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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 19871 | . . . . . 6 ⊢ (𝑉 ∈ LVec → 𝑉 ∈ LMod) | |
2 | 1 | 3ad2ant1 1128 | . . . . 5 ⊢ ((𝑉 ∈ LVec ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ (𝐾 ∖ { 0 })) → 𝑉 ∈ LMod) |
3 | eldifi 4096 | . . . . . 6 ⊢ (𝑁 ∈ (𝐾 ∖ { 0 }) → 𝑁 ∈ 𝐾) | |
4 | 3 | 3ad2ant3 1130 | . . . . 5 ⊢ ((𝑉 ∈ LVec ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ (𝐾 ∖ { 0 })) → 𝑁 ∈ 𝐾) |
5 | prjspertr.b | . . . . . . . 8 ⊢ 𝐵 = ((Base‘𝑉) ∖ {(0g‘𝑉)}) | |
6 | difss 4101 | . . . . . . . 8 ⊢ ((Base‘𝑉) ∖ {(0g‘𝑉)}) ⊆ (Base‘𝑉) | |
7 | 5, 6 | eqsstri 3994 | . . . . . . 7 ⊢ 𝐵 ⊆ (Base‘𝑉) |
8 | 7 | sseli 3956 | . . . . . 6 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ (Base‘𝑉)) |
9 | 8 | 3ad2ant2 1129 | . . . . 5 ⊢ ((𝑉 ∈ LVec ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ (𝐾 ∖ { 0 })) → 𝑋 ∈ (Base‘𝑉)) |
10 | eqid 2820 | . . . . . 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 19644 | . . . . 5 ⊢ ((𝑉 ∈ LMod ∧ 𝑁 ∈ 𝐾 ∧ 𝑋 ∈ (Base‘𝑉)) → (𝑁 · 𝑋) ∈ (Base‘𝑉)) |
15 | 2, 4, 9, 14 | syl3anc 1366 | . . . 4 ⊢ ((𝑉 ∈ LVec ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ (𝐾 ∖ { 0 })) → (𝑁 · 𝑋) ∈ (Base‘𝑉)) |
16 | eldifsni 4715 | . . . . . . 7 ⊢ (𝑁 ∈ (𝐾 ∖ { 0 }) → 𝑁 ≠ 0 ) | |
17 | 16 | 3ad2ant3 1130 | . . . . . 6 ⊢ ((𝑉 ∈ LVec ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ (𝐾 ∖ { 0 })) → 𝑁 ≠ 0 ) |
18 | eldifsni 4715 | . . . . . . . 8 ⊢ (𝑋 ∈ ((Base‘𝑉) ∖ {(0g‘𝑉)}) → 𝑋 ≠ (0g‘𝑉)) | |
19 | 18, 5 | eleq2s 2930 | . . . . . . 7 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ≠ (0g‘𝑉)) |
20 | 19 | 3ad2ant2 1129 | . . . . . 6 ⊢ ((𝑉 ∈ LVec ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ (𝐾 ∖ { 0 })) → 𝑋 ≠ (0g‘𝑉)) |
21 | prjspreln0.z | . . . . . . 7 ⊢ 0 = (0g‘𝑆) | |
22 | eqid 2820 | . . . . . . 7 ⊢ (0g‘𝑉) = (0g‘𝑉) | |
23 | simp1 1131 | . . . . . . 7 ⊢ ((𝑉 ∈ LVec ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ (𝐾 ∖ { 0 })) → 𝑉 ∈ LVec) | |
24 | 10, 12, 11, 13, 21, 22, 23, 4, 9 | lvecvsn0 19874 | . . . . . 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 4598 | . . . . 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 3940 | . . 3 ⊢ ((𝑉 ∈ LVec ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ (𝐾 ∖ { 0 })) → (𝑁 · 𝑋) ∈ ((Base‘𝑉) ∖ {(0g‘𝑉)})) |
29 | 28, 5 | eleqtrrdi 2923 | . 2 ⊢ ((𝑉 ∈ LVec ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ (𝐾 ∖ { 0 })) → (𝑁 · 𝑋) ∈ 𝐵) |
30 | simp2 1132 | . 2 ⊢ ((𝑉 ∈ LVec ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ (𝐾 ∖ { 0 })) → 𝑋 ∈ 𝐵) | |
31 | oveq1 7156 | . . . . . 6 ⊢ (𝑁 = 𝑚 → (𝑁 · 𝑋) = (𝑚 · 𝑋)) | |
32 | 31 | eqcoms 2828 | . . . . 5 ⊢ (𝑚 = 𝑁 → (𝑁 · 𝑋) = (𝑚 · 𝑋)) |
33 | tbtru 1544 | . . . . 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 1546 | . . 3 ⊢ ((𝑉 ∈ LVec ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ (𝐾 ∖ { 0 })) → ⊤) | |
37 | 4, 35, 36 | rspcedvd 3623 | . 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 39331 | . 2 ⊢ ((𝑁 · 𝑋) ∼ 𝑋 ↔ (((𝑁 · 𝑋) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ ∃𝑚 ∈ 𝐾 (𝑁 · 𝑋) = (𝑚 · 𝑋))) |
40 | 29, 30, 37, 39 | syl21anbrc 1339 | 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 1082 = wceq 1536 ⊤wtru 1537 ∈ wcel 2113 ≠ wne 3015 ∃wrex 3138 ∖ cdif 3926 {csn 4560 class class class wbr 5059 {copab 5121 ‘cfv 6348 (class class class)co 7149 Basecbs 16476 Scalarcsca 16561 ·𝑠 cvsca 16562 0gc0g 16706 LModclmod 19627 LVecclvec 19867 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-tpos 7885 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-er 8282 df-en 8503 df-dom 8504 df-sdom 8505 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-nn 11632 df-2 11694 df-3 11695 df-ndx 16479 df-slot 16480 df-base 16482 df-sets 16483 df-ress 16484 df-plusg 16571 df-mulr 16572 df-0g 16708 df-mgm 17845 df-sgrp 17894 df-mnd 17905 df-grp 18099 df-minusg 18100 df-mgp 19233 df-ur 19245 df-ring 19292 df-oppr 19366 df-dvdsr 19384 df-unit 19385 df-invr 19415 df-drng 19497 df-lmod 19629 df-lvec 19868 |
This theorem is referenced by: (None) |
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