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Mirrors > Home > MPE Home > Th. List > lmodvsubval2 | Structured version Visualization version GIF version |
Description: Value of vector subtraction in terms of addition. (hvsubval 30944 analog.) (Contributed by NM, 31-Mar-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.) |
Ref | Expression |
---|---|
lmodvsubval2.v | ⊢ 𝑉 = (Base‘𝑊) |
lmodvsubval2.p | ⊢ + = (+g‘𝑊) |
lmodvsubval2.m | ⊢ − = (-g‘𝑊) |
lmodvsubval2.f | ⊢ 𝐹 = (Scalar‘𝑊) |
lmodvsubval2.s | ⊢ · = ( ·𝑠 ‘𝑊) |
lmodvsubval2.n | ⊢ 𝑁 = (invg‘𝐹) |
lmodvsubval2.u | ⊢ 1 = (1r‘𝐹) |
Ref | Expression |
---|---|
lmodvsubval2 | ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 − 𝐵) = (𝐴 + ((𝑁‘ 1 ) · 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmodvsubval2.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
2 | lmodvsubval2.p | . . . 4 ⊢ + = (+g‘𝑊) | |
3 | eqid 2726 | . . . 4 ⊢ (invg‘𝑊) = (invg‘𝑊) | |
4 | lmodvsubval2.m | . . . 4 ⊢ − = (-g‘𝑊) | |
5 | 1, 2, 3, 4 | grpsubval 18973 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 − 𝐵) = (𝐴 + ((invg‘𝑊)‘𝐵))) |
6 | 5 | 3adant1 1127 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 − 𝐵) = (𝐴 + ((invg‘𝑊)‘𝐵))) |
7 | lmodvsubval2.f | . . . . 5 ⊢ 𝐹 = (Scalar‘𝑊) | |
8 | lmodvsubval2.s | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑊) | |
9 | lmodvsubval2.u | . . . . 5 ⊢ 1 = (1r‘𝐹) | |
10 | lmodvsubval2.n | . . . . 5 ⊢ 𝑁 = (invg‘𝐹) | |
11 | 1, 3, 7, 8, 9, 10 | lmodvneg1 20875 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝑉) → ((𝑁‘ 1 ) · 𝐵) = ((invg‘𝑊)‘𝐵)) |
12 | 11 | 3adant2 1128 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((𝑁‘ 1 ) · 𝐵) = ((invg‘𝑊)‘𝐵)) |
13 | 12 | oveq2d 7430 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 + ((𝑁‘ 1 ) · 𝐵)) = (𝐴 + ((invg‘𝑊)‘𝐵))) |
14 | 6, 13 | eqtr4d 2769 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 − 𝐵) = (𝐴 + ((𝑁‘ 1 ) · 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ‘cfv 6544 (class class class)co 7414 Basecbs 17206 +gcplusg 17259 Scalarcsca 17262 ·𝑠 cvsca 17263 invgcminusg 18922 -gcsg 18923 1rcur 20158 LModclmod 20830 |
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 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 ax-cnex 11203 ax-resscn 11204 ax-1cn 11205 ax-icn 11206 ax-addcl 11207 ax-addrcl 11208 ax-mulcl 11209 ax-mulrcl 11210 ax-mulcom 11211 ax-addass 11212 ax-mulass 11213 ax-distr 11214 ax-i2m1 11215 ax-1ne0 11216 ax-1rid 11217 ax-rnegex 11218 ax-rrecex 11219 ax-cnre 11220 ax-pre-lttri 11221 ax-pre-lttrn 11222 ax-pre-ltadd 11223 ax-pre-mulgt0 11224 |
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 3365 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-iun 4996 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6303 df-ord 6369 df-on 6370 df-lim 6371 df-suc 6372 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-er 8724 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11289 df-mnf 11290 df-xr 11291 df-ltxr 11292 df-le 11293 df-sub 11485 df-neg 11486 df-nn 12257 df-2 12319 df-sets 17159 df-slot 17177 df-ndx 17189 df-base 17207 df-plusg 17272 df-0g 17449 df-mgm 18626 df-sgrp 18705 df-mnd 18721 df-grp 18924 df-minusg 18925 df-sbg 18926 df-mgp 20112 df-ur 20159 df-ring 20212 df-lmod 20832 |
This theorem is referenced by: lmodsubvs 20888 lmodsubdi 20889 lmodsubdir 20890 lssvsubcl 20915 clmvsubval 25122 lflsub 38776 ldualvsub 38864 ldualvsubval 38866 lcdvsub 41327 lcdvsubval 41328 baerlem3lem1 41417 zlmodzxzsubm 47772 |
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