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Mirrors > Home > MPE Home > Th. List > lvecvscan | Structured version Visualization version GIF version |
Description: Cancellation law for scalar multiplication. (hvmulcan 30593 analog.) (Contributed by NM, 2-Jul-2014.) |
Ref | Expression |
---|---|
lvecmulcan.v | ⊢ 𝑉 = (Base‘𝑊) |
lvecmulcan.s | ⊢ · = ( ·𝑠 ‘𝑊) |
lvecmulcan.f | ⊢ 𝐹 = (Scalar‘𝑊) |
lvecmulcan.k | ⊢ 𝐾 = (Base‘𝐹) |
lvecmulcan.o | ⊢ 0 = (0g‘𝐹) |
lvecmulcan.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
lvecmulcan.a | ⊢ (𝜑 → 𝐴 ∈ 𝐾) |
lvecmulcan.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
lvecmulcan.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
lvecmulcan.n | ⊢ (𝜑 → 𝐴 ≠ 0 ) |
Ref | Expression |
---|---|
lvecvscan | ⊢ (𝜑 → ((𝐴 · 𝑋) = (𝐴 · 𝑌) ↔ 𝑋 = 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lvecmulcan.n | . . 3 ⊢ (𝜑 → 𝐴 ≠ 0 ) | |
2 | df-ne 2940 | . . . 4 ⊢ (𝐴 ≠ 0 ↔ ¬ 𝐴 = 0 ) | |
3 | biorf 934 | . . . 4 ⊢ (¬ 𝐴 = 0 → ((𝑋(-g‘𝑊)𝑌) = (0g‘𝑊) ↔ (𝐴 = 0 ∨ (𝑋(-g‘𝑊)𝑌) = (0g‘𝑊)))) | |
4 | 2, 3 | sylbi 216 | . . 3 ⊢ (𝐴 ≠ 0 → ((𝑋(-g‘𝑊)𝑌) = (0g‘𝑊) ↔ (𝐴 = 0 ∨ (𝑋(-g‘𝑊)𝑌) = (0g‘𝑊)))) |
5 | 1, 4 | syl 17 | . 2 ⊢ (𝜑 → ((𝑋(-g‘𝑊)𝑌) = (0g‘𝑊) ↔ (𝐴 = 0 ∨ (𝑋(-g‘𝑊)𝑌) = (0g‘𝑊)))) |
6 | lvecmulcan.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
7 | lveclmod 20862 | . . . 4 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
8 | 6, 7 | syl 17 | . . 3 ⊢ (𝜑 → 𝑊 ∈ LMod) |
9 | lvecmulcan.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
10 | lvecmulcan.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
11 | lvecmulcan.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
12 | eqid 2731 | . . . 4 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
13 | eqid 2731 | . . . 4 ⊢ (-g‘𝑊) = (-g‘𝑊) | |
14 | 11, 12, 13 | lmodsubeq0 20676 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((𝑋(-g‘𝑊)𝑌) = (0g‘𝑊) ↔ 𝑋 = 𝑌)) |
15 | 8, 9, 10, 14 | syl3anc 1370 | . 2 ⊢ (𝜑 → ((𝑋(-g‘𝑊)𝑌) = (0g‘𝑊) ↔ 𝑋 = 𝑌)) |
16 | lvecmulcan.s | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑊) | |
17 | lvecmulcan.f | . . . . 5 ⊢ 𝐹 = (Scalar‘𝑊) | |
18 | lvecmulcan.k | . . . . 5 ⊢ 𝐾 = (Base‘𝐹) | |
19 | lvecmulcan.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝐾) | |
20 | 11, 16, 17, 18, 13, 8, 19, 9, 10 | lmodsubdi 20674 | . . . 4 ⊢ (𝜑 → (𝐴 · (𝑋(-g‘𝑊)𝑌)) = ((𝐴 · 𝑋)(-g‘𝑊)(𝐴 · 𝑌))) |
21 | 20 | eqeq1d 2733 | . . 3 ⊢ (𝜑 → ((𝐴 · (𝑋(-g‘𝑊)𝑌)) = (0g‘𝑊) ↔ ((𝐴 · 𝑋)(-g‘𝑊)(𝐴 · 𝑌)) = (0g‘𝑊))) |
22 | lvecmulcan.o | . . . 4 ⊢ 0 = (0g‘𝐹) | |
23 | 11, 13 | lmodvsubcl 20662 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋(-g‘𝑊)𝑌) ∈ 𝑉) |
24 | 8, 9, 10, 23 | syl3anc 1370 | . . . 4 ⊢ (𝜑 → (𝑋(-g‘𝑊)𝑌) ∈ 𝑉) |
25 | 11, 16, 17, 18, 22, 12, 6, 19, 24 | lvecvs0or 20867 | . . 3 ⊢ (𝜑 → ((𝐴 · (𝑋(-g‘𝑊)𝑌)) = (0g‘𝑊) ↔ (𝐴 = 0 ∨ (𝑋(-g‘𝑊)𝑌) = (0g‘𝑊)))) |
26 | 11, 17, 16, 18 | lmodvscl 20633 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝐴 · 𝑋) ∈ 𝑉) |
27 | 8, 19, 9, 26 | syl3anc 1370 | . . . 4 ⊢ (𝜑 → (𝐴 · 𝑋) ∈ 𝑉) |
28 | 11, 17, 16, 18 | lmodvscl 20633 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉) → (𝐴 · 𝑌) ∈ 𝑉) |
29 | 8, 19, 10, 28 | syl3anc 1370 | . . . 4 ⊢ (𝜑 → (𝐴 · 𝑌) ∈ 𝑉) |
30 | 11, 12, 13 | lmodsubeq0 20676 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ (𝐴 · 𝑋) ∈ 𝑉 ∧ (𝐴 · 𝑌) ∈ 𝑉) → (((𝐴 · 𝑋)(-g‘𝑊)(𝐴 · 𝑌)) = (0g‘𝑊) ↔ (𝐴 · 𝑋) = (𝐴 · 𝑌))) |
31 | 8, 27, 29, 30 | syl3anc 1370 | . . 3 ⊢ (𝜑 → (((𝐴 · 𝑋)(-g‘𝑊)(𝐴 · 𝑌)) = (0g‘𝑊) ↔ (𝐴 · 𝑋) = (𝐴 · 𝑌))) |
32 | 21, 25, 31 | 3bitr3d 309 | . 2 ⊢ (𝜑 → ((𝐴 = 0 ∨ (𝑋(-g‘𝑊)𝑌) = (0g‘𝑊)) ↔ (𝐴 · 𝑋) = (𝐴 · 𝑌))) |
33 | 5, 15, 32 | 3bitr3rd 310 | 1 ⊢ (𝜑 → ((𝐴 · 𝑋) = (𝐴 · 𝑌) ↔ 𝑋 = 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∨ wo 844 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 ‘cfv 6543 (class class class)co 7412 Basecbs 17149 Scalarcsca 17205 ·𝑠 cvsca 17206 0gc0g 17390 -gcsg 18858 LModclmod 20615 LVecclvec 20858 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-tpos 8215 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-er 8707 df-en 8944 df-dom 8945 df-sdom 8946 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-2 12280 df-3 12281 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-0g 17392 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-grp 18859 df-minusg 18860 df-sbg 18861 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-oppr 20226 df-dvdsr 20249 df-unit 20250 df-invr 20280 df-drng 20503 df-lmod 20617 df-lvec 20859 |
This theorem is referenced by: (None) |
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