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Mirrors > Home > MPE Home > Th. List > Mathboxes > prjspvs | Structured version Visualization version GIF version |
Description: A nonzero multiple of a vector is equivalent to the vector. (Contributed by Steven Nguyen, 6-Jun-2023.) |
Ref | Expression |
---|---|
prjsprel.1 | ⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))} |
prjspertr.b | ⊢ 𝐵 = ((Base‘𝑉) ∖ {(0g‘𝑉)}) |
prjspertr.s | ⊢ 𝑆 = (Scalar‘𝑉) |
prjspertr.x | ⊢ · = ( ·𝑠 ‘𝑉) |
prjspertr.k | ⊢ 𝐾 = (Base‘𝑆) |
prjspreln0.z | ⊢ 0 = (0g‘𝑆) |
Ref | Expression |
---|---|
prjspvs | ⊢ ((𝑉 ∈ LVec ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ (𝐾 ∖ { 0 })) → (𝑁 · 𝑋) ∼ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lveclmod 19878 | . . . . . 6 ⊢ (𝑉 ∈ LVec → 𝑉 ∈ LMod) | |
2 | 1 | 3ad2ant1 1129 | . . . . 5 ⊢ ((𝑉 ∈ LVec ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ (𝐾 ∖ { 0 })) → 𝑉 ∈ LMod) |
3 | eldifi 4103 | . . . . . 6 ⊢ (𝑁 ∈ (𝐾 ∖ { 0 }) → 𝑁 ∈ 𝐾) | |
4 | 3 | 3ad2ant3 1131 | . . . . 5 ⊢ ((𝑉 ∈ LVec ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ (𝐾 ∖ { 0 })) → 𝑁 ∈ 𝐾) |
5 | prjspertr.b | . . . . . . . 8 ⊢ 𝐵 = ((Base‘𝑉) ∖ {(0g‘𝑉)}) | |
6 | difss 4108 | . . . . . . . 8 ⊢ ((Base‘𝑉) ∖ {(0g‘𝑉)}) ⊆ (Base‘𝑉) | |
7 | 5, 6 | eqsstri 4001 | . . . . . . 7 ⊢ 𝐵 ⊆ (Base‘𝑉) |
8 | 7 | sseli 3963 | . . . . . 6 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ (Base‘𝑉)) |
9 | 8 | 3ad2ant2 1130 | . . . . 5 ⊢ ((𝑉 ∈ LVec ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ (𝐾 ∖ { 0 })) → 𝑋 ∈ (Base‘𝑉)) |
10 | eqid 2821 | . . . . . 6 ⊢ (Base‘𝑉) = (Base‘𝑉) | |
11 | prjspertr.s | . . . . . 6 ⊢ 𝑆 = (Scalar‘𝑉) | |
12 | prjspertr.x | . . . . . 6 ⊢ · = ( ·𝑠 ‘𝑉) | |
13 | prjspertr.k | . . . . . 6 ⊢ 𝐾 = (Base‘𝑆) | |
14 | 10, 11, 12, 13 | lmodvscl 19651 | . . . . 5 ⊢ ((𝑉 ∈ LMod ∧ 𝑁 ∈ 𝐾 ∧ 𝑋 ∈ (Base‘𝑉)) → (𝑁 · 𝑋) ∈ (Base‘𝑉)) |
15 | 2, 4, 9, 14 | syl3anc 1367 | . . . 4 ⊢ ((𝑉 ∈ LVec ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ (𝐾 ∖ { 0 })) → (𝑁 · 𝑋) ∈ (Base‘𝑉)) |
16 | eldifsni 4722 | . . . . . . 7 ⊢ (𝑁 ∈ (𝐾 ∖ { 0 }) → 𝑁 ≠ 0 ) | |
17 | 16 | 3ad2ant3 1131 | . . . . . 6 ⊢ ((𝑉 ∈ LVec ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ (𝐾 ∖ { 0 })) → 𝑁 ≠ 0 ) |
18 | eldifsni 4722 | . . . . . . . 8 ⊢ (𝑋 ∈ ((Base‘𝑉) ∖ {(0g‘𝑉)}) → 𝑋 ≠ (0g‘𝑉)) | |
19 | 18, 5 | eleq2s 2931 | . . . . . . 7 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ≠ (0g‘𝑉)) |
20 | 19 | 3ad2ant2 1130 | . . . . . 6 ⊢ ((𝑉 ∈ LVec ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ (𝐾 ∖ { 0 })) → 𝑋 ≠ (0g‘𝑉)) |
21 | prjspreln0.z | . . . . . . 7 ⊢ 0 = (0g‘𝑆) | |
22 | eqid 2821 | . . . . . . 7 ⊢ (0g‘𝑉) = (0g‘𝑉) | |
23 | simp1 1132 | . . . . . . 7 ⊢ ((𝑉 ∈ LVec ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ (𝐾 ∖ { 0 })) → 𝑉 ∈ LVec) | |
24 | 10, 12, 11, 13, 21, 22, 23, 4, 9 | lvecvsn0 19881 | . . . . . 6 ⊢ ((𝑉 ∈ LVec ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ (𝐾 ∖ { 0 })) → ((𝑁 · 𝑋) ≠ (0g‘𝑉) ↔ (𝑁 ≠ 0 ∧ 𝑋 ≠ (0g‘𝑉)))) |
25 | 17, 20, 24 | mpbir2and 711 | . . . . 5 ⊢ ((𝑉 ∈ LVec ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ (𝐾 ∖ { 0 })) → (𝑁 · 𝑋) ≠ (0g‘𝑉)) |
26 | nelsn 4605 | . . . . 5 ⊢ ((𝑁 · 𝑋) ≠ (0g‘𝑉) → ¬ (𝑁 · 𝑋) ∈ {(0g‘𝑉)}) | |
27 | 25, 26 | syl 17 | . . . 4 ⊢ ((𝑉 ∈ LVec ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ (𝐾 ∖ { 0 })) → ¬ (𝑁 · 𝑋) ∈ {(0g‘𝑉)}) |
28 | 15, 27 | eldifd 3947 | . . 3 ⊢ ((𝑉 ∈ LVec ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ (𝐾 ∖ { 0 })) → (𝑁 · 𝑋) ∈ ((Base‘𝑉) ∖ {(0g‘𝑉)})) |
29 | 28, 5 | eleqtrrdi 2924 | . 2 ⊢ ((𝑉 ∈ LVec ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ (𝐾 ∖ { 0 })) → (𝑁 · 𝑋) ∈ 𝐵) |
30 | simp2 1133 | . 2 ⊢ ((𝑉 ∈ LVec ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ (𝐾 ∖ { 0 })) → 𝑋 ∈ 𝐵) | |
31 | oveq1 7163 | . . . . . 6 ⊢ (𝑁 = 𝑚 → (𝑁 · 𝑋) = (𝑚 · 𝑋)) | |
32 | 31 | eqcoms 2829 | . . . . 5 ⊢ (𝑚 = 𝑁 → (𝑁 · 𝑋) = (𝑚 · 𝑋)) |
33 | tbtru 1545 | . . . . 5 ⊢ ((𝑁 · 𝑋) = (𝑚 · 𝑋) ↔ ((𝑁 · 𝑋) = (𝑚 · 𝑋) ↔ ⊤)) | |
34 | 32, 33 | sylib 220 | . . . 4 ⊢ (𝑚 = 𝑁 → ((𝑁 · 𝑋) = (𝑚 · 𝑋) ↔ ⊤)) |
35 | 34 | adantl 484 | . . 3 ⊢ (((𝑉 ∈ LVec ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ (𝐾 ∖ { 0 })) ∧ 𝑚 = 𝑁) → ((𝑁 · 𝑋) = (𝑚 · 𝑋) ↔ ⊤)) |
36 | trud 1547 | . . 3 ⊢ ((𝑉 ∈ LVec ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ (𝐾 ∖ { 0 })) → ⊤) | |
37 | 4, 35, 36 | rspcedvd 3626 | . 2 ⊢ ((𝑉 ∈ LVec ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ (𝐾 ∖ { 0 })) → ∃𝑚 ∈ 𝐾 (𝑁 · 𝑋) = (𝑚 · 𝑋)) |
38 | prjsprel.1 | . . 3 ⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))} | |
39 | 38 | prjsprel 39303 | . 2 ⊢ ((𝑁 · 𝑋) ∼ 𝑋 ↔ (((𝑁 · 𝑋) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ ∃𝑚 ∈ 𝐾 (𝑁 · 𝑋) = (𝑚 · 𝑋))) |
40 | 29, 30, 37, 39 | syl21anbrc 1340 | 1 ⊢ ((𝑉 ∈ LVec ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ (𝐾 ∖ { 0 })) → (𝑁 · 𝑋) ∼ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ⊤wtru 1538 ∈ wcel 2114 ≠ wne 3016 ∃wrex 3139 ∖ cdif 3933 {csn 4567 class class class wbr 5066 {copab 5128 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 Scalarcsca 16568 ·𝑠 cvsca 16569 0gc0g 16713 LModclmod 19634 LVecclvec 19874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-tpos 7892 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 df-mgp 19240 df-ur 19252 df-ring 19299 df-oppr 19373 df-dvdsr 19391 df-unit 19392 df-invr 19422 df-drng 19504 df-lmod 19636 df-lvec 19875 |
This theorem is referenced by: (None) |
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