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Mirrors > Home > MPE Home > Th. List > lvecvscan2 | Structured version Visualization version GIF version |
Description: Cancellation law for scalar multiplication. (hvmulcan2 28849 analog.) (Contributed by NM, 2-Jul-2014.) |
Ref | Expression |
---|---|
lvecmulcan2.v | ⊢ 𝑉 = (Base‘𝑊) |
lvecmulcan2.s | ⊢ · = ( ·𝑠 ‘𝑊) |
lvecmulcan2.f | ⊢ 𝐹 = (Scalar‘𝑊) |
lvecmulcan2.k | ⊢ 𝐾 = (Base‘𝐹) |
lvecmulcan2.o | ⊢ 0 = (0g‘𝑊) |
lvecmulcan2.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
lvecmulcan2.a | ⊢ (𝜑 → 𝐴 ∈ 𝐾) |
lvecmulcan2.b | ⊢ (𝜑 → 𝐵 ∈ 𝐾) |
lvecmulcan2.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
lvecmulcan2.n | ⊢ (𝜑 → 𝑋 ≠ 0 ) |
Ref | Expression |
---|---|
lvecvscan2 | ⊢ (𝜑 → ((𝐴 · 𝑋) = (𝐵 · 𝑋) ↔ 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lvecmulcan2.n | . . . . 5 ⊢ (𝜑 → 𝑋 ≠ 0 ) | |
2 | 1 | neneqd 3021 | . . . 4 ⊢ (𝜑 → ¬ 𝑋 = 0 ) |
3 | biorf 933 | . . . . 5 ⊢ (¬ 𝑋 = 0 → ((𝐴(-g‘𝐹)𝐵) = (0g‘𝐹) ↔ (𝑋 = 0 ∨ (𝐴(-g‘𝐹)𝐵) = (0g‘𝐹)))) | |
4 | orcom 866 | . . . . 5 ⊢ ((𝑋 = 0 ∨ (𝐴(-g‘𝐹)𝐵) = (0g‘𝐹)) ↔ ((𝐴(-g‘𝐹)𝐵) = (0g‘𝐹) ∨ 𝑋 = 0 )) | |
5 | 3, 4 | syl6bb 289 | . . . 4 ⊢ (¬ 𝑋 = 0 → ((𝐴(-g‘𝐹)𝐵) = (0g‘𝐹) ↔ ((𝐴(-g‘𝐹)𝐵) = (0g‘𝐹) ∨ 𝑋 = 0 ))) |
6 | 2, 5 | syl 17 | . . 3 ⊢ (𝜑 → ((𝐴(-g‘𝐹)𝐵) = (0g‘𝐹) ↔ ((𝐴(-g‘𝐹)𝐵) = (0g‘𝐹) ∨ 𝑋 = 0 ))) |
7 | lvecmulcan2.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
8 | lvecmulcan2.s | . . . 4 ⊢ · = ( ·𝑠 ‘𝑊) | |
9 | lvecmulcan2.f | . . . 4 ⊢ 𝐹 = (Scalar‘𝑊) | |
10 | lvecmulcan2.k | . . . 4 ⊢ 𝐾 = (Base‘𝐹) | |
11 | eqid 2821 | . . . 4 ⊢ (0g‘𝐹) = (0g‘𝐹) | |
12 | lvecmulcan2.o | . . . 4 ⊢ 0 = (0g‘𝑊) | |
13 | lvecmulcan2.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
14 | lveclmod 19877 | . . . . . . 7 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
15 | 13, 14 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LMod) |
16 | 9 | lmodfgrp 19642 | . . . . . 6 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Grp) |
17 | 15, 16 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ Grp) |
18 | lvecmulcan2.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝐾) | |
19 | lvecmulcan2.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝐾) | |
20 | eqid 2821 | . . . . . 6 ⊢ (-g‘𝐹) = (-g‘𝐹) | |
21 | 10, 20 | grpsubcl 18178 | . . . . 5 ⊢ ((𝐹 ∈ Grp ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) → (𝐴(-g‘𝐹)𝐵) ∈ 𝐾) |
22 | 17, 18, 19, 21 | syl3anc 1367 | . . . 4 ⊢ (𝜑 → (𝐴(-g‘𝐹)𝐵) ∈ 𝐾) |
23 | lvecmulcan2.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
24 | 7, 8, 9, 10, 11, 12, 13, 22, 23 | lvecvs0or 19879 | . . 3 ⊢ (𝜑 → (((𝐴(-g‘𝐹)𝐵) · 𝑋) = 0 ↔ ((𝐴(-g‘𝐹)𝐵) = (0g‘𝐹) ∨ 𝑋 = 0 ))) |
25 | eqid 2821 | . . . . 5 ⊢ (-g‘𝑊) = (-g‘𝑊) | |
26 | 7, 8, 9, 10, 25, 20, 15, 18, 19, 23 | lmodsubdir 19691 | . . . 4 ⊢ (𝜑 → ((𝐴(-g‘𝐹)𝐵) · 𝑋) = ((𝐴 · 𝑋)(-g‘𝑊)(𝐵 · 𝑋))) |
27 | 26 | eqeq1d 2823 | . . 3 ⊢ (𝜑 → (((𝐴(-g‘𝐹)𝐵) · 𝑋) = 0 ↔ ((𝐴 · 𝑋)(-g‘𝑊)(𝐵 · 𝑋)) = 0 )) |
28 | 6, 24, 27 | 3bitr2rd 310 | . 2 ⊢ (𝜑 → (((𝐴 · 𝑋)(-g‘𝑊)(𝐵 · 𝑋)) = 0 ↔ (𝐴(-g‘𝐹)𝐵) = (0g‘𝐹))) |
29 | 7, 9, 8, 10 | lmodvscl 19650 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝐴 · 𝑋) ∈ 𝑉) |
30 | 15, 18, 23, 29 | syl3anc 1367 | . . 3 ⊢ (𝜑 → (𝐴 · 𝑋) ∈ 𝑉) |
31 | 7, 9, 8, 10 | lmodvscl 19650 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝐵 · 𝑋) ∈ 𝑉) |
32 | 15, 19, 23, 31 | syl3anc 1367 | . . 3 ⊢ (𝜑 → (𝐵 · 𝑋) ∈ 𝑉) |
33 | 7, 12, 25 | lmodsubeq0 19692 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝐴 · 𝑋) ∈ 𝑉 ∧ (𝐵 · 𝑋) ∈ 𝑉) → (((𝐴 · 𝑋)(-g‘𝑊)(𝐵 · 𝑋)) = 0 ↔ (𝐴 · 𝑋) = (𝐵 · 𝑋))) |
34 | 15, 30, 32, 33 | syl3anc 1367 | . 2 ⊢ (𝜑 → (((𝐴 · 𝑋)(-g‘𝑊)(𝐵 · 𝑋)) = 0 ↔ (𝐴 · 𝑋) = (𝐵 · 𝑋))) |
35 | 10, 11, 20 | grpsubeq0 18184 | . . 3 ⊢ ((𝐹 ∈ Grp ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) → ((𝐴(-g‘𝐹)𝐵) = (0g‘𝐹) ↔ 𝐴 = 𝐵)) |
36 | 17, 18, 19, 35 | syl3anc 1367 | . 2 ⊢ (𝜑 → ((𝐴(-g‘𝐹)𝐵) = (0g‘𝐹) ↔ 𝐴 = 𝐵)) |
37 | 28, 34, 36 | 3bitr3d 311 | 1 ⊢ (𝜑 → ((𝐴 · 𝑋) = (𝐵 · 𝑋) ↔ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∨ wo 843 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ‘cfv 6354 (class class class)co 7155 Basecbs 16482 Scalarcsca 16567 ·𝑠 cvsca 16568 0gc0g 16712 Grpcgrp 18102 -gcsg 18104 LModclmod 19633 LVecclvec 19873 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-tpos 7891 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-3 11700 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-mulr 16578 df-0g 16714 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-grp 18105 df-minusg 18106 df-sbg 18107 df-mgp 19239 df-ur 19251 df-ring 19298 df-oppr 19372 df-dvdsr 19390 df-unit 19391 df-invr 19421 df-drng 19503 df-lmod 19635 df-lvec 19874 |
This theorem is referenced by: lspsneu 19894 lvecindp 19909 lvecindp2 19910 linds2eq 30941 lshpsmreu 36244 lshpkrlem5 36249 hgmapval1 39028 hgmapadd 39029 hgmapmul 39030 hgmaprnlem1N 39031 hgmap11 39037 |
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