prjspnval2 - Mathbox for Steven Nguyen

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Theorem prjspnval2 39342

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: 𝑥,𝑙,𝑦,𝑊 𝑥,𝑁,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(𝑙)

**Proof of Theorem prjspnval2**
Step	Hyp	Ref	Expression
1		prjspnval 39341	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐾 ∈ DivRing) → (𝑁ℙ𝕣𝕠𝕛_n𝐾) = (ℙ𝕣𝕠𝕛‘(𝐾 freeLMod (0...𝑁))))
2		simpr 487	. . . . . 6 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐾 ∈ DivRing) → 𝐾 ∈ DivRing)
3		ovexd 7188	. . . . . 6 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐾 ∈ DivRing) → (0...𝑁) ∈ V)
4		prjspnval2.w	. . . . . . 7 ⊢ 𝑊 = (𝐾 freeLMod (0...𝑁))
5	4	frlmlvec 20901	. . . . . 6 ⊢ ((𝐾 ∈ DivRing ∧ (0...𝑁) ∈ V) → 𝑊 ∈ LVec)
6	2, 3, 5	syl2anc 586	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐾 ∈ DivRing) → 𝑊 ∈ LVec)
7		prjspnval2.b	. . . . . 6 ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0_g‘𝑊)})
8		prjspnval2.x	. . . . . 6 ⊢ · = ( ·_𝑠 ‘𝑊)
9		eqid 2820	. . . . . 6 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
10		eqid 2820	. . . . . 6 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
11	7, 8, 9, 10	prjspval 39328	. . . . 5 ⊢ (𝑊 ∈ LVec → (ℙ𝕣𝕠𝕛‘𝑊) = (𝐵 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙 · 𝑦))}))
12	6, 11	syl 17	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐾 ∈ DivRing) → (ℙ𝕣𝕠𝕛‘𝑊) = (𝐵 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙 · 𝑦))}))
13		prjspnval2.s	. . . . . . . . 9 ⊢ 𝑆 = (Base‘𝐾)
14	4	frlmsca 20893	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ DivRing ∧ (0...𝑁) ∈ V) → 𝐾 = (Scalar‘𝑊))
15	2, 3, 14	syl2anc 586	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐾 ∈ DivRing) → 𝐾 = (Scalar‘𝑊))
16	15	fveq2d 6671	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐾 ∈ DivRing) → (Base‘𝐾) = (Base‘(Scalar‘𝑊)))
17	13, 16	syl5req 2868	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐾 ∈ DivRing) → (Base‘(Scalar‘𝑊)) = 𝑆)
18	17	rexeqdv 3415	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐾 ∈ DivRing) → (∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙 · 𝑦) ↔ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦)))
19	18	anbi2d 630	. . . . . 6 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐾 ∈ DivRing) → (((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙 · 𝑦)) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))))
20	19	opabbidv 5129	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐾 ∈ DivRing) → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙 · 𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))})
21	20	qseq2d 8343	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐾 ∈ DivRing) → (𝐵 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙 · 𝑦))}) = (𝐵 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))}))
22	12, 21	eqtrd 2855	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐾 ∈ DivRing) → (ℙ𝕣𝕠𝕛‘𝑊) = (𝐵 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))}))
23	4	eqcomi 2829	. . . 4 ⊢ (𝐾 freeLMod (0...𝑁)) = 𝑊
24	23	fveq2i 6670	. . 3 ⊢ (ℙ𝕣𝕠𝕛‘(𝐾 freeLMod (0...𝑁))) = (ℙ𝕣𝕠𝕛‘𝑊)
25		prjspnval2.e	. . . 4 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))}
26	25	qseq2i 8342	. . 3 ⊢ (𝐵 / ∼ ) = (𝐵 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))})
27	22, 24, 26	3eqtr4g 2880	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐾 ∈ DivRing) → (ℙ𝕣𝕠𝕛‘(𝐾 freeLMod (0...𝑁))) = (𝐵 / ∼ ))
28	1, 27	eqtrd 2855	1 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐾 ∈ DivRing) → (𝑁ℙ𝕣𝕠𝕛_n𝐾) = (𝐵 / ∼ ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∃wrex 3138 Vcvv 3493 ∖ cdif 3930 {csn 4564 {copab 5125 ‘cfv 6352 (class class class)co 7153 / cqs 8285 0cc0 10534 ℕ₀cn0 11895 ...cfz 12890 Basecbs 16479 Scalarcsca 16564 ·_𝑠 cvsca 16565 0_gc0g 16709 DivRingcdr 19498 LVecclvec 19870 freeLMod cfrlm 20886 ℙ𝕣𝕠𝕛cprjsp 39326 ℙ𝕣𝕠𝕛_ncprjspn 39339

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5187 ax-sep 5200 ax-nul 5207 ax-pow 5263 ax-pr 5327 ax-un 7458 ax-cnex 10590 ax-resscn 10591 ax-1cn 10592 ax-icn 10593 ax-addcl 10594 ax-addrcl 10595 ax-mulcl 10596 ax-mulrcl 10597 ax-mulcom 10598 ax-addass 10599 ax-mulass 10600 ax-distr 10601 ax-i2m1 10602 ax-1ne0 10603 ax-1rid 10604 ax-rnegex 10605 ax-rrecex 10606 ax-cnre 10607 ax-pre-lttri 10608 ax-pre-lttrn 10609 ax-pre-ltadd 10610 ax-pre-mulgt0 10611

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3495 df-sbc 3771 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4465 df-pw 4538 df-sn 4565 df-pr 4567 df-tp 4569 df-op 4571 df-uni 4836 df-int 4874 df-iun 4918 df-br 5064 df-opab 5126 df-mpt 5144 df-tr 5170 df-id 5457 df-eprel 5462 df-po 5471 df-so 5472 df-fr 5511 df-we 5513 df-xp 5558 df-rel 5559 df-cnv 5560 df-co 5561 df-dm 5562 df-rn 5563 df-res 5564 df-ima 5565 df-pred 6145 df-ord 6191 df-on 6192 df-lim 6193 df-suc 6194 df-iota 6311 df-fun 6354 df-fn 6355 df-f 6356 df-f1 6357 df-fo 6358 df-f1o 6359 df-fv 6360 df-riota 7111 df-ov 7156 df-oprab 7157 df-mpo 7158 df-om 7578 df-1st 7686 df-2nd 7687 df-wrecs 7944 df-recs 8005 df-rdg 8043 df-1o 8099 df-oadd 8103 df-er 8286 df-ec 8288 df-qs 8292 df-map 8405 df-ixp 8459 df-en 8507 df-dom 8508 df-sdom 8509 df-fin 8510 df-sup 8903 df-pnf 10674 df-mnf 10675 df-xr 10676 df-ltxr 10677 df-le 10678 df-sub 10869 df-neg 10870 df-nn 11636 df-2 11698 df-3 11699 df-4 11700 df-5 11701 df-6 11702 df-7 11703 df-8 11704 df-9 11705 df-n0 11896 df-z 11980 df-dec 12097 df-uz 12242 df-fz 12891 df-struct 16481 df-ndx 16482 df-slot 16483 df-base 16485 df-sets 16486 df-ress 16487 df-plusg 16574 df-mulr 16575 df-sca 16577 df-vsca 16578 df-ip 16579 df-tset 16580 df-ple 16581 df-ds 16583 df-hom 16585 df-cco 16586 df-0g 16711 df-prds 16717 df-pws 16719 df-mgm 17848 df-sgrp 17897 df-mnd 17908 df-grp 18102 df-minusg 18103 df-sbg 18104 df-subg 18272 df-mgp 19236 df-ur 19248 df-ring 19295 df-drng 19500 df-subrg 19529 df-lmod 19632 df-lss 19700 df-lvec 19871 df-sra 19940 df-rgmod 19941 df-dsmm 20872 df-frlm 20887 df-prjsp 39327 df-prjspn 39340

This theorem is referenced by: 0prjspn 39345

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