prjspnequivnorm - Mathbox for Steven Nguyen

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Theorem prjspnequivnorm 43640

Description: In a free module, a nonzero vector is equivalent to its normalized vector. (Contributed by SN, 24-Sep-2026.)

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

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

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

**Proof of Theorem prjspnequivnorm**
Step	Hyp	Ref	Expression
1		prjspnnorm.e	. . . 4 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))}
2		prjspnnorm.w	. . . 4 ⊢ 𝑊 = (𝐾 freeLMod (0...𝑁))
3		prjspnnorm.b	. . . 4 ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0_g‘𝑊)})
4		prjspnnorm.s	. . . 4 ⊢ 𝑆 = (Base‘𝐾)
5		prjspnnorm.t	. . . 4 ⊢ · = ( ·_𝑠 ‘𝑊)
6		prjspnnorm.k	. . . 4 ⊢ (𝜑 → 𝐾 ∈ DivRing)
7	1, 2, 3, 4, 5, 6	prjspner 43627	. . 3 ⊢ (𝜑 → ∼ Er 𝐵)
8		eqid 2761	. . . 4 ⊢ (0_g‘𝐾) = (0_g‘𝐾)
9		prjspnnorm.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
10		prjspnnorm.i	. . . . 5 ⊢ 𝐼 = (inv_r‘𝐾)
11		prjspnnorm.j	. . . . . 6 ⊢ 𝐽 = (𝑏 ∈ 𝐵 ↦ inf({𝑖 ∈ (0...𝑁) ∣ (𝑏‘𝑖) ≠ (0_g‘𝐾)}, ℝ, < ))
12	6	drngringd 20981	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ Ring)
13		prjspnnorm.n	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
14	11, 2, 3, 12, 13, 9, 4	frlmnzcoordcl2 43636	. . . . 5 ⊢ (𝜑 → (𝑋‘(𝐽‘𝑋)) ∈ 𝑆)
15	11, 2, 3, 12, 13, 9	frlmnzcoordn0 43637	. . . . 5 ⊢ (𝜑 → (𝑋‘(𝐽‘𝑋)) ≠ (0_g‘𝐾))
16	4, 8, 10, 6, 14, 15	drnginvrcld 21006	. . . 4 ⊢ (𝜑 → (𝐼‘(𝑋‘(𝐽‘𝑋))) ∈ 𝑆)
17	4, 8, 10, 6, 14, 15	drnginvrn0d 43565	. . . 4 ⊢ (𝜑 → (𝐼‘(𝑋‘(𝐽‘𝑋))) ≠ (0_g‘𝐾))
18	1, 2, 3, 4, 5, 8, 6, 9, 16, 17	prjspnvs 43628	. . 3 ⊢ (𝜑 → ((𝐼‘(𝑋‘(𝐽‘𝑋))) · 𝑋) ∼ 𝑋)
19	7, 18	ersym 8723	. 2 ⊢ (𝜑 → 𝑋 ∼ ((𝐼‘(𝑋‘(𝐽‘𝑋))) · 𝑋))
20		prjspnnorm.f	. . 3 ⊢ 𝐹 = (𝑣 ∈ 𝐵 ↦ ((𝐼‘(𝑣‘(𝐽‘𝑣))) · 𝑣))
21	20, 9	prjspnnormval 43639	. 2 ⊢ (𝜑 → (𝐹‘𝑋) = ((𝐼‘(𝑋‘(𝐽‘𝑋))) · 𝑋))
22	19, 21	breqtrrd 5133	1 ⊢ (𝜑 → 𝑋 ∼ (𝐹‘𝑋))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ≠ wne 2956 ∃wrex 3087 {crab 3413 ∖ cdif 3896 {csn 4584 class class class wbr 5103 {copab 5167 ↦ cmpt 5186 ‘cfv 6537 (class class class)co 7418 infcinf 9426 ℝcr 11192 0cc0 11193 < clt 11336 ℕ₀cn0 12599 ...cfz 13632 Basecbs 17380 ·_𝑠 cvsca 17425 0_gc0g 17603 inv_rcinvr 20610 DivRingcdr 20973 freeLMod cfrlm 22045

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 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7749 ax-cnex 11249 ax-resscn 11250 ax-1cn 11251 ax-icn 11252 ax-addcl 11253 ax-addrcl 11254 ax-mulcl 11255 ax-mulrcl 11256 ax-mulcom 11257 ax-addass 11258 ax-mulass 11259 ax-distr 11260 ax-i2m1 11261 ax-1ne0 11262 ax-1rid 11263 ax-rnegex 11264 ax-rrecex 11265 ax-cnre 11266 ax-pre-lttri 11267 ax-pre-lttrn 11268 ax-pre-ltadd 11269 ax-pre-mulgt0 11270

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7375 df-ov 7421 df-oprab 7422 df-mpo 7423 df-om 7876 df-1st 7999 df-2nd 8000 df-supp 8171 df-tpos 8236 df-frecs 8292 df-wrecs 8323 df-recs 8372 df-rdg 8411 df-1o 8469 df-er 8710 df-map 8842 df-ixp 8919 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-fsupp 9347 df-sup 9427 df-inf 9428 df-pnf 11338 df-mnf 11339 df-xr 11340 df-ltxr 11341 df-le 11342 df-sub 11536 df-neg 11537 df-nn 12329 df-2 12398 df-3 12399 df-4 12400 df-5 12401 df-6 12402 df-7 12403 df-8 12404 df-9 12405 df-n0 12600 df-z 12687 df-dec 12808 df-uz 12959 df-fz 13633 df-struct 17318 df-sets 17335 df-slot 17353 df-ndx 17365 df-base 17381 df-ress 17402 df-plusg 17434 df-mulr 17435 df-sca 17437 df-vsca 17438 df-ip 17439 df-tset 17440 df-ple 17441 df-ds 17443 df-hom 17445 df-cco 17446 df-0g 17605 df-prds 17611 df-pws 17613 df-mgm 18809 df-sgrp 18901 df-mnd 18917 df-grp 19140 df-minusg 19141 df-sbg 19142 df-subg 19326 df-cmn 19989 df-abl 19990 df-mgp 20354 df-rng 20368 df-ur 20401 df-ring 20454 df-oppr 20560 df-dvdsr 20580 df-unit 20581 df-invr 20611 df-subrg 20815 df-drng 20975 df-lmod 21130 df-lss 21200 df-lvec 21371 df-sra 21441 df-rgmod 21442 df-dsmm 22031 df-frlm 22046

This theorem is used by: prjspnnorm 43641

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