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| Mirrors > Home > MPE Home > Th. List > Mathboxes > prjspnvs | Structured version Visualization version GIF version | ||
| Description: A nonzero multiple of a vector is equivalent to the vector. This converts the equivalence relation used in prjspvs 42620 (see prjspnerlem 42627). (Contributed by SN, 8-Aug-2024.) |
| Ref | Expression |
|---|---|
| prjspnvs.e | ⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))} |
| prjspnvs.w | ⊢ 𝑊 = (𝐾 freeLMod (0...𝑁)) |
| prjspnvs.b | ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)}) |
| prjspnvs.s | ⊢ 𝑆 = (Base‘𝐾) |
| prjspnvs.x | ⊢ · = ( ·𝑠 ‘𝑊) |
| prjspnvs.0 | ⊢ 0 = (0g‘𝐾) |
| prjspnvs.k | ⊢ (𝜑 → 𝐾 ∈ DivRing) |
| prjspnvs.1 | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
| prjspnvs.2 | ⊢ (𝜑 → 𝐶 ∈ 𝑆) |
| prjspnvs.3 | ⊢ (𝜑 → 𝐶 ≠ 0 ) |
| Ref | Expression |
|---|---|
| prjspnvs | ⊢ (𝜑 → (𝐶 · 𝑋) ∼ 𝑋) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | prjspnvs.k | . . . 4 ⊢ (𝜑 → 𝐾 ∈ DivRing) | |
| 2 | ovexd 7466 | . . . 4 ⊢ (𝜑 → (0...𝑁) ∈ V) | |
| 3 | prjspnvs.w | . . . . 5 ⊢ 𝑊 = (𝐾 freeLMod (0...𝑁)) | |
| 4 | 3 | frlmlvec 21781 | . . . 4 ⊢ ((𝐾 ∈ DivRing ∧ (0...𝑁) ∈ V) → 𝑊 ∈ LVec) |
| 5 | 1, 2, 4 | syl2anc 584 | . . 3 ⊢ (𝜑 → 𝑊 ∈ LVec) |
| 6 | prjspnvs.1 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
| 7 | prjspnvs.2 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑆) | |
| 8 | prjspnvs.3 | . . . . . 6 ⊢ (𝜑 → 𝐶 ≠ 0 ) | |
| 9 | nelsn 4666 | . . . . . 6 ⊢ (𝐶 ≠ 0 → ¬ 𝐶 ∈ { 0 }) | |
| 10 | 8, 9 | syl 17 | . . . . 5 ⊢ (𝜑 → ¬ 𝐶 ∈ { 0 }) |
| 11 | 7, 10 | eldifd 3962 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝑆 ∖ { 0 })) |
| 12 | prjspnvs.s | . . . . . 6 ⊢ 𝑆 = (Base‘𝐾) | |
| 13 | 3 | frlmsca 21773 | . . . . . . . 8 ⊢ ((𝐾 ∈ DivRing ∧ (0...𝑁) ∈ V) → 𝐾 = (Scalar‘𝑊)) |
| 14 | 1, 2, 13 | syl2anc 584 | . . . . . . 7 ⊢ (𝜑 → 𝐾 = (Scalar‘𝑊)) |
| 15 | 14 | fveq2d 6910 | . . . . . 6 ⊢ (𝜑 → (Base‘𝐾) = (Base‘(Scalar‘𝑊))) |
| 16 | 12, 15 | eqtrid 2789 | . . . . 5 ⊢ (𝜑 → 𝑆 = (Base‘(Scalar‘𝑊))) |
| 17 | prjspnvs.0 | . . . . . . 7 ⊢ 0 = (0g‘𝐾) | |
| 18 | 14 | fveq2d 6910 | . . . . . . 7 ⊢ (𝜑 → (0g‘𝐾) = (0g‘(Scalar‘𝑊))) |
| 19 | 17, 18 | eqtrid 2789 | . . . . . 6 ⊢ (𝜑 → 0 = (0g‘(Scalar‘𝑊))) |
| 20 | 19 | sneqd 4638 | . . . . 5 ⊢ (𝜑 → { 0 } = {(0g‘(Scalar‘𝑊))}) |
| 21 | 16, 20 | difeq12d 4127 | . . . 4 ⊢ (𝜑 → (𝑆 ∖ { 0 }) = ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))})) |
| 22 | 11, 21 | eleqtrd 2843 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))})) |
| 23 | eqid 2737 | . . . 4 ⊢ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙 · 𝑦))} = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙 · 𝑦))} | |
| 24 | prjspnvs.b | . . . 4 ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)}) | |
| 25 | eqid 2737 | . . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
| 26 | prjspnvs.x | . . . 4 ⊢ · = ( ·𝑠 ‘𝑊) | |
| 27 | eqid 2737 | . . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
| 28 | eqid 2737 | . . . 4 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊)) | |
| 29 | 23, 24, 25, 26, 27, 28 | prjspvs 42620 | . . 3 ⊢ ((𝑊 ∈ LVec ∧ 𝑋 ∈ 𝐵 ∧ 𝐶 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))})) → (𝐶 · 𝑋){〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙 · 𝑦))}𝑋) |
| 30 | 5, 6, 22, 29 | syl3anc 1373 | . 2 ⊢ (𝜑 → (𝐶 · 𝑋){〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙 · 𝑦))}𝑋) |
| 31 | prjspnvs.e | . . . . 5 ⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))} | |
| 32 | 31, 3, 24, 12, 26 | prjspnerlem 42627 | . . . 4 ⊢ (𝐾 ∈ DivRing → ∼ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙 · 𝑦))}) |
| 33 | 1, 32 | syl 17 | . . 3 ⊢ (𝜑 → ∼ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙 · 𝑦))}) |
| 34 | 33 | breqd 5154 | . 2 ⊢ (𝜑 → ((𝐶 · 𝑋) ∼ 𝑋 ↔ (𝐶 · 𝑋){〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙 · 𝑦))}𝑋)) |
| 35 | 30, 34 | mpbird 257 | 1 ⊢ (𝜑 → (𝐶 · 𝑋) ∼ 𝑋) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∃wrex 3070 Vcvv 3480 ∖ cdif 3948 {csn 4626 class class class wbr 5143 {copab 5205 ‘cfv 6561 (class class class)co 7431 0cc0 11155 ...cfz 13547 Basecbs 17247 Scalarcsca 17300 ·𝑠 cvsca 17301 0gc0g 17484 DivRingcdr 20729 LVecclvec 21101 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-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-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-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: prjspner01 42635 |
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