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| Mirrors > Home > MPE Home > Th. List > Mathboxes > prjspner01 | Structured version Visualization version GIF version | ||
| 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.) |
| 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) ∖ {(0g‘𝑊)}) |
| prjspner01.w | ⊢ 𝑊 = (𝐾 freeLMod (0...𝑁)) |
| prjspner01.t | ⊢ · = ( ·𝑠 ‘𝑊) |
| prjspner01.s | ⊢ 𝑆 = (Base‘𝐾) |
| prjspner01.0 | ⊢ 0 = (0g‘𝐾) |
| prjspner01.i | ⊢ 𝐼 = (invr‘𝐾) |
| prjspner01.k | ⊢ (𝜑 → 𝐾 ∈ DivRing) |
| prjspner01.n | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
| prjspner01.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
| Ref | Expression |
|---|---|
| prjspner01 | ⊢ (𝜑 → 𝑋 ∼ (𝐹‘𝑋)) |
| 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) ∖ {(0g‘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 42629 | . . . . . 6 ⊢ (𝜑 → ∼ Er 𝐵) |
| 8 | prjspner01.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
| 9 | 7, 8 | erref 8765 | . . . . 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 = (0g‘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 7466 | . . . . . . . . 9 ⊢ (𝜑 → (0...𝑁) ∈ V) | |
| 16 | 8, 3 | eleqtrdi 2851 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ ((Base‘𝑊) ∖ {(0g‘𝑊)})) |
| 17 | 16 | eldifad 3963 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑊)) |
| 18 | eqid 2737 | . . . . . . . . . 10 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
| 19 | 2, 4, 18 | frlmbasf 21780 | . . . . . . . . 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 13664 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → 0 ∈ (0...𝑁)) | |
| 23 | 21, 22 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 0 ∈ (0...𝑁)) |
| 24 | 20, 23 | ffvelcdmd 7105 | . . . . . . 7 ⊢ (𝜑 → (𝑋‘0) ∈ 𝑆) |
| 25 | neqne 2948 | . . . . . . 7 ⊢ (¬ (𝑋‘0) = 0 → (𝑋‘0) ≠ 0 ) | |
| 26 | prjspner01.i | . . . . . . . 8 ⊢ 𝐼 = (invr‘𝐾) | |
| 27 | 4, 12, 26 | drnginvrcl 20753 | . . . . . . 7 ⊢ ((𝐾 ∈ DivRing ∧ (𝑋‘0) ∈ 𝑆 ∧ (𝑋‘0) ≠ 0 ) → (𝐼‘(𝑋‘0)) ∈ 𝑆) |
| 28 | 6, 24, 25, 27 | syl2an3an 1424 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝑋‘0) = 0 ) → (𝐼‘(𝑋‘0)) ∈ 𝑆) |
| 29 | 4, 12, 26 | drnginvrn0 20754 | . . . . . . 7 ⊢ ((𝐾 ∈ DivRing ∧ (𝑋‘0) ∈ 𝑆 ∧ (𝑋‘0) ≠ 0 ) → (𝐼‘(𝑋‘0)) ≠ 0 ) |
| 30 | 6, 24, 25, 29 | syl2an3an 1424 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝑋‘0) = 0 ) → (𝐼‘(𝑋‘0)) ≠ 0 ) |
| 31 | 1, 2, 3, 4, 5, 12, 13, 14, 28, 30 | prjspnvs 42630 | . . . . 5 ⊢ ((𝜑 ∧ ¬ (𝑋‘0) = 0 ) → ((𝐼‘(𝑋‘0)) · 𝑋) ∼ 𝑋) |
| 32 | 11, 31 | ersym 8757 | . . . 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 42263 | . . 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 6905 | . . . . 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 6910 | . . . . 5 ⊢ (𝑏 = 𝑋 → (𝐼‘(𝑏‘0)) = (𝐼‘(𝑋‘0))) |
| 41 | 40, 39 | oveq12d 7449 | . . . 4 ⊢ (𝑏 = 𝑋 → ((𝐼‘(𝑏‘0)) · 𝑏) = ((𝐼‘(𝑋‘0)) · 𝑋)) |
| 42 | 38, 39, 41 | ifbieq12d 4554 | . . 3 ⊢ (𝑏 = 𝑋 → if((𝑏‘0) = 0 , 𝑏, ((𝐼‘(𝑏‘0)) · 𝑏)) = if((𝑋‘0) = 0 , 𝑋, ((𝐼‘(𝑋‘0)) · 𝑋))) |
| 43 | ovexd 7466 | . . . 4 ⊢ (𝜑 → ((𝐼‘(𝑋‘0)) · 𝑋) ∈ V) | |
| 44 | 8, 43 | ifexd 4574 | . . 3 ⊢ (𝜑 → if((𝑋‘0) = 0 , 𝑋, ((𝐼‘(𝑋‘0)) · 𝑋)) ∈ V) |
| 45 | 36, 42, 8, 44 | fvmptd3 7039 | . 2 ⊢ (𝜑 → (𝐹‘𝑋) = if((𝑋‘0) = 0 , 𝑋, ((𝐼‘(𝑋‘0)) · 𝑋))) |
| 46 | 35, 45 | breqtrrd 5171 | 1 ⊢ (𝜑 → 𝑋 ∼ (𝐹‘𝑋)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 if-wif 1063 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∃wrex 3070 Vcvv 3480 ∖ cdif 3948 ifcif 4525 {csn 4626 class class class wbr 5143 {copab 5205 ↦ cmpt 5225 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 Er wer 8742 0cc0 11155 ℕ0cn0 12526 ...cfz 13547 Basecbs 17247 ·𝑠 cvsca 17301 0gc0g 17484 invrcinvr 20387 DivRingcdr 20729 freeLMod cfrlm 21766 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| 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 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-sup 9482 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-hom 17321 df-cco 17322 df-0g 17486 df-prds 17492 df-pws 17494 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-sbg 18956 df-subg 19141 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-subrg 20570 df-drng 20731 df-lmod 20860 df-lss 20930 df-lvec 21102 df-sra 21172 df-rgmod 21173 df-dsmm 21752 df-frlm 21767 |
| This theorem is referenced by: prjspner1 42636 |
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