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Mirrors > Home > MPE Home > Th. List > clmvsubval2 | Structured version Visualization version GIF version |
Description: Value of vector subtraction on a subcomplex module. (Contributed by Mario Carneiro, 19-Nov-2013.) (Revised by AV, 7-Oct-2021.) |
Ref | Expression |
---|---|
clmvsubval.v | ⊢ 𝑉 = (Base‘𝑊) |
clmvsubval.p | ⊢ + = (+g‘𝑊) |
clmvsubval.m | ⊢ − = (-g‘𝑊) |
clmvsubval.f | ⊢ 𝐹 = (Scalar‘𝑊) |
clmvsubval.s | ⊢ · = ( ·𝑠 ‘𝑊) |
Ref | Expression |
---|---|
clmvsubval2 | ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 − 𝐵) = ((-1 · 𝐵) + 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clmvsubval.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
2 | clmvsubval.p | . . 3 ⊢ + = (+g‘𝑊) | |
3 | clmvsubval.m | . . 3 ⊢ − = (-g‘𝑊) | |
4 | clmvsubval.f | . . 3 ⊢ 𝐹 = (Scalar‘𝑊) | |
5 | clmvsubval.s | . . 3 ⊢ · = ( ·𝑠 ‘𝑊) | |
6 | 1, 2, 3, 4, 5 | clmvsubval 23978 | . 2 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 − 𝐵) = (𝐴 + (-1 · 𝐵))) |
7 | clmabl 23938 | . . . 4 ⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ Abel) | |
8 | 7 | 3ad2ant1 1135 | . . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝑊 ∈ Abel) |
9 | simp2 1139 | . . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ 𝑉) | |
10 | simpl 486 | . . . . 5 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉) → 𝑊 ∈ ℂMod) | |
11 | eqid 2734 | . . . . . . 7 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
12 | 4, 11 | clmneg1 23951 | . . . . . 6 ⊢ (𝑊 ∈ ℂMod → -1 ∈ (Base‘𝐹)) |
13 | 12 | adantr 484 | . . . . 5 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉) → -1 ∈ (Base‘𝐹)) |
14 | simpr 488 | . . . . 5 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ 𝑉) | |
15 | 1, 4, 5, 11 | clmvscl 23957 | . . . . 5 ⊢ ((𝑊 ∈ ℂMod ∧ -1 ∈ (Base‘𝐹) ∧ 𝐵 ∈ 𝑉) → (-1 · 𝐵) ∈ 𝑉) |
16 | 10, 13, 14, 15 | syl3anc 1373 | . . . 4 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉) → (-1 · 𝐵) ∈ 𝑉) |
17 | 16 | 3adant2 1133 | . . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (-1 · 𝐵) ∈ 𝑉) |
18 | 1, 2 | ablcom 19160 | . . 3 ⊢ ((𝑊 ∈ Abel ∧ 𝐴 ∈ 𝑉 ∧ (-1 · 𝐵) ∈ 𝑉) → (𝐴 + (-1 · 𝐵)) = ((-1 · 𝐵) + 𝐴)) |
19 | 8, 9, 17, 18 | syl3anc 1373 | . 2 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 + (-1 · 𝐵)) = ((-1 · 𝐵) + 𝐴)) |
20 | 6, 19 | eqtrd 2774 | 1 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 − 𝐵) = ((-1 · 𝐵) + 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ‘cfv 6369 (class class class)co 7202 1c1 10713 -cneg 11046 Basecbs 16684 +gcplusg 16767 Scalarcsca 16770 ·𝑠 cvsca 16771 -gcsg 18339 Abelcabl 19143 ℂModcclm 23931 |
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 ax-addf 10791 ax-mulf 10792 |
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-1o 8191 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 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-3 11877 df-4 11878 df-5 11879 df-6 11880 df-7 11881 df-8 11882 df-9 11883 df-n0 12074 df-z 12160 df-dec 12277 df-uz 12422 df-fz 13079 df-seq 13558 df-struct 16686 df-ndx 16687 df-slot 16688 df-base 16690 df-sets 16691 df-ress 16692 df-plusg 16780 df-mulr 16781 df-starv 16782 df-tset 16786 df-ple 16787 df-ds 16789 df-unif 16790 df-0g 16918 df-mgm 18086 df-sgrp 18135 df-mnd 18146 df-grp 18340 df-minusg 18341 df-sbg 18342 df-mulg 18461 df-subg 18512 df-cmn 19144 df-abl 19145 df-mgp 19477 df-ur 19489 df-ring 19536 df-cring 19537 df-subrg 19770 df-lmod 19873 df-cnfld 20336 df-clm 23932 |
This theorem is referenced by: clmvz 23980 |
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