prjspner01 - Mathbox for Steven Nguyen

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Theorem prjspner01 40462

Description: Any vector is equivalent to a vector whose zeroth coordinate is 0 or 1 (proof of the equivalence). (Contributed by SN, 13-Aug-2023.)

**Hypotheses**
Ref	Expression
prjspner01.e	⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))}
prjspner01.f	⊢ 𝐹 = (𝑏 ∈ 𝐵 ↦ if((𝑏‘0) = 0 , 𝑏, ((𝐼‘(𝑏‘0)) · 𝑏)))
prjspner01.b	⊢ 𝐵 = ((Base‘𝑊) ∖ {(0_g‘𝑊)})
prjspner01.w	⊢ 𝑊 = (𝐾 freeLMod (0...𝑁))
prjspner01.t	⊢ · = ( ·_𝑠 ‘𝑊)
prjspner01.s	⊢ 𝑆 = (Base‘𝐾)
prjspner01.0	⊢ 0 = (0_g‘𝐾)
prjspner01.i	⊢ 𝐼 = (inv_r‘𝐾)
prjspner01.k	⊢ (𝜑 → 𝐾 ∈ DivRing)
prjspner01.n	⊢ (𝜑 → 𝑁 ∈ ℕ₀)
prjspner01.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)

**Assertion**
Ref	Expression
prjspner01	⊢ (𝜑 → 𝑋 ∼ (𝐹‘𝑋))

Distinct variable groups: 𝑥,𝐵,𝑦 𝑋,𝑙,𝑥,𝑦 𝑊,𝑙,𝑥,𝑦 · ,𝑙,𝑥,𝑦 𝑆,𝑙 𝐼,𝑙,𝑥,𝑦 𝑥,𝐾,𝑦 𝑥, 0 ,𝑦 𝐵,𝑏 𝑋,𝑏 0 ,𝑏 · ,𝑏 𝐼,𝑏 𝜑,𝑏 Allowed substitution hints: 𝜑(𝑥,𝑦,𝑙) 𝐵(𝑙) ∼ (𝑥,𝑦,𝑏,𝑙) 𝑆(𝑥,𝑦,𝑏) 𝐹(𝑥,𝑦,𝑏,𝑙) 𝐾(𝑏,𝑙) 𝑁(𝑥,𝑦,𝑏,𝑙) 𝑊(𝑏) 0 (𝑙)

**Proof of Theorem prjspner01**
Step	Hyp	Ref	Expression
1		prjspner01.e	. . . . . . 7 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))}
2		prjspner01.w	. . . . . . 7 ⊢ 𝑊 = (𝐾 freeLMod (0...𝑁))
3		prjspner01.b	. . . . . . 7 ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0_g‘𝑊)})
4		prjspner01.s	. . . . . . 7 ⊢ 𝑆 = (Base‘𝐾)
5		prjspner01.t	. . . . . . 7 ⊢ · = ( ·_𝑠 ‘𝑊)
6		prjspner01.k	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ DivRing)
7	1, 2, 3, 4, 5, 6	prjspner 40458	. . . . . 6 ⊢ (𝜑 → ∼ Er 𝐵)
8		prjspner01.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
9	7, 8	erref 8518	. . . . 5 ⊢ (𝜑 → 𝑋 ∼ 𝑋)
10	9	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝑋‘0) = 0 ) → 𝑋 ∼ 𝑋)
11	7	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ ¬ (𝑋‘0) = 0 ) → ∼ Er 𝐵)
12		prjspner01.0	. . . . . 6 ⊢ 0 = (0_g‘𝐾)
13	6	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝑋‘0) = 0 ) → 𝐾 ∈ DivRing)
14	8	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝑋‘0) = 0 ) → 𝑋 ∈ 𝐵)
15		ovexd 7310	. . . . . . . . 9 ⊢ (𝜑 → (0...𝑁) ∈ V)
16	8, 3	eleqtrdi 2849	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ ((Base‘𝑊) ∖ {(0_g‘𝑊)}))
17	16	eldifad 3899	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑊))
18		eqid 2738	. . . . . . . . . 10 ⊢ (Base‘𝑊) = (Base‘𝑊)
19	2, 4, 18	frlmbasf 20967	. . . . . . . . 9 ⊢ (((0...𝑁) ∈ V ∧ 𝑋 ∈ (Base‘𝑊)) → 𝑋:(0...𝑁)⟶𝑆)
20	15, 17, 19	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → 𝑋:(0...𝑁)⟶𝑆)
21		prjspner01.n	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
22		0elfz 13353	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ₀ → 0 ∈ (0...𝑁))
23	21, 22	syl 17	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ (0...𝑁))
24	20, 23	ffvelrnd 6962	. . . . . . 7 ⊢ (𝜑 → (𝑋‘0) ∈ 𝑆)
25		neqne 2951	. . . . . . 7 ⊢ (¬ (𝑋‘0) = 0 → (𝑋‘0) ≠ 0 )
26		prjspner01.i	. . . . . . . 8 ⊢ 𝐼 = (inv_r‘𝐾)
27	4, 12, 26	drnginvrcl 20008	. . . . . . 7 ⊢ ((𝐾 ∈ DivRing ∧ (𝑋‘0) ∈ 𝑆 ∧ (𝑋‘0) ≠ 0 ) → (𝐼‘(𝑋‘0)) ∈ 𝑆)
28	6, 24, 25, 27	syl2an3an 1421	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝑋‘0) = 0 ) → (𝐼‘(𝑋‘0)) ∈ 𝑆)
29	4, 12, 26	drnginvrn0 20009	. . . . . . 7 ⊢ ((𝐾 ∈ DivRing ∧ (𝑋‘0) ∈ 𝑆 ∧ (𝑋‘0) ≠ 0 ) → (𝐼‘(𝑋‘0)) ≠ 0 )
30	6, 24, 25, 29	syl2an3an 1421	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝑋‘0) = 0 ) → (𝐼‘(𝑋‘0)) ≠ 0 )
31	1, 2, 3, 4, 5, 12, 13, 14, 28, 30	prjspnvs 40459	. . . . 5 ⊢ ((𝜑 ∧ ¬ (𝑋‘0) = 0 ) → ((𝐼‘(𝑋‘0)) · 𝑋) ∼ 𝑋)
32	11, 31	ersym 8510	. . . 4 ⊢ ((𝜑 ∧ ¬ (𝑋‘0) = 0 ) → 𝑋 ∼ ((𝐼‘(𝑋‘0)) · 𝑋))
33	10, 32	ifpimpda 1080	. . 3 ⊢ (𝜑 → if-((𝑋‘0) = 0 , 𝑋 ∼ 𝑋, 𝑋 ∼ ((𝐼‘(𝑋‘0)) · 𝑋)))
34		brif2 40199	. . 3 ⊢ (𝑋 ∼ if((𝑋‘0) = 0 , 𝑋, ((𝐼‘(𝑋‘0)) · 𝑋)) ↔ if-((𝑋‘0) = 0 , 𝑋 ∼ 𝑋, 𝑋 ∼ ((𝐼‘(𝑋‘0)) · 𝑋)))
35	33, 34	sylibr 233	. 2 ⊢ (𝜑 → 𝑋 ∼ if((𝑋‘0) = 0 , 𝑋, ((𝐼‘(𝑋‘0)) · 𝑋)))
36		prjspner01.f	. . 3 ⊢ 𝐹 = (𝑏 ∈ 𝐵 ↦ if((𝑏‘0) = 0 , 𝑏, ((𝐼‘(𝑏‘0)) · 𝑏)))
37		fveq1 6773	. . . . 5 ⊢ (𝑏 = 𝑋 → (𝑏‘0) = (𝑋‘0))
38	37	eqeq1d 2740	. . . 4 ⊢ (𝑏 = 𝑋 → ((𝑏‘0) = 0 ↔ (𝑋‘0) = 0 ))
39		id 22	. . . 4 ⊢ (𝑏 = 𝑋 → 𝑏 = 𝑋)
40	37	fveq2d 6778	. . . . 5 ⊢ (𝑏 = 𝑋 → (𝐼‘(𝑏‘0)) = (𝐼‘(𝑋‘0)))
41	40, 39	oveq12d 7293	. . . 4 ⊢ (𝑏 = 𝑋 → ((𝐼‘(𝑏‘0)) · 𝑏) = ((𝐼‘(𝑋‘0)) · 𝑋))
42	38, 39, 41	ifbieq12d 4487	. . 3 ⊢ (𝑏 = 𝑋 → if((𝑏‘0) = 0 , 𝑏, ((𝐼‘(𝑏‘0)) · 𝑏)) = if((𝑋‘0) = 0 , 𝑋, ((𝐼‘(𝑋‘0)) · 𝑋)))
43		ovexd 7310	. . . 4 ⊢ (𝜑 → ((𝐼‘(𝑋‘0)) · 𝑋) ∈ V)
44	8, 43	ifexd 4507	. . 3 ⊢ (𝜑 → if((𝑋‘0) = 0 , 𝑋, ((𝐼‘(𝑋‘0)) · 𝑋)) ∈ V)
45	36, 42, 8, 44	fvmptd3 6898	. 2 ⊢ (𝜑 → (𝐹‘𝑋) = if((𝑋‘0) = 0 , 𝑋, ((𝐼‘(𝑋‘0)) · 𝑋)))
46	35, 45	breqtrrd 5102	1 ⊢ (𝜑 → 𝑋 ∼ (𝐹‘𝑋))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 if-wif 1060 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∃wrex 3065 Vcvv 3432 ∖ cdif 3884 ifcif 4459 {csn 4561 class class class wbr 5074 {copab 5136 ↦ cmpt 5157 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 Er wer 8495 0cc0 10871 ℕ₀cn0 12233 ...cfz 13239 Basecbs 16912 ·_𝑠 cvsca 16966 0_gc0g 17150 inv_rcinvr 19913 DivRingcdr 19991 freeLMod cfrlm 20953

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-ifp 1061 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-tpos 8042 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-sup 9201 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-fz 13240 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-hom 16986 df-cco 16987 df-0g 17152 df-prds 17158 df-pws 17160 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-grp 18580 df-minusg 18581 df-sbg 18582 df-subg 18752 df-mgp 19721 df-ur 19738 df-ring 19785 df-oppr 19862 df-dvdsr 19883 df-unit 19884 df-invr 19914 df-drng 19993 df-subrg 20022 df-lmod 20125 df-lss 20194 df-lvec 20365 df-sra 20434 df-rgmod 20435 df-dsmm 20939 df-frlm 20954

This theorem is referenced by: prjspner1 40463

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