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Mirrors > Home > MPE Home > Th. List > lvecvscan2 | Structured version Visualization version GIF version |
Description: Cancellation law for scalar multiplication. (hvmulcan2 28260 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 2937 | . . . 4 ⊢ (𝜑 → ¬ 𝑋 = 0 ) |
3 | biorf 419 | . . . . 5 ⊢ (¬ 𝑋 = 0 → ((𝐴(-g‘𝐹)𝐵) = (0g‘𝐹) ↔ (𝑋 = 0 ∨ (𝐴(-g‘𝐹)𝐵) = (0g‘𝐹)))) | |
4 | orcom 401 | . . . . 5 ⊢ ((𝑋 = 0 ∨ (𝐴(-g‘𝐹)𝐵) = (0g‘𝐹)) ↔ ((𝐴(-g‘𝐹)𝐵) = (0g‘𝐹) ∨ 𝑋 = 0 )) | |
5 | 3, 4 | syl6bb 276 | . . . 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 2760 | . . . 4 ⊢ (0g‘𝐹) = (0g‘𝐹) | |
12 | lvecmulcan2.o | . . . 4 ⊢ 0 = (0g‘𝑊) | |
13 | lvecmulcan2.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
14 | lveclmod 19328 | . . . . . . 7 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
15 | 13, 14 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LMod) |
16 | 9 | lmodfgrp 19094 | . . . . . 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 2760 | . . . . . 6 ⊢ (-g‘𝐹) = (-g‘𝐹) | |
21 | 10, 20 | grpsubcl 17716 | . . . . 5 ⊢ ((𝐹 ∈ Grp ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) → (𝐴(-g‘𝐹)𝐵) ∈ 𝐾) |
22 | 17, 18, 19, 21 | syl3anc 1477 | . . . 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 19330 | . . 3 ⊢ (𝜑 → (((𝐴(-g‘𝐹)𝐵) · 𝑋) = 0 ↔ ((𝐴(-g‘𝐹)𝐵) = (0g‘𝐹) ∨ 𝑋 = 0 ))) |
25 | eqid 2760 | . . . . 5 ⊢ (-g‘𝑊) = (-g‘𝑊) | |
26 | 7, 8, 9, 10, 25, 20, 15, 18, 19, 23 | lmodsubdir 19143 | . . . 4 ⊢ (𝜑 → ((𝐴(-g‘𝐹)𝐵) · 𝑋) = ((𝐴 · 𝑋)(-g‘𝑊)(𝐵 · 𝑋))) |
27 | 26 | eqeq1d 2762 | . . 3 ⊢ (𝜑 → (((𝐴(-g‘𝐹)𝐵) · 𝑋) = 0 ↔ ((𝐴 · 𝑋)(-g‘𝑊)(𝐵 · 𝑋)) = 0 )) |
28 | 6, 24, 27 | 3bitr2rd 297 | . 2 ⊢ (𝜑 → (((𝐴 · 𝑋)(-g‘𝑊)(𝐵 · 𝑋)) = 0 ↔ (𝐴(-g‘𝐹)𝐵) = (0g‘𝐹))) |
29 | 7, 9, 8, 10 | lmodvscl 19102 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝐴 · 𝑋) ∈ 𝑉) |
30 | 15, 18, 23, 29 | syl3anc 1477 | . . 3 ⊢ (𝜑 → (𝐴 · 𝑋) ∈ 𝑉) |
31 | 7, 9, 8, 10 | lmodvscl 19102 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝐵 · 𝑋) ∈ 𝑉) |
32 | 15, 19, 23, 31 | syl3anc 1477 | . . 3 ⊢ (𝜑 → (𝐵 · 𝑋) ∈ 𝑉) |
33 | 7, 12, 25 | lmodsubeq0 19144 | . . 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 1477 | . 2 ⊢ (𝜑 → (((𝐴 · 𝑋)(-g‘𝑊)(𝐵 · 𝑋)) = 0 ↔ (𝐴 · 𝑋) = (𝐵 · 𝑋))) |
35 | 10, 11, 20 | grpsubeq0 17722 | . . 3 ⊢ ((𝐹 ∈ Grp ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) → ((𝐴(-g‘𝐹)𝐵) = (0g‘𝐹) ↔ 𝐴 = 𝐵)) |
36 | 17, 18, 19, 35 | syl3anc 1477 | . 2 ⊢ (𝜑 → ((𝐴(-g‘𝐹)𝐵) = (0g‘𝐹) ↔ 𝐴 = 𝐵)) |
37 | 28, 34, 36 | 3bitr3d 298 | 1 ⊢ (𝜑 → ((𝐴 · 𝑋) = (𝐵 · 𝑋) ↔ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∨ wo 382 = wceq 1632 ∈ wcel 2139 ≠ wne 2932 ‘cfv 6049 (class class class)co 6814 Basecbs 16079 Scalarcsca 16166 ·𝑠 cvsca 16167 0gc0g 16322 Grpcgrp 17643 -gcsg 17645 LModclmod 19085 LVecclvec 19324 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-cnex 10204 ax-resscn 10205 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-addrcl 10209 ax-mulcl 10210 ax-mulrcl 10211 ax-mulcom 10212 ax-addass 10213 ax-mulass 10214 ax-distr 10215 ax-i2m1 10216 ax-1ne0 10217 ax-1rid 10218 ax-rnegex 10219 ax-rrecex 10220 ax-cnre 10221 ax-pre-lttri 10222 ax-pre-lttrn 10223 ax-pre-ltadd 10224 ax-pre-mulgt0 10225 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6775 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-om 7232 df-1st 7334 df-2nd 7335 df-tpos 7522 df-wrecs 7577 df-recs 7638 df-rdg 7676 df-er 7913 df-en 8124 df-dom 8125 df-sdom 8126 df-pnf 10288 df-mnf 10289 df-xr 10290 df-ltxr 10291 df-le 10292 df-sub 10480 df-neg 10481 df-nn 11233 df-2 11291 df-3 11292 df-ndx 16082 df-slot 16083 df-base 16085 df-sets 16086 df-ress 16087 df-plusg 16176 df-mulr 16177 df-0g 16324 df-mgm 17463 df-sgrp 17505 df-mnd 17516 df-grp 17646 df-minusg 17647 df-sbg 17648 df-mgp 18710 df-ur 18722 df-ring 18769 df-oppr 18843 df-dvdsr 18861 df-unit 18862 df-invr 18892 df-drng 18971 df-lmod 19087 df-lvec 19325 |
This theorem is referenced by: lspsneu 19345 lvecindp 19360 lvecindp2 19361 lshpsmreu 34917 lshpkrlem5 34922 hgmapval1 37705 hgmapadd 37706 hgmapmul 37707 hgmaprnlem1N 37708 hgmap11 37714 |
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