prjspnval2 - Mathbox for Steven Nguyen

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Theorem prjspnval2 42633

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 42631	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐾 ∈ DivRing) → (𝑁ℙ𝕣𝕠𝕛_n𝐾) = (ℙ𝕣𝕠𝕛‘(𝐾 freeLMod (0...𝑁))))
2		prjspnval2.w	. . . . 5 ⊢ 𝑊 = (𝐾 freeLMod (0...𝑁))
3	2	fveq2i 6908	. . . 4 ⊢ (ℙ𝕣𝕠𝕛‘𝑊) = (ℙ𝕣𝕠𝕛‘(𝐾 freeLMod (0...𝑁)))
4		ovex 7465	. . . . . . 7 ⊢ (0...𝑁) ∈ V
5	2	frlmlvec 21782	. . . . . . 7 ⊢ ((𝐾 ∈ DivRing ∧ (0...𝑁) ∈ V) → 𝑊 ∈ LVec)
6	4, 5	mpan2 691	. . . . . 6 ⊢ (𝐾 ∈ DivRing → 𝑊 ∈ LVec)
7		prjspnval2.b	. . . . . . 7 ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0_g‘𝑊)})
8		prjspnval2.x	. . . . . . 7 ⊢ · = ( ·_𝑠 ‘𝑊)
9		eqid 2736	. . . . . . 7 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
10		eqid 2736	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
11	7, 8, 9, 10	prjspval 42618	. . . . . 6 ⊢ (𝑊 ∈ LVec → (ℙ𝕣𝕠𝕛‘𝑊) = (𝐵 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙 · 𝑦))}))
12	6, 11	syl 17	. . . . 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 42632	. . . . . 6 ⊢ (𝐾 ∈ DivRing → ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙 · 𝑦))})
16	15	qseq2d 8806	. . . . 5 ⊢ (𝐾 ∈ DivRing → (𝐵 / ∼ ) = (𝐵 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙 · 𝑦))}))
17	12, 16	eqtr4d 2779	. . . 4 ⊢ (𝐾 ∈ DivRing → (ℙ𝕣𝕠𝕛‘𝑊) = (𝐵 / ∼ ))
18	3, 17	eqtr3id 2790	. . 3 ⊢ (𝐾 ∈ DivRing → (ℙ𝕣𝕠𝕛‘(𝐾 freeLMod (0...𝑁))) = (𝐵 / ∼ ))
19	18	adantl 481	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐾 ∈ DivRing) → (ℙ𝕣𝕠𝕛‘(𝐾 freeLMod (0...𝑁))) = (𝐵 / ∼ ))
20	1, 19	eqtrd 2776	1 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐾 ∈ DivRing) → (𝑁ℙ𝕣𝕠𝕛_n𝐾) = (𝐵 / ∼ ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∃wrex 3069 Vcvv 3479 ∖ cdif 3947 {csn 4625 {copab 5204 ‘cfv 6560 (class class class)co 7432 / cqs 8745 0cc0 11156 ℕ₀cn0 12528 ...cfz 13548 Basecbs 17248 Scalarcsca 17301 ·_𝑠 cvsca 17302 0_gc0g 17485 DivRingcdr 20730 LVecclvec 21102 freeLMod cfrlm 21767 ℙ𝕣𝕠𝕛cprjsp 42616 ℙ𝕣𝕠𝕛_ncprjspn 42629

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-er 8746 df-ec 8748 df-qs 8752 df-map 8869 df-ixp 8939 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-sup 9483 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-uz 12880 df-fz 13549 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-hom 17322 df-cco 17323 df-0g 17487 df-prds 17493 df-pws 17495 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-grp 18955 df-minusg 18956 df-sbg 18957 df-subg 19142 df-cmn 19801 df-abl 19802 df-mgp 20139 df-rng 20151 df-ur 20180 df-ring 20233 df-subrg 20571 df-drng 20732 df-lmod 20861 df-lss 20931 df-lvec 21103 df-sra 21173 df-rgmod 21174 df-dsmm 21753 df-frlm 21768 df-prjsp 42617 df-prjspn 42630

This theorem is referenced by: prjspnssbas 42636 prjspnn0 42637 0prjspn 42643

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