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Mirrors > Home > MPE Home > Th. List > lmodvsneg | Structured version Visualization version GIF version |
Description: Multiplication of a vector by a negated scalar. (Contributed by Stefan O'Rear, 28-Feb-2015.) |
Ref | Expression |
---|---|
lmodvsneg.b | ⊢ 𝐵 = (Base‘𝑊) |
lmodvsneg.f | ⊢ 𝐹 = (Scalar‘𝑊) |
lmodvsneg.s | ⊢ · = ( ·𝑠 ‘𝑊) |
lmodvsneg.n | ⊢ 𝑁 = (invg‘𝑊) |
lmodvsneg.k | ⊢ 𝐾 = (Base‘𝐹) |
lmodvsneg.m | ⊢ 𝑀 = (invg‘𝐹) |
lmodvsneg.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lmodvsneg.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
lmodvsneg.r | ⊢ (𝜑 → 𝑅 ∈ 𝐾) |
Ref | Expression |
---|---|
lmodvsneg | ⊢ (𝜑 → (𝑁‘(𝑅 · 𝑋)) = ((𝑀‘𝑅) · 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmodvsneg.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
2 | lmodvsneg.f | . . . . . . 7 ⊢ 𝐹 = (Scalar‘𝑊) | |
3 | 2 | lmodring 18919 | . . . . . 6 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Ring) |
4 | 1, 3 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ Ring) |
5 | ringgrp 18598 | . . . . 5 ⊢ (𝐹 ∈ Ring → 𝐹 ∈ Grp) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ Grp) |
7 | lmodvsneg.k | . . . . . 6 ⊢ 𝐾 = (Base‘𝐹) | |
8 | eqid 2651 | . . . . . 6 ⊢ (1r‘𝐹) = (1r‘𝐹) | |
9 | 7, 8 | ringidcl 18614 | . . . . 5 ⊢ (𝐹 ∈ Ring → (1r‘𝐹) ∈ 𝐾) |
10 | 4, 9 | syl 17 | . . . 4 ⊢ (𝜑 → (1r‘𝐹) ∈ 𝐾) |
11 | lmodvsneg.m | . . . . 5 ⊢ 𝑀 = (invg‘𝐹) | |
12 | 7, 11 | grpinvcl 17514 | . . . 4 ⊢ ((𝐹 ∈ Grp ∧ (1r‘𝐹) ∈ 𝐾) → (𝑀‘(1r‘𝐹)) ∈ 𝐾) |
13 | 6, 10, 12 | syl2anc 694 | . . 3 ⊢ (𝜑 → (𝑀‘(1r‘𝐹)) ∈ 𝐾) |
14 | lmodvsneg.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ 𝐾) | |
15 | lmodvsneg.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
16 | lmodvsneg.b | . . . 4 ⊢ 𝐵 = (Base‘𝑊) | |
17 | lmodvsneg.s | . . . 4 ⊢ · = ( ·𝑠 ‘𝑊) | |
18 | eqid 2651 | . . . 4 ⊢ (.r‘𝐹) = (.r‘𝐹) | |
19 | 16, 2, 17, 7, 18 | lmodvsass 18936 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ ((𝑀‘(1r‘𝐹)) ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵)) → (((𝑀‘(1r‘𝐹))(.r‘𝐹)𝑅) · 𝑋) = ((𝑀‘(1r‘𝐹)) · (𝑅 · 𝑋))) |
20 | 1, 13, 14, 15, 19 | syl13anc 1368 | . 2 ⊢ (𝜑 → (((𝑀‘(1r‘𝐹))(.r‘𝐹)𝑅) · 𝑋) = ((𝑀‘(1r‘𝐹)) · (𝑅 · 𝑋))) |
21 | 7, 18, 8, 11, 4, 14 | ringnegl 18640 | . . 3 ⊢ (𝜑 → ((𝑀‘(1r‘𝐹))(.r‘𝐹)𝑅) = (𝑀‘𝑅)) |
22 | 21 | oveq1d 6705 | . 2 ⊢ (𝜑 → (((𝑀‘(1r‘𝐹))(.r‘𝐹)𝑅) · 𝑋) = ((𝑀‘𝑅) · 𝑋)) |
23 | 16, 2, 17, 7 | lmodvscl 18928 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵) → (𝑅 · 𝑋) ∈ 𝐵) |
24 | 1, 14, 15, 23 | syl3anc 1366 | . . 3 ⊢ (𝜑 → (𝑅 · 𝑋) ∈ 𝐵) |
25 | lmodvsneg.n | . . . 4 ⊢ 𝑁 = (invg‘𝑊) | |
26 | 16, 25, 2, 17, 8, 11 | lmodvneg1 18954 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑅 · 𝑋) ∈ 𝐵) → ((𝑀‘(1r‘𝐹)) · (𝑅 · 𝑋)) = (𝑁‘(𝑅 · 𝑋))) |
27 | 1, 24, 26 | syl2anc 694 | . 2 ⊢ (𝜑 → ((𝑀‘(1r‘𝐹)) · (𝑅 · 𝑋)) = (𝑁‘(𝑅 · 𝑋))) |
28 | 20, 22, 27 | 3eqtr3rd 2694 | 1 ⊢ (𝜑 → (𝑁‘(𝑅 · 𝑋)) = ((𝑀‘𝑅) · 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1523 ∈ wcel 2030 ‘cfv 5926 (class class class)co 6690 Basecbs 15904 .rcmulr 15989 Scalarcsca 15991 ·𝑠 cvsca 15992 Grpcgrp 17469 invgcminusg 17470 1rcur 18547 Ringcrg 18593 LModclmod 18911 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-plusg 16001 df-0g 16149 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-grp 17472 df-minusg 17473 df-mgp 18536 df-ur 18548 df-ring 18595 df-lmod 18913 |
This theorem is referenced by: lmodnegadd 18960 clmvsneg 22946 baerlem5alem1 37314 lincext3 42570 lindslinindimp2lem4 42575 lincresunit3 42595 |
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