Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > lvecvscan2 | Structured version Visualization version GIF version |
Description: Cancellation law for scalar multiplication. (hvmulcan2 27314 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 2787 | . . . 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 275 | . . . 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 2610 | . . . 4 ⊢ (0g‘𝐹) = (0g‘𝐹) | |
12 | lvecmulcan2.o | . . . 4 ⊢ 0 = (0g‘𝑊) | |
13 | lvecmulcan2.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
14 | lveclmod 18927 | . . . . . . 7 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
15 | 13, 14 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LMod) |
16 | 9 | lmodfgrp 18695 | . . . . . 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 2610 | . . . . . 6 ⊢ (-g‘𝐹) = (-g‘𝐹) | |
21 | 10, 20 | grpsubcl 17318 | . . . . 5 ⊢ ((𝐹 ∈ Grp ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) → (𝐴(-g‘𝐹)𝐵) ∈ 𝐾) |
22 | 17, 18, 19, 21 | syl3anc 1318 | . . . 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 18929 | . . 3 ⊢ (𝜑 → (((𝐴(-g‘𝐹)𝐵) · 𝑋) = 0 ↔ ((𝐴(-g‘𝐹)𝐵) = (0g‘𝐹) ∨ 𝑋 = 0 ))) |
25 | eqid 2610 | . . . . 5 ⊢ (-g‘𝑊) = (-g‘𝑊) | |
26 | 7, 8, 9, 10, 25, 20, 15, 18, 19, 23 | lmodsubdir 18744 | . . . 4 ⊢ (𝜑 → ((𝐴(-g‘𝐹)𝐵) · 𝑋) = ((𝐴 · 𝑋)(-g‘𝑊)(𝐵 · 𝑋))) |
27 | 26 | eqeq1d 2612 | . . 3 ⊢ (𝜑 → (((𝐴(-g‘𝐹)𝐵) · 𝑋) = 0 ↔ ((𝐴 · 𝑋)(-g‘𝑊)(𝐵 · 𝑋)) = 0 )) |
28 | 6, 24, 27 | 3bitr2rd 296 | . 2 ⊢ (𝜑 → (((𝐴 · 𝑋)(-g‘𝑊)(𝐵 · 𝑋)) = 0 ↔ (𝐴(-g‘𝐹)𝐵) = (0g‘𝐹))) |
29 | 7, 9, 8, 10 | lmodvscl 18703 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝐴 · 𝑋) ∈ 𝑉) |
30 | 15, 18, 23, 29 | syl3anc 1318 | . . 3 ⊢ (𝜑 → (𝐴 · 𝑋) ∈ 𝑉) |
31 | 7, 9, 8, 10 | lmodvscl 18703 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝐵 · 𝑋) ∈ 𝑉) |
32 | 15, 19, 23, 31 | syl3anc 1318 | . . 3 ⊢ (𝜑 → (𝐵 · 𝑋) ∈ 𝑉) |
33 | 7, 12, 25 | lmodsubeq0 18745 | . . 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 1318 | . 2 ⊢ (𝜑 → (((𝐴 · 𝑋)(-g‘𝑊)(𝐵 · 𝑋)) = 0 ↔ (𝐴 · 𝑋) = (𝐵 · 𝑋))) |
35 | 10, 11, 20 | grpsubeq0 17324 | . . 3 ⊢ ((𝐹 ∈ Grp ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) → ((𝐴(-g‘𝐹)𝐵) = (0g‘𝐹) ↔ 𝐴 = 𝐵)) |
36 | 17, 18, 19, 35 | syl3anc 1318 | . 2 ⊢ (𝜑 → ((𝐴(-g‘𝐹)𝐵) = (0g‘𝐹) ↔ 𝐴 = 𝐵)) |
37 | 28, 34, 36 | 3bitr3d 297 | 1 ⊢ (𝜑 → ((𝐴 · 𝑋) = (𝐵 · 𝑋) ↔ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 Scalarcsca 15771 ·𝑠 cvsca 15772 0gc0g 15923 Grpcgrp 17245 -gcsg 17247 LModclmod 18686 LVecclvec 18923 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mgp 18313 df-ur 18325 df-ring 18372 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-invr 18495 df-drng 18572 df-lmod 18688 df-lvec 18924 |
This theorem is referenced by: lspsneu 18944 lvecindp 18959 lvecindp2 18960 lshpsmreu 33414 lshpkrlem5 33419 hgmapval1 36203 hgmapadd 36204 hgmapmul 36205 hgmaprnlem1N 36206 hgmap11 36212 |
Copyright terms: Public domain | W3C validator |