prjspner01 - Mathbox for Steven Nguyen

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Theorem prjspner01 43398

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 43392	. . . . . 6 ⊢ (𝜑 → ∼ Er 𝐵)
8		prjspner01.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
9	7, 8	erref 8724	. . . . 5 ⊢ (𝜑 → 𝑋 ∼ 𝑋)
10	9	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝑋‘0) = 0 ) → 𝑋 ∼ 𝑋)
11	7	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ ¬ (𝑋‘0) = 0 ) → ∼ Er 𝐵)
12		prjspner01.0	. . . . . 6 ⊢ 0 = (0_g‘𝐾)
13	6	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝑋‘0) = 0 ) → 𝐾 ∈ DivRing)
14	8	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝑋‘0) = 0 ) → 𝑋 ∈ 𝐵)
15		ovexd 7458	. . . . . . . . 9 ⊢ (𝜑 → (0...𝑁) ∈ V)
16	8, 3	eleqtrdi 2876	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ ((Base‘𝑊) ∖ {(0_g‘𝑊)}))
17	16	eldifad 3920	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑊))
18		eqid 2766	. . . . . . . . . 10 ⊢ (Base‘𝑊) = (Base‘𝑊)
19	2, 4, 18	frlmbasf 21947	. . . . . . . . 9 ⊢ (((0...𝑁) ∈ V ∧ 𝑋 ∈ (Base‘𝑊)) → 𝑋:(0...𝑁)⟶𝑆)
20	15, 17, 19	syl2anc 596	. . . . . . . 8 ⊢ (𝜑 → 𝑋:(0...𝑁)⟶𝑆)
21		prjspner01.n	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
22		0elfz 13671	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ₀ → 0 ∈ (0...𝑁))
23	21, 22	syl 18	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ (0...𝑁))
24	20, 23	ffvelcdmd 7087	. . . . . . 7 ⊢ (𝜑 → (𝑋‘0) ∈ 𝑆)
25		neqne 2969	. . . . . . 7 ⊢ (¬ (𝑋‘0) = 0 → (𝑋‘0) ≠ 0 )
26		prjspner01.i	. . . . . . . 8 ⊢ 𝐼 = (inv_r‘𝐾)
27	4, 12, 26	drnginvrcl 20894	. . . . . . 7 ⊢ ((𝐾 ∈ DivRing ∧ (𝑋‘0) ∈ 𝑆 ∧ (𝑋‘0) ≠ 0 ) → (𝐼‘(𝑋‘0)) ∈ 𝑆)
28	6, 24, 25, 27	syl2an3an 1449	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝑋‘0) = 0 ) → (𝐼‘(𝑋‘0)) ∈ 𝑆)
29	4, 12, 26	drnginvrn0 20895	. . . . . . 7 ⊢ ((𝐾 ∈ DivRing ∧ (𝑋‘0) ∈ 𝑆 ∧ (𝑋‘0) ≠ 0 ) → (𝐼‘(𝑋‘0)) ≠ 0 )
30	6, 24, 25, 29	syl2an3an 1449	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝑋‘0) = 0 ) → (𝐼‘(𝑋‘0)) ≠ 0 )
31	1, 2, 3, 4, 5, 12, 13, 14, 28, 30	prjspnvs 43393	. . . . 5 ⊢ ((𝜑 ∧ ¬ (𝑋‘0) = 0 ) → ((𝐼‘(𝑋‘0)) · 𝑋) ∼ 𝑋)
32	11, 31	ersym 8716	. . . 4 ⊢ ((𝜑 ∧ ¬ (𝑋‘0) = 0 ) → 𝑋 ∼ ((𝐼‘(𝑋‘0)) · 𝑋))
33	10, 32	ifpimpda 1097	. . 3 ⊢ (𝜑 → if-((𝑋‘0) = 0 , 𝑋 ∼ 𝑋, 𝑋 ∼ ((𝐼‘(𝑋‘0)) · 𝑋)))
34		brif2 43036	. . 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 237	. 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 6887	. . . . 5 ⊢ (𝑏 = 𝑋 → (𝑏‘0) = (𝑋‘0))
38	37	eqeq1d 2768	. . . 4 ⊢ (𝑏 = 𝑋 → ((𝑏‘0) = 0 ↔ (𝑋‘0) = 0 ))
39		id 23	. . . 4 ⊢ (𝑏 = 𝑋 → 𝑏 = 𝑋)
40	37	fveq2d 6892	. . . . 5 ⊢ (𝑏 = 𝑋 → (𝐼‘(𝑏‘0)) = (𝐼‘(𝑋‘0)))
41	40, 39	oveq12d 7441	. . . 4 ⊢ (𝑏 = 𝑋 → ((𝐼‘(𝑏‘0)) · 𝑏) = ((𝐼‘(𝑋‘0)) · 𝑋))
42	38, 39, 41	ifbieq12d 4521	. . 3 ⊢ (𝑏 = 𝑋 → if((𝑏‘0) = 0 , 𝑏, ((𝐼‘(𝑏‘0)) · 𝑏)) = if((𝑋‘0) = 0 , 𝑋, ((𝐼‘(𝑋‘0)) · 𝑋)))
43		ovexd 7458	. . . 4 ⊢ (𝜑 → ((𝐼‘(𝑋‘0)) · 𝑋) ∈ V)
44	8, 43	ifexd 4541	. . 3 ⊢ (𝜑 → if((𝑋‘0) = 0 , 𝑋, ((𝐼‘(𝑋‘0)) · 𝑋)) ∈ V)
45	36, 42, 8, 44	fvmptd3 7020	. 2 ⊢ (𝜑 → (𝐹‘𝑋) = if((𝑋‘0) = 0 , 𝑋, ((𝐼‘(𝑋‘0)) · 𝑋)))
46	35, 45	breqtrrd 5144	1 ⊢ (𝜑 → 𝑋 ∼ (𝐹‘𝑋))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 if-wif 1078 = wceq 1570 ∈ wcel 2146 ≠ wne 2961 ∃wrex 3092 Vcvv 3458 ∖ cdif 3905 ifcif 4492 {csn 4594 class class class wbr 5114 {copab 5178 ↦ cmpt 5197 ⟶wf 6539 ‘cfv 6543 (class class class)co 7423 Er wer 8700 0cc0 11118 ℕ₀cn0 12522 ...cfz 13553 Basecbs 17294 ·_𝑠 cvsca 17339 0_gc0g 17517 inv_rcinvr 20502 DivRingcdr 20864 freeLMod cfrlm 21933

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-ifp 1079 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4878 df-iun 4963 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7872 df-1st 7995 df-2nd 7996 df-supp 8166 df-tpos 8231 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8703 df-map 8835 df-ixp 8905 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-fsupp 9332 df-sup 9412 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-nn 12252 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12523 df-z 12610 df-dec 12730 df-uz 12881 df-fz 13554 df-struct 17232 df-sets 17249 df-slot 17267 df-ndx 17279 df-base 17295 df-ress 17316 df-plusg 17348 df-mulr 17349 df-sca 17351 df-vsca 17352 df-ip 17353 df-tset 17354 df-ple 17355 df-ds 17357 df-hom 17359 df-cco 17360 df-0g 17519 df-prds 17525 df-pws 17527 df-mgm 18723 df-sgrp 18806 df-mnd 18822 df-grp 19034 df-minusg 19035 df-sbg 19036 df-subg 19220 df-cmn 19883 df-abl 19884 df-mgp 20248 df-rng 20262 df-ur 20295 df-ring 20348 df-oppr 20452 df-dvdsr 20472 df-unit 20473 df-invr 20503 df-subrg 20706 df-drng 20866 df-lmod 21020 df-lss 21090 df-lvec 21261 df-sra 21331 df-rgmod 21332 df-dsmm 21919 df-frlm 21934

This theorem is used by: prjspner1 43399

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