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Mirrors > Home > MPE Home > Th. List > lvecvscan2 | Structured version Visualization version GIF version |
Description: Cancellation law for scalar multiplication. (hvmulcan2 28777 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 3018 | . . . 4 ⊢ (𝜑 → ¬ 𝑋 = 0 ) |
3 | biorf 930 | . . . . 5 ⊢ (¬ 𝑋 = 0 → ((𝐴(-g‘𝐹)𝐵) = (0g‘𝐹) ↔ (𝑋 = 0 ∨ (𝐴(-g‘𝐹)𝐵) = (0g‘𝐹)))) | |
4 | orcom 864 | . . . . 5 ⊢ ((𝑋 = 0 ∨ (𝐴(-g‘𝐹)𝐵) = (0g‘𝐹)) ↔ ((𝐴(-g‘𝐹)𝐵) = (0g‘𝐹) ∨ 𝑋 = 0 )) | |
5 | 3, 4 | syl6bb 288 | . . . 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 2818 | . . . 4 ⊢ (0g‘𝐹) = (0g‘𝐹) | |
12 | lvecmulcan2.o | . . . 4 ⊢ 0 = (0g‘𝑊) | |
13 | lvecmulcan2.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
14 | lveclmod 19807 | . . . . . . 7 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
15 | 13, 14 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LMod) |
16 | 9 | lmodfgrp 19572 | . . . . . 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 2818 | . . . . . 6 ⊢ (-g‘𝐹) = (-g‘𝐹) | |
21 | 10, 20 | grpsubcl 18117 | . . . . 5 ⊢ ((𝐹 ∈ Grp ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) → (𝐴(-g‘𝐹)𝐵) ∈ 𝐾) |
22 | 17, 18, 19, 21 | syl3anc 1363 | . . . 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 19809 | . . 3 ⊢ (𝜑 → (((𝐴(-g‘𝐹)𝐵) · 𝑋) = 0 ↔ ((𝐴(-g‘𝐹)𝐵) = (0g‘𝐹) ∨ 𝑋 = 0 ))) |
25 | eqid 2818 | . . . . 5 ⊢ (-g‘𝑊) = (-g‘𝑊) | |
26 | 7, 8, 9, 10, 25, 20, 15, 18, 19, 23 | lmodsubdir 19621 | . . . 4 ⊢ (𝜑 → ((𝐴(-g‘𝐹)𝐵) · 𝑋) = ((𝐴 · 𝑋)(-g‘𝑊)(𝐵 · 𝑋))) |
27 | 26 | eqeq1d 2820 | . . 3 ⊢ (𝜑 → (((𝐴(-g‘𝐹)𝐵) · 𝑋) = 0 ↔ ((𝐴 · 𝑋)(-g‘𝑊)(𝐵 · 𝑋)) = 0 )) |
28 | 6, 24, 27 | 3bitr2rd 309 | . 2 ⊢ (𝜑 → (((𝐴 · 𝑋)(-g‘𝑊)(𝐵 · 𝑋)) = 0 ↔ (𝐴(-g‘𝐹)𝐵) = (0g‘𝐹))) |
29 | 7, 9, 8, 10 | lmodvscl 19580 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝐴 · 𝑋) ∈ 𝑉) |
30 | 15, 18, 23, 29 | syl3anc 1363 | . . 3 ⊢ (𝜑 → (𝐴 · 𝑋) ∈ 𝑉) |
31 | 7, 9, 8, 10 | lmodvscl 19580 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝐵 · 𝑋) ∈ 𝑉) |
32 | 15, 19, 23, 31 | syl3anc 1363 | . . 3 ⊢ (𝜑 → (𝐵 · 𝑋) ∈ 𝑉) |
33 | 7, 12, 25 | lmodsubeq0 19622 | . . 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 1363 | . 2 ⊢ (𝜑 → (((𝐴 · 𝑋)(-g‘𝑊)(𝐵 · 𝑋)) = 0 ↔ (𝐴 · 𝑋) = (𝐵 · 𝑋))) |
35 | 10, 11, 20 | grpsubeq0 18123 | . . 3 ⊢ ((𝐹 ∈ Grp ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) → ((𝐴(-g‘𝐹)𝐵) = (0g‘𝐹) ↔ 𝐴 = 𝐵)) |
36 | 17, 18, 19, 35 | syl3anc 1363 | . 2 ⊢ (𝜑 → ((𝐴(-g‘𝐹)𝐵) = (0g‘𝐹) ↔ 𝐴 = 𝐵)) |
37 | 28, 34, 36 | 3bitr3d 310 | 1 ⊢ (𝜑 → ((𝐴 · 𝑋) = (𝐵 · 𝑋) ↔ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∨ wo 841 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 Scalarcsca 16556 ·𝑠 cvsca 16557 0gc0g 16701 Grpcgrp 18041 -gcsg 18043 LModclmod 19563 LVecclvec 19803 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 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 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-tpos 7881 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-minusg 18045 df-sbg 18046 df-mgp 19169 df-ur 19181 df-ring 19228 df-oppr 19302 df-dvdsr 19320 df-unit 19321 df-invr 19351 df-drng 19433 df-lmod 19565 df-lvec 19804 |
This theorem is referenced by: lspsneu 19824 lvecindp 19839 lvecindp2 19840 linds2eq 30868 lshpsmreu 36125 lshpkrlem5 36130 hgmapval1 38909 hgmapadd 38910 hgmapmul 38911 hgmaprnlem1N 38912 hgmap11 38918 |
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