval2.e	⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))}
prjspnval2.w	⊢ 𝑊 = (𝐾 freeLMod (0...𝑁))
prjspnval2.b	⊢ 𝐵 = ((Base‘𝑊) ∖ {(0_g‘𝑊)})
prjspnval2.s	⊢ 𝑆 = (Base‘𝐾)
prjspnval2.x	⊢ · = ( ·_𝑠 ‘𝑊)

**Assertion**
Ref	Expression
prjspnval2	⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐾 ∈ DivRing) → (𝑁ℙ𝕣𝕠𝕛_n𝐾) = (𝐵 / ∼ ))

Distinct variable groups: 𝑊,𝑙,𝑥,𝑦 𝑥,𝐾,𝑦 𝑆,𝑙 Allowed substitution hints: 𝐵(𝑥,𝑦,𝑙) ∼ (𝑥,𝑦,𝑙) 𝑆(𝑥,𝑦) · (𝑥,𝑦,𝑙) 𝐾(𝑙) 𝑁(𝑥,𝑦,𝑙)

**Proof of Theorem prjspnval2**
Step	Hyp	Ref	Expression
1		prjspnval 43239	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐾 ∈ DivRing) → (𝑁ℙ𝕣𝕠𝕛_n𝐾) = (ℙ𝕣𝕠𝕛‘(𝐾 freeLMod (0...𝑁))))
2		prjspnval2.w	. . . . 5 ⊢ 𝑊 = (𝐾 freeLMod (0...𝑁))
3	2	fveq2i 6885	. . . 4 ⊢ (ℙ𝕣𝕠𝕛‘𝑊) = (ℙ𝕣𝕠𝕛‘(𝐾 freeLMod (0...𝑁)))
4		ovex 7444	. . . . . . 7 ⊢ (0...𝑁) ∈ V
5	2	frlmlvec 21879	. . . . . . 7 ⊢ ((𝐾 ∈ DivRing ∧ (0...𝑁) ∈ V) → 𝑊 ∈ LVec)
6	4, 5	mpan2 703	. . . . . 6 ⊢ (𝐾 ∈ DivRing → 𝑊 ∈ LVec)
7		prjspnval2.b	. . . . . . 7 ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0_g‘𝑊)})
8		prjspnval2.x	. . . . . . 7 ⊢ · = ( ·_𝑠 ‘𝑊)
9		eqid 2769	. . . . . . 7 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
10		eqid 2769	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
11	7, 8, 9, 10	prjspval 43226	. . . . . 6 ⊢ (𝑊 ∈ LVec → (ℙ𝕣𝕠𝕛‘𝑊) = (𝐵 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙 · 𝑦))}))
12	6, 11	syl 18	. . . . 5 ⊢ (𝐾 ∈ DivRing → (ℙ𝕣𝕠𝕛‘𝑊) = (𝐵 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙 · 𝑦))}))
13		prjspnval2.e	. . . . . . 7 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))}
14		prjspnval2.s	. . . . . . 7 ⊢ 𝑆 = (Base‘𝐾)
15	13, 2, 7, 14, 8	prjspnerlem 43240	. . . . . 6 ⊢ (𝐾 ∈ DivRing → ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙 · 𝑦))})
16	15	qseq2d 8757	. . . . 5 ⊢ (𝐾 ∈ DivRing → (𝐵 / ∼ ) = (𝐵 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙 · 𝑦))}))
17	12, 16	eqtr4d 2807	. . . 4 ⊢ (𝐾 ∈ DivRing → (ℙ𝕣𝕠𝕛‘𝑊) = (𝐵 / ∼ ))
18	3, 17	eqtr3id 2818	. . 3 ⊢ (𝐾 ∈ DivRing → (ℙ𝕣𝕠𝕛‘(𝐾 freeLMod (0...𝑁))) = (𝐵 / ∼ ))
19	18	adantl 486	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐾 ∈ DivRing) → (ℙ𝕣𝕠𝕛‘(𝐾 freeLMod (0...𝑁))) = (𝐵 / ∼ ))
20	1, 19	eqtrd 2804	1 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐾 ∈ DivRing) → (𝑁ℙ𝕣𝕠𝕛_n𝐾) = (𝐵 / ∼ ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1567 ∈ wcel 2149 ∃wrex 3095 Vcvv 3463 ∖ cdif 3910 {csn 4594 {copab 5177 ‘cfv 6537 (class class class)co 7411 / cqs 8692 0cc0 11099 ℕ₀cn0 12503 ...cfz 13534 Basecbs 17268 Scalarcsca 17312 ·_𝑠 cvsca 17313 0_gc0g 17491 DivRingcdr 20812 LVecclvec 21200 freeLMod cfrlm 21864 ℙ𝕣𝕠𝕛cprjsp 43224 ℙ𝕣𝕠𝕛_ncprjspn 43237

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-cnex 11155 ax-resscn 11156 ax-1cn 11157 ax-icn 11158 ax-addcl 11159 ax-addrcl 11160 ax-mulcl 11161 ax-mulrcl 11162 ax-mulcom 11163 ax-addass 11164 ax-mulass 11165 ax-distr 11166 ax-i2m1 11167 ax-1ne0 11168 ax-1rid 11169 ax-rnegex 11170 ax-rrecex 11171 ax-cnre 11172 ax-pre-lttri 11173 ax-pre-lttrn 11174 ax-pre-ltadd 11175 ax-pre-mulgt0 11176

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4877 df-iun 4962 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7862 df-1st 7985 df-2nd 7986 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8452 df-er 8693 df-ec 8695 df-qs 8699 df-map 8825 df-ixp 8895 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-sup 9401 df-pnf 11244 df-mnf 11245 df-xr 11246 df-ltxr 11247 df-le 11248 df-sub 11442 df-neg 11443 df-nn 12233 df-2 12302 df-3 12303 df-4 12304 df-5 12305 df-6 12306 df-7 12307 df-8 12308 df-9 12309 df-n0 12504 df-z 12591 df-dec 12711 df-uz 12862 df-fz 13535 df-struct 17206 df-sets 17223 df-slot 17241 df-ndx 17253 df-base 17269 df-ress 17290 df-plusg 17322 df-mulr 17323 df-sca 17325 df-vsca 17326 df-ip 17327 df-tset 17328 df-ple 17329 df-ds 17331 df-hom 17333 df-cco 17334 df-0g 17493 df-prds 17499 df-pws 17501 df-mgm 18697 df-sgrp 18776 df-mnd 18792 df-grp 19002 df-minusg 19003 df-sbg 19004 df-subg 19188 df-cmn 19851 df-abl 19852 df-mgp 20216 df-rng 20230 df-ur 20263 df-ring 20316 df-subrg 20654 df-drng 20814 df-lmod 20960 df-lss 21030 df-lvec 21201 df-sra 21271 df-rgmod 21272 df-dsmm 21850 df-frlm 21865 df-prjsp 43225 df-prjspn 43238

This theorem is referenced by: prjspnssbas 43244 prjspnn0 43245 0prjspn 43251

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