prjspner01 - Mathbox for Steven Nguyen

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Theorem prjspner01 40949

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 40943	. . . . . 6 ⊢ (𝜑 → ∼ Er 𝐵)
8		prjspner01.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
9	7, 8	erref 8668	. . . . 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 7392	. . . . . . . . 9 ⊢ (𝜑 → (0...𝑁) ∈ V)
16	8, 3	eleqtrdi 2848	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ ((Base‘𝑊) ∖ {(0_g‘𝑊)}))
17	16	eldifad 3922	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑊))
18		eqid 2736	. . . . . . . . . 10 ⊢ (Base‘𝑊) = (Base‘𝑊)
19	2, 4, 18	frlmbasf 21166	. . . . . . . . 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 13538	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ₀ → 0 ∈ (0...𝑁))
23	21, 22	syl 17	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ (0...𝑁))
24	20, 23	ffvelcdmd 7036	. . . . . . 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 20205	. . . . . . 7 ⊢ ((𝐾 ∈ DivRing ∧ (𝑋‘0) ∈ 𝑆 ∧ (𝑋‘0) ≠ 0 ) → (𝐼‘(𝑋‘0)) ∈ 𝑆)
28	6, 24, 25, 27	syl2an3an 1422	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝑋‘0) = 0 ) → (𝐼‘(𝑋‘0)) ∈ 𝑆)
29	4, 12, 26	drnginvrn0 20206	. . . . . . 7 ⊢ ((𝐾 ∈ DivRing ∧ (𝑋‘0) ∈ 𝑆 ∧ (𝑋‘0) ≠ 0 ) → (𝐼‘(𝑋‘0)) ≠ 0 )
30	6, 24, 25, 29	syl2an3an 1422	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝑋‘0) = 0 ) → (𝐼‘(𝑋‘0)) ≠ 0 )
31	1, 2, 3, 4, 5, 12, 13, 14, 28, 30	prjspnvs 40944	. . . . 5 ⊢ ((𝜑 ∧ ¬ (𝑋‘0) = 0 ) → ((𝐼‘(𝑋‘0)) · 𝑋) ∼ 𝑋)
32	11, 31	ersym 8660	. . . 4 ⊢ ((𝜑 ∧ ¬ (𝑋‘0) = 0 ) → 𝑋 ∼ ((𝐼‘(𝑋‘0)) · 𝑋))
33	10, 32	ifpimpda 1081	. . 3 ⊢ (𝜑 → if-((𝑋‘0) = 0 , 𝑋 ∼ 𝑋, 𝑋 ∼ ((𝐼‘(𝑋‘0)) · 𝑋)))
34		brif2 40646	. . 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 6841	. . . . 5 ⊢ (𝑏 = 𝑋 → (𝑏‘0) = (𝑋‘0))
38	37	eqeq1d 2738	. . . 4 ⊢ (𝑏 = 𝑋 → ((𝑏‘0) = 0 ↔ (𝑋‘0) = 0 ))
39		id 22	. . . 4 ⊢ (𝑏 = 𝑋 → 𝑏 = 𝑋)
40	37	fveq2d 6846	. . . . 5 ⊢ (𝑏 = 𝑋 → (𝐼‘(𝑏‘0)) = (𝐼‘(𝑋‘0)))
41	40, 39	oveq12d 7375	. . . 4 ⊢ (𝑏 = 𝑋 → ((𝐼‘(𝑏‘0)) · 𝑏) = ((𝐼‘(𝑋‘0)) · 𝑋))
42	38, 39, 41	ifbieq12d 4514	. . 3 ⊢ (𝑏 = 𝑋 → if((𝑏‘0) = 0 , 𝑏, ((𝐼‘(𝑏‘0)) · 𝑏)) = if((𝑋‘0) = 0 , 𝑋, ((𝐼‘(𝑋‘0)) · 𝑋)))
43		ovexd 7392	. . . 4 ⊢ (𝜑 → ((𝐼‘(𝑋‘0)) · 𝑋) ∈ V)
44	8, 43	ifexd 4534	. . 3 ⊢ (𝜑 → if((𝑋‘0) = 0 , 𝑋, ((𝐼‘(𝑋‘0)) · 𝑋)) ∈ V)
45	36, 42, 8, 44	fvmptd3 6971	. 2 ⊢ (𝜑 → (𝐹‘𝑋) = if((𝑋‘0) = 0 , 𝑋, ((𝐼‘(𝑋‘0)) · 𝑋)))
46	35, 45	breqtrrd 5133	1 ⊢ (𝜑 → 𝑋 ∼ (𝐹‘𝑋))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 if-wif 1061 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∃wrex 3073 Vcvv 3445 ∖ cdif 3907 ifcif 4486 {csn 4586 class class class wbr 5105 {copab 5167 ↦ cmpt 5188 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 Er wer 8645 0cc0 11051 ℕ₀cn0 12413 ...cfz 13424 Basecbs 17083 ·_𝑠 cvsca 17137 0_gc0g 17321 inv_rcinvr 20100 DivRingcdr 20185 freeLMod cfrlm 21152

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-ifp 1062 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-tpos 8157 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-map 8767 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-sup 9378 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-fz 13425 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-hom 17157 df-cco 17158 df-0g 17323 df-prds 17329 df-pws 17331 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-grp 18751 df-minusg 18752 df-sbg 18753 df-subg 18925 df-mgp 19897 df-ur 19914 df-ring 19966 df-oppr 20049 df-dvdsr 20070 df-unit 20071 df-invr 20101 df-drng 20187 df-subrg 20220 df-lmod 20324 df-lss 20393 df-lvec 20564 df-sra 20633 df-rgmod 20634 df-dsmm 21138 df-frlm 21153

This theorem is referenced by: prjspner1 40950

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