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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 40010 (see prjspnerlem 40017). (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 7199 | . . . 4 ⊢ (𝜑 → (0...𝑁) ∈ V) | |
3 | prjspnvs.w | . . . . 5 ⊢ 𝑊 = (𝐾 freeLMod (0...𝑁)) | |
4 | 3 | frlmlvec 20570 | . . . 4 ⊢ ((𝐾 ∈ DivRing ∧ (0...𝑁) ∈ V) → 𝑊 ∈ LVec) |
5 | 1, 2, 4 | syl2anc 587 | . . 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 4553 | . . . . . 6 ⊢ (𝐶 ≠ 0 → ¬ 𝐶 ∈ { 0 }) | |
10 | 8, 9 | syl 17 | . . . . 5 ⊢ (𝜑 → ¬ 𝐶 ∈ { 0 }) |
11 | 7, 10 | eldifd 3852 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝑆 ∖ { 0 })) |
12 | prjspnvs.s | . . . . . 6 ⊢ 𝑆 = (Base‘𝐾) | |
13 | 3 | frlmsca 20562 | . . . . . . . 8 ⊢ ((𝐾 ∈ DivRing ∧ (0...𝑁) ∈ V) → 𝐾 = (Scalar‘𝑊)) |
14 | 1, 2, 13 | syl2anc 587 | . . . . . . 7 ⊢ (𝜑 → 𝐾 = (Scalar‘𝑊)) |
15 | 14 | fveq2d 6672 | . . . . . 6 ⊢ (𝜑 → (Base‘𝐾) = (Base‘(Scalar‘𝑊))) |
16 | 12, 15 | syl5eq 2785 | . . . . 5 ⊢ (𝜑 → 𝑆 = (Base‘(Scalar‘𝑊))) |
17 | prjspnvs.0 | . . . . . . 7 ⊢ 0 = (0g‘𝐾) | |
18 | 14 | fveq2d 6672 | . . . . . . 7 ⊢ (𝜑 → (0g‘𝐾) = (0g‘(Scalar‘𝑊))) |
19 | 17, 18 | syl5eq 2785 | . . . . . 6 ⊢ (𝜑 → 0 = (0g‘(Scalar‘𝑊))) |
20 | 19 | sneqd 4525 | . . . . 5 ⊢ (𝜑 → { 0 } = {(0g‘(Scalar‘𝑊))}) |
21 | 16, 20 | difeq12d 4012 | . . . 4 ⊢ (𝜑 → (𝑆 ∖ { 0 }) = ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))})) |
22 | 11, 21 | eleqtrd 2835 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))})) |
23 | eqid 2738 | . . . 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 2738 | . . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
26 | prjspnvs.x | . . . 4 ⊢ · = ( ·𝑠 ‘𝑊) | |
27 | eqid 2738 | . . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
28 | eqid 2738 | . . . 4 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊)) | |
29 | 23, 24, 25, 26, 27, 28 | prjspvs 40010 | . . 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 1372 | . 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 40017 | . . . 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 5038 | . 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 260 | 1 ⊢ (𝜑 → (𝐶 · 𝑋) ∼ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2113 ≠ wne 2934 ∃wrex 3054 Vcvv 3397 ∖ cdif 3838 {csn 4513 class class class wbr 5027 {copab 5089 ‘cfv 6333 (class class class)co 7164 0cc0 10608 ...cfz 12974 Basecbs 16579 Scalarcsca 16664 ·𝑠 cvsca 16665 0gc0g 16809 DivRingcdr 19614 LVecclvec 19986 freeLMod cfrlm 20555 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-om 7594 df-1st 7707 df-2nd 7708 df-tpos 7914 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-1o 8124 df-er 8313 df-map 8432 df-ixp 8501 df-en 8549 df-dom 8550 df-sdom 8551 df-fin 8552 df-sup 8972 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-nn 11710 df-2 11772 df-3 11773 df-4 11774 df-5 11775 df-6 11776 df-7 11777 df-8 11778 df-9 11779 df-n0 11970 df-z 12056 df-dec 12173 df-uz 12318 df-fz 12975 df-struct 16581 df-ndx 16582 df-slot 16583 df-base 16585 df-sets 16586 df-ress 16587 df-plusg 16674 df-mulr 16675 df-sca 16677 df-vsca 16678 df-ip 16679 df-tset 16680 df-ple 16681 df-ds 16683 df-hom 16685 df-cco 16686 df-0g 16811 df-prds 16817 df-pws 16819 df-mgm 17961 df-sgrp 18010 df-mnd 18021 df-grp 18215 df-minusg 18216 df-sbg 18217 df-subg 18387 df-mgp 19352 df-ur 19364 df-ring 19411 df-oppr 19488 df-dvdsr 19506 df-unit 19507 df-invr 19537 df-drng 19616 df-subrg 19645 df-lmod 19748 df-lss 19816 df-lvec 19987 df-sra 20056 df-rgmod 20057 df-dsmm 20541 df-frlm 20556 |
This theorem is referenced by: prjspner01 40023 |
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