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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 29069 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 2734 | . . . 4 ⊢ (invg‘𝑊) = (invg‘𝑊) | |
4 | lmodvsubval2.m | . . . 4 ⊢ − = (-g‘𝑊) | |
5 | 1, 2, 3, 4 | grpsubval 18385 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 − 𝐵) = (𝐴 + ((invg‘𝑊)‘𝐵))) |
6 | 5 | 3adant1 1132 | . 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 19914 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝑉) → ((𝑁‘ 1 ) · 𝐵) = ((invg‘𝑊)‘𝐵)) |
12 | 11 | 3adant2 1133 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((𝑁‘ 1 ) · 𝐵) = ((invg‘𝑊)‘𝐵)) |
13 | 12 | oveq2d 7218 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 + ((𝑁‘ 1 ) · 𝐵)) = (𝐴 + ((invg‘𝑊)‘𝐵))) |
14 | 6, 13 | eqtr4d 2777 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 − 𝐵) = (𝐴 + ((𝑁‘ 1 ) · 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ‘cfv 6369 (class class class)co 7202 Basecbs 16684 +gcplusg 16767 Scalarcsca 16770 ·𝑠 cvsca 16771 invgcminusg 18338 -gcsg 18339 1rcur 19488 LModclmod 19871 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-2 11876 df-ndx 16687 df-slot 16688 df-base 16690 df-sets 16691 df-plusg 16780 df-0g 16918 df-mgm 18086 df-sgrp 18135 df-mnd 18146 df-grp 18340 df-minusg 18341 df-sbg 18342 df-mgp 19477 df-ur 19489 df-ring 19536 df-lmod 19873 |
This theorem is referenced by: lmodsubvs 19927 lmodsubdi 19928 lmodsubdir 19929 lssvsubcl 19952 clmvsubval 23978 lflsub 36775 ldualvsub 36863 ldualvsubval 36865 lcdvsub 39325 lcdvsubval 39326 baerlem3lem1 39415 zlmodzxzsubm 45322 |
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