prjspnval2 - Mathbox for Steven Nguyen

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Theorem prjspnval2 43068

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 43066	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐾 ∈ DivRing) → (𝑁ℙ𝕣𝕠𝕛_n𝐾) = (ℙ𝕣𝕠𝕛‘(𝐾 freeLMod (0...𝑁))))
2		prjspnval2.w	. . . . 5 ⊢ 𝑊 = (𝐾 freeLMod (0...𝑁))
3	2	fveq2i 6830	. . . 4 ⊢ (ℙ𝕣𝕠𝕛‘𝑊) = (ℙ𝕣𝕠𝕛‘(𝐾 freeLMod (0...𝑁)))
4		ovex 7389	. . . . . . 7 ⊢ (0...𝑁) ∈ V
5	2	frlmlvec 21736	. . . . . . 7 ⊢ ((𝐾 ∈ DivRing ∧ (0...𝑁) ∈ V) → 𝑊 ∈ LVec)
6	4, 5	mpan2 697	. . . . . 6 ⊢ (𝐾 ∈ DivRing → 𝑊 ∈ LVec)
7		prjspnval2.b	. . . . . . 7 ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0_g‘𝑊)})
8		prjspnval2.x	. . . . . . 7 ⊢ · = ( ·_𝑠 ‘𝑊)
9		eqid 2739	. . . . . . 7 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
10		eqid 2739	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
11	7, 8, 9, 10	prjspval 43053	. . . . . 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 43067	. . . . . 6 ⊢ (𝐾 ∈ DivRing → ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙 · 𝑦))})
16	15	qseq2d 8697	. . . . 5 ⊢ (𝐾 ∈ DivRing → (𝐵 / ∼ ) = (𝐵 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙 · 𝑦))}))
17	12, 16	eqtr4d 2777	. . . 4 ⊢ (𝐾 ∈ DivRing → (ℙ𝕣𝕠𝕛‘𝑊) = (𝐵 / ∼ ))
18	3, 17	eqtr3id 2788	. . 3 ⊢ (𝐾 ∈ DivRing → (ℙ𝕣𝕠𝕛‘(𝐾 freeLMod (0...𝑁))) = (𝐵 / ∼ ))
19	18	adantl 482	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐾 ∈ DivRing) → (ℙ𝕣𝕠𝕛‘(𝐾 freeLMod (0...𝑁))) = (𝐵 / ∼ ))
20	1, 19	eqtrd 2774	1 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐾 ∈ DivRing) → (𝑁ℙ𝕣𝕠𝕛_n𝐾) = (𝐵 / ∼ ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1547 ∈ wcel 2119 ∃wrex 3063 Vcvv 3431 ∖ cdif 3880 {csn 4555 {copab 5134 ‘cfv 6485 (class class class)co 7356 / cqs 8632 0cc0 11029 ℕ₀cn0 12428 ...cfz 13452 Basecbs 17170 Scalarcsca 17214 ·_𝑠 cvsca 17215 0_gc0g 17393 DivRingcdr 20701 LVecclvec 21092 freeLMod cfrlm 21721 ℙ𝕣𝕠𝕛cprjsp 43051 ℙ𝕣𝕠𝕛_ncprjspn 43064

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 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 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 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4839 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8633 df-ec 8635 df-qs 8639 df-map 8765 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9345 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-fz 13453 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ds 17233 df-hom 17235 df-cco 17236 df-0g 17395 df-prds 17401 df-pws 17403 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18903 df-minusg 18904 df-sbg 18905 df-subg 19090 df-cmn 19748 df-abl 19749 df-mgp 20113 df-rng 20125 df-ur 20154 df-ring 20207 df-subrg 20542 df-drng 20703 df-lmod 20852 df-lss 20922 df-lvec 21093 df-sra 21163 df-rgmod 21164 df-dsmm 21707 df-frlm 21722 df-prjsp 43052 df-prjspn 43065

This theorem is referenced by: prjspnssbas 43071 prjspnn0 43072 0prjspn 43078

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