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Mirrors > Home > MPE Home > Th. List > lvecvscan2 | Structured version Visualization version GIF version |
Description: Cancellation law for scalar multiplication. (hvmulcan2 30304 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 2946 | . . . 4 ⊢ (𝜑 → ¬ 𝑋 = 0 ) |
3 | biorf 936 | . . . . 5 ⊢ (¬ 𝑋 = 0 → ((𝐴(-g‘𝐹)𝐵) = (0g‘𝐹) ↔ (𝑋 = 0 ∨ (𝐴(-g‘𝐹)𝐵) = (0g‘𝐹)))) | |
4 | orcom 869 | . . . . 5 ⊢ ((𝑋 = 0 ∨ (𝐴(-g‘𝐹)𝐵) = (0g‘𝐹)) ↔ ((𝐴(-g‘𝐹)𝐵) = (0g‘𝐹) ∨ 𝑋 = 0 )) | |
5 | 3, 4 | bitrdi 287 | . . . 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 2733 | . . . 4 ⊢ (0g‘𝐹) = (0g‘𝐹) | |
12 | lvecmulcan2.o | . . . 4 ⊢ 0 = (0g‘𝑊) | |
13 | lvecmulcan2.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
14 | lveclmod 20705 | . . . . . . 7 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
15 | 13, 14 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LMod) |
16 | 9 | lmodfgrp 20468 | . . . . . 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 2733 | . . . . . 6 ⊢ (-g‘𝐹) = (-g‘𝐹) | |
21 | 10, 20 | grpsubcl 18899 | . . . . 5 ⊢ ((𝐹 ∈ Grp ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) → (𝐴(-g‘𝐹)𝐵) ∈ 𝐾) |
22 | 17, 18, 19, 21 | syl3anc 1372 | . . . 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 20709 | . . 3 ⊢ (𝜑 → (((𝐴(-g‘𝐹)𝐵) · 𝑋) = 0 ↔ ((𝐴(-g‘𝐹)𝐵) = (0g‘𝐹) ∨ 𝑋 = 0 ))) |
25 | eqid 2733 | . . . . 5 ⊢ (-g‘𝑊) = (-g‘𝑊) | |
26 | 7, 8, 9, 10, 25, 20, 15, 18, 19, 23 | lmodsubdir 20518 | . . . 4 ⊢ (𝜑 → ((𝐴(-g‘𝐹)𝐵) · 𝑋) = ((𝐴 · 𝑋)(-g‘𝑊)(𝐵 · 𝑋))) |
27 | 26 | eqeq1d 2735 | . . 3 ⊢ (𝜑 → (((𝐴(-g‘𝐹)𝐵) · 𝑋) = 0 ↔ ((𝐴 · 𝑋)(-g‘𝑊)(𝐵 · 𝑋)) = 0 )) |
28 | 6, 24, 27 | 3bitr2rd 308 | . 2 ⊢ (𝜑 → (((𝐴 · 𝑋)(-g‘𝑊)(𝐵 · 𝑋)) = 0 ↔ (𝐴(-g‘𝐹)𝐵) = (0g‘𝐹))) |
29 | 7, 9, 8, 10 | lmodvscl 20477 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝐴 · 𝑋) ∈ 𝑉) |
30 | 15, 18, 23, 29 | syl3anc 1372 | . . 3 ⊢ (𝜑 → (𝐴 · 𝑋) ∈ 𝑉) |
31 | 7, 9, 8, 10 | lmodvscl 20477 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝐵 · 𝑋) ∈ 𝑉) |
32 | 15, 19, 23, 31 | syl3anc 1372 | . . 3 ⊢ (𝜑 → (𝐵 · 𝑋) ∈ 𝑉) |
33 | 7, 12, 25 | lmodsubeq0 20519 | . . 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 1372 | . 2 ⊢ (𝜑 → (((𝐴 · 𝑋)(-g‘𝑊)(𝐵 · 𝑋)) = 0 ↔ (𝐴 · 𝑋) = (𝐵 · 𝑋))) |
35 | 10, 11, 20 | grpsubeq0 18905 | . . 3 ⊢ ((𝐹 ∈ Grp ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) → ((𝐴(-g‘𝐹)𝐵) = (0g‘𝐹) ↔ 𝐴 = 𝐵)) |
36 | 17, 18, 19, 35 | syl3anc 1372 | . 2 ⊢ (𝜑 → ((𝐴(-g‘𝐹)𝐵) = (0g‘𝐹) ↔ 𝐴 = 𝐵)) |
37 | 28, 34, 36 | 3bitr3d 309 | 1 ⊢ (𝜑 → ((𝐴 · 𝑋) = (𝐵 · 𝑋) ↔ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ‘cfv 6540 (class class class)co 7404 Basecbs 17140 Scalarcsca 17196 ·𝑠 cvsca 17197 0gc0g 17381 Grpcgrp 18815 -gcsg 18817 LModclmod 20459 LVecclvec 20701 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7851 df-1st 7970 df-2nd 7971 df-tpos 8206 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-0g 17383 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-grp 18818 df-minusg 18819 df-sbg 18820 df-mgp 19980 df-ur 19997 df-ring 20049 df-oppr 20139 df-dvdsr 20160 df-unit 20161 df-invr 20191 df-drng 20306 df-lmod 20461 df-lvec 20702 |
This theorem is referenced by: lspsneu 20724 lvecindp 20739 lvecindp2 20740 linds2eq 32462 lshpsmreu 37917 lshpkrlem5 37922 hgmapval1 40702 hgmapadd 40703 hgmapmul 40704 hgmaprnlem1N 40705 hgmap11 40711 |
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