prjspner01 - Mathbox for Steven Nguyen

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Theorem prjspner01 42977

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 42971	. . . . . 6 ⊢ (𝜑 → ∼ Er 𝐵)
8		prjspner01.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
9	7, 8	erref 8666	. . . . 5 ⊢ (𝜑 → 𝑋 ∼ 𝑋)
10	9	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑋‘0) = 0 ) → 𝑋 ∼ 𝑋)
11	7	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ¬ (𝑋‘0) = 0 ) → ∼ Er 𝐵)
12		prjspner01.0	. . . . . 6 ⊢ 0 = (0_g‘𝐾)
13	6	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝑋‘0) = 0 ) → 𝐾 ∈ DivRing)
14	8	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝑋‘0) = 0 ) → 𝑋 ∈ 𝐵)
15		ovexd 7403	. . . . . . . . 9 ⊢ (𝜑 → (0...𝑁) ∈ V)
16	8, 3	eleqtrdi 2847	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ ((Base‘𝑊) ∖ {(0_g‘𝑊)}))
17	16	eldifad 3915	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑊))
18		eqid 2737	. . . . . . . . . 10 ⊢ (Base‘𝑊) = (Base‘𝑊)
19	2, 4, 18	frlmbasf 21727	. . . . . . . . 9 ⊢ (((0...𝑁) ∈ V ∧ 𝑋 ∈ (Base‘𝑊)) → 𝑋:(0...𝑁)⟶𝑆)
20	15, 17, 19	syl2anc 585	. . . . . . . 8 ⊢ (𝜑 → 𝑋:(0...𝑁)⟶𝑆)
21		prjspner01.n	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
22		0elfz 13552	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ₀ → 0 ∈ (0...𝑁))
23	21, 22	syl 17	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ (0...𝑁))
24	20, 23	ffvelcdmd 7039	. . . . . . 7 ⊢ (𝜑 → (𝑋‘0) ∈ 𝑆)
25		neqne 2941	. . . . . . 7 ⊢ (¬ (𝑋‘0) = 0 → (𝑋‘0) ≠ 0 )
26		prjspner01.i	. . . . . . . 8 ⊢ 𝐼 = (inv_r‘𝐾)
27	4, 12, 26	drnginvrcl 20698	. . . . . . 7 ⊢ ((𝐾 ∈ DivRing ∧ (𝑋‘0) ∈ 𝑆 ∧ (𝑋‘0) ≠ 0 ) → (𝐼‘(𝑋‘0)) ∈ 𝑆)
28	6, 24, 25, 27	syl2an3an 1425	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝑋‘0) = 0 ) → (𝐼‘(𝑋‘0)) ∈ 𝑆)
29	4, 12, 26	drnginvrn0 20699	. . . . . . 7 ⊢ ((𝐾 ∈ DivRing ∧ (𝑋‘0) ∈ 𝑆 ∧ (𝑋‘0) ≠ 0 ) → (𝐼‘(𝑋‘0)) ≠ 0 )
30	6, 24, 25, 29	syl2an3an 1425	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝑋‘0) = 0 ) → (𝐼‘(𝑋‘0)) ≠ 0 )
31	1, 2, 3, 4, 5, 12, 13, 14, 28, 30	prjspnvs 42972	. . . . 5 ⊢ ((𝜑 ∧ ¬ (𝑋‘0) = 0 ) → ((𝐼‘(𝑋‘0)) · 𝑋) ∼ 𝑋)
32	11, 31	ersym 8658	. . . 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 42590	. . 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 234	. 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 2739	. . . 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 7386	. . . 4 ⊢ (𝑏 = 𝑋 → ((𝐼‘(𝑏‘0)) · 𝑏) = ((𝐼‘(𝑋‘0)) · 𝑋))
42	38, 39, 41	ifbieq12d 4510	. . 3 ⊢ (𝑏 = 𝑋 → if((𝑏‘0) = 0 , 𝑏, ((𝐼‘(𝑏‘0)) · 𝑏)) = if((𝑋‘0) = 0 , 𝑋, ((𝐼‘(𝑋‘0)) · 𝑋)))
43		ovexd 7403	. . . 4 ⊢ (𝜑 → ((𝐼‘(𝑋‘0)) · 𝑋) ∈ V)
44	8, 43	ifexd 4530	. . 3 ⊢ (𝜑 → if((𝑋‘0) = 0 , 𝑋, ((𝐼‘(𝑋‘0)) · 𝑋)) ∈ V)
45	36, 42, 8, 44	fvmptd3 6973	. 2 ⊢ (𝜑 → (𝐹‘𝑋) = if((𝑋‘0) = 0 , 𝑋, ((𝐼‘(𝑋‘0)) · 𝑋)))
46	35, 45	breqtrrd 5128	1 ⊢ (𝜑 → 𝑋 ∼ (𝐹‘𝑋))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 if-wif 1063 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∃wrex 3062 Vcvv 3442 ∖ cdif 3900 ifcif 4481 {csn 4582 class class class wbr 5100 {copab 5162 ↦ cmpt 5181 ⟶wf 6496 ‘cfv 6500 (class class class)co 7368 Er wer 8642 0cc0 11038 ℕ₀cn0 12413 ...cfz 13435 Basecbs 17148 ·_𝑠 cvsca 17193 0_gc0g 17371 inv_rcinvr 20335 DivRingcdr 20674 freeLMod cfrlm 21713

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ifp 1064 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-supp 8113 df-tpos 8178 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-er 8645 df-map 8777 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9277 df-sup 9357 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-z 12501 df-dec 12620 df-uz 12764 df-fz 13436 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-sca 17205 df-vsca 17206 df-ip 17207 df-tset 17208 df-ple 17209 df-ds 17211 df-hom 17213 df-cco 17214 df-0g 17373 df-prds 17379 df-pws 17381 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-grp 18878 df-minusg 18879 df-sbg 18880 df-subg 19065 df-cmn 19723 df-abl 19724 df-mgp 20088 df-rng 20100 df-ur 20129 df-ring 20182 df-oppr 20285 df-dvdsr 20305 df-unit 20306 df-invr 20336 df-subrg 20515 df-drng 20676 df-lmod 20825 df-lss 20895 df-lvec 21067 df-sra 21137 df-rgmod 21138 df-dsmm 21699 df-frlm 21714

This theorem is referenced by: prjspner1 42978

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