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Mirrors > Home > MPE Home > Th. List > clmvsubval | Structured version Visualization version GIF version |
Description: Value of vector subtraction in terms of addition in a subcomplex module. Analogue of lmodvsubval2 19967. (Contributed by NM, 31-Mar-2014.) (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 |
---|---|
clmvsubval | ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 − 𝐵) = (𝐴 + (-1 · 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clmlmod 23977 | . . 3 ⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ LMod) | |
2 | clmvsubval.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
3 | clmvsubval.p | . . . 4 ⊢ + = (+g‘𝑊) | |
4 | clmvsubval.m | . . . 4 ⊢ − = (-g‘𝑊) | |
5 | clmvsubval.f | . . . 4 ⊢ 𝐹 = (Scalar‘𝑊) | |
6 | clmvsubval.s | . . . 4 ⊢ · = ( ·𝑠 ‘𝑊) | |
7 | eqid 2738 | . . . 4 ⊢ (invg‘𝐹) = (invg‘𝐹) | |
8 | eqid 2738 | . . . 4 ⊢ (1r‘𝐹) = (1r‘𝐹) | |
9 | 2, 3, 4, 5, 6, 7, 8 | lmodvsubval2 19967 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 − 𝐵) = (𝐴 + (((invg‘𝐹)‘(1r‘𝐹)) · 𝐵))) |
10 | 1, 9 | syl3an1 1165 | . 2 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 − 𝐵) = (𝐴 + (((invg‘𝐹)‘(1r‘𝐹)) · 𝐵))) |
11 | 5 | clm1 23983 | . . . . . . . 8 ⊢ (𝑊 ∈ ℂMod → 1 = (1r‘𝐹)) |
12 | 11 | eqcomd 2744 | . . . . . . 7 ⊢ (𝑊 ∈ ℂMod → (1r‘𝐹) = 1) |
13 | 12 | fveq2d 6730 | . . . . . 6 ⊢ (𝑊 ∈ ℂMod → ((invg‘𝐹)‘(1r‘𝐹)) = ((invg‘𝐹)‘1)) |
14 | 5 | clmring 23980 | . . . . . . . . 9 ⊢ (𝑊 ∈ ℂMod → 𝐹 ∈ Ring) |
15 | eqid 2738 | . . . . . . . . . 10 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
16 | 15, 8 | ringidcl 19599 | . . . . . . . . 9 ⊢ (𝐹 ∈ Ring → (1r‘𝐹) ∈ (Base‘𝐹)) |
17 | 14, 16 | syl 17 | . . . . . . . 8 ⊢ (𝑊 ∈ ℂMod → (1r‘𝐹) ∈ (Base‘𝐹)) |
18 | 11, 17 | eqeltrd 2839 | . . . . . . 7 ⊢ (𝑊 ∈ ℂMod → 1 ∈ (Base‘𝐹)) |
19 | 5, 15 | clmneg 23991 | . . . . . . 7 ⊢ ((𝑊 ∈ ℂMod ∧ 1 ∈ (Base‘𝐹)) → -1 = ((invg‘𝐹)‘1)) |
20 | 18, 19 | mpdan 687 | . . . . . 6 ⊢ (𝑊 ∈ ℂMod → -1 = ((invg‘𝐹)‘1)) |
21 | 13, 20 | eqtr4d 2781 | . . . . 5 ⊢ (𝑊 ∈ ℂMod → ((invg‘𝐹)‘(1r‘𝐹)) = -1) |
22 | 21 | 3ad2ant1 1135 | . . . 4 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((invg‘𝐹)‘(1r‘𝐹)) = -1) |
23 | 22 | oveq1d 7237 | . . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (((invg‘𝐹)‘(1r‘𝐹)) · 𝐵) = (-1 · 𝐵)) |
24 | 23 | oveq2d 7238 | . 2 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 + (((invg‘𝐹)‘(1r‘𝐹)) · 𝐵)) = (𝐴 + (-1 · 𝐵))) |
25 | 10, 24 | eqtrd 2778 | 1 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 − 𝐵) = (𝐴 + (-1 · 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1089 = wceq 1543 ∈ wcel 2111 ‘cfv 6389 (class class class)co 7222 1c1 10743 -cneg 11076 Basecbs 16773 +gcplusg 16815 Scalarcsca 16818 ·𝑠 cvsca 16819 invgcminusg 18379 -gcsg 18380 1rcur 19529 Ringcrg 19575 LModclmod 19912 ℂModcclm 23972 |
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 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5201 ax-nul 5208 ax-pow 5267 ax-pr 5331 ax-un 7532 ax-cnex 10798 ax-resscn 10799 ax-1cn 10800 ax-icn 10801 ax-addcl 10802 ax-addrcl 10803 ax-mulcl 10804 ax-mulrcl 10805 ax-mulcom 10806 ax-addass 10807 ax-mulass 10808 ax-distr 10809 ax-i2m1 10810 ax-1ne0 10811 ax-1rid 10812 ax-rnegex 10813 ax-rrecex 10814 ax-cnre 10815 ax-pre-lttri 10816 ax-pre-lttrn 10817 ax-pre-ltadd 10818 ax-pre-mulgt0 10819 ax-addf 10821 ax-mulf 10822 |
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 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-reu 3069 df-rmo 3070 df-rab 3071 df-v 3417 df-sbc 3704 df-csb 3821 df-dif 3878 df-un 3880 df-in 3882 df-ss 3892 df-pss 3894 df-nul 4247 df-if 4449 df-pw 4524 df-sn 4551 df-pr 4553 df-tp 4555 df-op 4557 df-uni 4829 df-iun 4915 df-br 5063 df-opab 5125 df-mpt 5145 df-tr 5171 df-id 5464 df-eprel 5469 df-po 5477 df-so 5478 df-fr 5518 df-we 5520 df-xp 5566 df-rel 5567 df-cnv 5568 df-co 5569 df-dm 5570 df-rn 5571 df-res 5572 df-ima 5573 df-pred 6169 df-ord 6225 df-on 6226 df-lim 6227 df-suc 6228 df-iota 6347 df-fun 6391 df-fn 6392 df-f 6393 df-f1 6394 df-fo 6395 df-f1o 6396 df-fv 6397 df-riota 7179 df-ov 7225 df-oprab 7226 df-mpo 7227 df-om 7654 df-1st 7770 df-2nd 7771 df-wrecs 8056 df-recs 8117 df-rdg 8155 df-1o 8211 df-er 8400 df-en 8636 df-dom 8637 df-sdom 8638 df-fin 8639 df-pnf 10882 df-mnf 10883 df-xr 10884 df-ltxr 10885 df-le 10886 df-sub 11077 df-neg 11078 df-nn 11844 df-2 11906 df-3 11907 df-4 11908 df-5 11909 df-6 11910 df-7 11911 df-8 11912 df-9 11913 df-n0 12104 df-z 12190 df-dec 12307 df-uz 12452 df-fz 13109 df-struct 16713 df-sets 16730 df-slot 16748 df-ndx 16758 df-base 16774 df-ress 16798 df-plusg 16828 df-mulr 16829 df-starv 16830 df-tset 16834 df-ple 16835 df-ds 16837 df-unif 16838 df-0g 16959 df-mgm 18127 df-sgrp 18176 df-mnd 18187 df-grp 18381 df-minusg 18382 df-sbg 18383 df-subg 18553 df-cmn 19185 df-mgp 19518 df-ur 19530 df-ring 19577 df-cring 19578 df-subrg 19811 df-lmod 19914 df-cnfld 20377 df-clm 23973 |
This theorem is referenced by: clmvsubval2 24020 ncvsdif 24065 ncvspds 24071 cphipval 24153 |
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