prjspnval2 - Mathbox for Steven Nguyen

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Theorem prjspnval2 43197

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 43195	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐾 ∈ DivRing) → (𝑁ℙ𝕣𝕠𝕛_n𝐾) = (ℙ𝕣𝕠𝕛‘(𝐾 freeLMod (0...𝑁))))
2		prjspnval2.w	. . . . 5 ⊢ 𝑊 = (𝐾 freeLMod (0...𝑁))
3	2	fveq2i 6870	. . . 4 ⊢ (ℙ𝕣𝕠𝕛‘𝑊) = (ℙ𝕣𝕠𝕛‘(𝐾 freeLMod (0...𝑁)))
4		ovex 7429	. . . . . . 7 ⊢ (0...𝑁) ∈ V
5	2	frlmlvec 21810	. . . . . . 7 ⊢ ((𝐾 ∈ DivRing ∧ (0...𝑁) ∈ V) → 𝑊 ∈ LVec)
6	4, 5	mpan2 701	. . . . . 6 ⊢ (𝐾 ∈ DivRing → 𝑊 ∈ LVec)
7		prjspnval2.b	. . . . . . 7 ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0_g‘𝑊)})
8		prjspnval2.x	. . . . . . 7 ⊢ · = ( ·_𝑠 ‘𝑊)
9		eqid 2762	. . . . . . 7 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
10		eqid 2762	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
11	7, 8, 9, 10	prjspval 43182	. . . . . 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 43196	. . . . . 6 ⊢ (𝐾 ∈ DivRing → ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙 · 𝑦))})
16	15	qseq2d 8742	. . . . 5 ⊢ (𝐾 ∈ DivRing → (𝐵 / ∼ ) = (𝐵 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙 · 𝑦))}))
17	12, 16	eqtr4d 2800	. . . 4 ⊢ (𝐾 ∈ DivRing → (ℙ𝕣𝕠𝕛‘𝑊) = (𝐵 / ∼ ))
18	3, 17	eqtr3id 2811	. . 3 ⊢ (𝐾 ∈ DivRing → (ℙ𝕣𝕠𝕛‘(𝐾 freeLMod (0...𝑁))) = (𝐵 / ∼ ))
19	18	adantl 485	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐾 ∈ DivRing) → (ℙ𝕣𝕠𝕛‘(𝐾 freeLMod (0...𝑁))) = (𝐵 / ∼ ))
20	1, 19	eqtrd 2797	1 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐾 ∈ DivRing) → (𝑁ℙ𝕣𝕠𝕛_n𝐾) = (𝐵 / ∼ ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1560 ∈ wcel 2142 ∃wrex 3086 Vcvv 3454 ∖ cdif 3901 {csn 4582 {copab 5162 ‘cfv 6521 (class class class)co 7396 / cqs 8677 0cc0 11073 ℕ₀cn0 12481 ...cfz 13512 Basecbs 17245 Scalarcsca 17289 ·_𝑠 cvsca 17290 0_gc0g 17468 DivRingcdr 20775 LVecclvec 21166 freeLMod cfrlm 21795 ℙ𝕣𝕠𝕛cprjsp 43180 ℙ𝕣𝕠𝕛_ncprjspn 43193

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5227 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718 ax-cnex 11129 ax-resscn 11130 ax-1cn 11131 ax-icn 11132 ax-addcl 11133 ax-addrcl 11134 ax-mulcl 11135 ax-mulrcl 11136 ax-mulcom 11137 ax-addass 11138 ax-mulass 11139 ax-distr 11140 ax-i2m1 11141 ax-1ne0 11142 ax-1rid 11143 ax-rnegex 11144 ax-rrecex 11145 ax-cnre 11146 ax-pre-lttri 11147 ax-pre-lttrn 11148 ax-pre-ltadd 11149 ax-pre-mulgt0 11150

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1099 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-iun 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7847 df-1st 7970 df-2nd 7971 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-1o 8437 df-er 8678 df-ec 8680 df-qs 8684 df-map 8810 df-ixp 8880 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-sup 9388 df-pnf 11218 df-mnf 11219 df-xr 11220 df-ltxr 11221 df-le 11222 df-sub 11416 df-neg 11417 df-nn 12211 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12482 df-z 12569 df-dec 12689 df-uz 12840 df-fz 13513 df-struct 17183 df-sets 17200 df-slot 17218 df-ndx 17230 df-base 17246 df-ress 17267 df-plusg 17299 df-mulr 17300 df-sca 17302 df-vsca 17303 df-ip 17304 df-tset 17305 df-ple 17306 df-ds 17308 df-hom 17310 df-cco 17311 df-0g 17470 df-prds 17476 df-pws 17478 df-mgm 18674 df-sgrp 18753 df-mnd 18769 df-grp 18978 df-minusg 18979 df-sbg 18980 df-subg 19165 df-cmn 19822 df-abl 19823 df-mgp 20187 df-rng 20199 df-ur 20228 df-ring 20281 df-subrg 20616 df-drng 20777 df-lmod 20926 df-lss 20996 df-lvec 21167 df-sra 21237 df-rgmod 21238 df-dsmm 21781 df-frlm 21796 df-prjsp 43181 df-prjspn 43194

This theorem is referenced by: prjspnssbas 43200 prjspnn0 43201 0prjspn 43207

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