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Mirrors > Home > MPE Home > Th. List > clmvneg1 | Structured version Visualization version GIF version |
Description: Minus 1 times a vector is the negative of the vector. Equation 2 of [Kreyszig] p. 51. (lmodvneg1 20795 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.) |
Ref | Expression |
---|---|
clmvneg1.v | ⊢ 𝑉 = (Base‘𝑊) |
clmvneg1.n | ⊢ 𝑁 = (invg‘𝑊) |
clmvneg1.f | ⊢ 𝐹 = (Scalar‘𝑊) |
clmvneg1.s | ⊢ · = ( ·𝑠 ‘𝑊) |
Ref | Expression |
---|---|
clmvneg1 | ⊢ ((𝑊 ∈ ℂMod ∧ 𝑋 ∈ 𝑉) → (-1 · 𝑋) = (𝑁‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clmvneg1.f | . . . . . . . 8 ⊢ 𝐹 = (Scalar‘𝑊) | |
2 | eqid 2728 | . . . . . . . 8 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
3 | 1, 2 | clmzss 25025 | . . . . . . 7 ⊢ (𝑊 ∈ ℂMod → ℤ ⊆ (Base‘𝐹)) |
4 | 1zzd 12631 | . . . . . . 7 ⊢ (𝑊 ∈ ℂMod → 1 ∈ ℤ) | |
5 | 3, 4 | sseldd 3983 | . . . . . 6 ⊢ (𝑊 ∈ ℂMod → 1 ∈ (Base‘𝐹)) |
6 | 1, 2 | clmneg 25028 | . . . . . 6 ⊢ ((𝑊 ∈ ℂMod ∧ 1 ∈ (Base‘𝐹)) → -1 = ((invg‘𝐹)‘1)) |
7 | 5, 6 | mpdan 685 | . . . . 5 ⊢ (𝑊 ∈ ℂMod → -1 = ((invg‘𝐹)‘1)) |
8 | 1 | clm1 25020 | . . . . . 6 ⊢ (𝑊 ∈ ℂMod → 1 = (1r‘𝐹)) |
9 | 8 | fveq2d 6906 | . . . . 5 ⊢ (𝑊 ∈ ℂMod → ((invg‘𝐹)‘1) = ((invg‘𝐹)‘(1r‘𝐹))) |
10 | 7, 9 | eqtrd 2768 | . . . 4 ⊢ (𝑊 ∈ ℂMod → -1 = ((invg‘𝐹)‘(1r‘𝐹))) |
11 | 10 | adantr 479 | . . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝑋 ∈ 𝑉) → -1 = ((invg‘𝐹)‘(1r‘𝐹))) |
12 | 11 | oveq1d 7441 | . 2 ⊢ ((𝑊 ∈ ℂMod ∧ 𝑋 ∈ 𝑉) → (-1 · 𝑋) = (((invg‘𝐹)‘(1r‘𝐹)) · 𝑋)) |
13 | clmlmod 25014 | . . 3 ⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ LMod) | |
14 | clmvneg1.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
15 | clmvneg1.n | . . . 4 ⊢ 𝑁 = (invg‘𝑊) | |
16 | clmvneg1.s | . . . 4 ⊢ · = ( ·𝑠 ‘𝑊) | |
17 | eqid 2728 | . . . 4 ⊢ (1r‘𝐹) = (1r‘𝐹) | |
18 | eqid 2728 | . . . 4 ⊢ (invg‘𝐹) = (invg‘𝐹) | |
19 | 14, 15, 1, 16, 17, 18 | lmodvneg1 20795 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (((invg‘𝐹)‘(1r‘𝐹)) · 𝑋) = (𝑁‘𝑋)) |
20 | 13, 19 | sylan 578 | . 2 ⊢ ((𝑊 ∈ ℂMod ∧ 𝑋 ∈ 𝑉) → (((invg‘𝐹)‘(1r‘𝐹)) · 𝑋) = (𝑁‘𝑋)) |
21 | 12, 20 | eqtrd 2768 | 1 ⊢ ((𝑊 ∈ ℂMod ∧ 𝑋 ∈ 𝑉) → (-1 · 𝑋) = (𝑁‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ‘cfv 6553 (class class class)co 7426 1c1 11147 -cneg 11483 ℤcz 12596 Basecbs 17187 Scalarcsca 17243 ·𝑠 cvsca 17244 invgcminusg 18898 1rcur 20128 LModclmod 20750 ℂModcclm 25009 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-addf 11225 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 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 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-fz 13525 df-seq 14007 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-starv 17255 df-tset 17259 df-ple 17260 df-ds 17262 df-unif 17263 df-0g 17430 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-grp 18900 df-minusg 18901 df-mulg 19031 df-subg 19085 df-cmn 19744 df-mgp 20082 df-ur 20129 df-ring 20182 df-cring 20183 df-subrg 20515 df-lmod 20752 df-cnfld 21287 df-clm 25010 |
This theorem is referenced by: clmpm1dir 25050 clmvsrinv 25054 clmvslinv 25055 |
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