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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 20844 | . . . . . 6 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Ring) |
4 | 1, 3 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ Ring) |
5 | ringgrp 20221 | . . . . 5 ⊢ (𝐹 ∈ Ring → 𝐹 ∈ Grp) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ Grp) |
7 | lmodvsneg.k | . . . . . 6 ⊢ 𝐾 = (Base‘𝐹) | |
8 | eqid 2726 | . . . . . 6 ⊢ (1r‘𝐹) = (1r‘𝐹) | |
9 | 7, 8 | ringidcl 20245 | . . . . 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 18982 | . . . 4 ⊢ ((𝐹 ∈ Grp ∧ (1r‘𝐹) ∈ 𝐾) → (𝑀‘(1r‘𝐹)) ∈ 𝐾) |
13 | 6, 10, 12 | syl2anc 582 | . . 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 2726 | . . . 4 ⊢ (.r‘𝐹) = (.r‘𝐹) | |
19 | 16, 2, 17, 7, 18 | lmodvsass 20863 | . . 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 1369 | . 2 ⊢ (𝜑 → (((𝑀‘(1r‘𝐹))(.r‘𝐹)𝑅) · 𝑋) = ((𝑀‘(1r‘𝐹)) · (𝑅 · 𝑋))) |
21 | 7, 18, 8, 11, 4, 14 | ringnegl 20281 | . . 3 ⊢ (𝜑 → ((𝑀‘(1r‘𝐹))(.r‘𝐹)𝑅) = (𝑀‘𝑅)) |
22 | 21 | oveq1d 7439 | . 2 ⊢ (𝜑 → (((𝑀‘(1r‘𝐹))(.r‘𝐹)𝑅) · 𝑋) = ((𝑀‘𝑅) · 𝑋)) |
23 | 16, 2, 17, 7 | lmodvscl 20854 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵) → (𝑅 · 𝑋) ∈ 𝐵) |
24 | 1, 14, 15, 23 | syl3anc 1368 | . . 3 ⊢ (𝜑 → (𝑅 · 𝑋) ∈ 𝐵) |
25 | lmodvsneg.n | . . . 4 ⊢ 𝑁 = (invg‘𝑊) | |
26 | 16, 25, 2, 17, 8, 11 | lmodvneg1 20881 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑅 · 𝑋) ∈ 𝐵) → ((𝑀‘(1r‘𝐹)) · (𝑅 · 𝑋)) = (𝑁‘(𝑅 · 𝑋))) |
27 | 1, 24, 26 | syl2anc 582 | . 2 ⊢ (𝜑 → ((𝑀‘(1r‘𝐹)) · (𝑅 · 𝑋)) = (𝑁‘(𝑅 · 𝑋))) |
28 | 20, 22, 27 | 3eqtr3rd 2775 | 1 ⊢ (𝜑 → (𝑁‘(𝑅 · 𝑋)) = ((𝑀‘𝑅) · 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ‘cfv 6554 (class class class)co 7424 Basecbs 17213 .rcmulr 17267 Scalarcsca 17269 ·𝑠 cvsca 17270 Grpcgrp 18928 invgcminusg 18929 1rcur 20164 Ringcrg 20216 LModclmod 20836 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-2 12327 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-plusg 17279 df-0g 17456 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-grp 18931 df-minusg 18932 df-cmn 19780 df-abl 19781 df-mgp 20118 df-rng 20136 df-ur 20165 df-ring 20218 df-lmod 20838 |
This theorem is referenced by: lmodnegadd 20887 clmvsneg 25118 linds2eq 33256 baerlem5alem1 41407 lincext3 47839 lindslinindimp2lem4 47844 lincresunit3 47864 |
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