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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 19796 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 2738 | . . . . . . . 8 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
3 | 1, 2 | clmzss 23830 | . . . . . . 7 ⊢ (𝑊 ∈ ℂMod → ℤ ⊆ (Base‘𝐹)) |
4 | 1zzd 12094 | . . . . . . 7 ⊢ (𝑊 ∈ ℂMod → 1 ∈ ℤ) | |
5 | 3, 4 | sseldd 3878 | . . . . . 6 ⊢ (𝑊 ∈ ℂMod → 1 ∈ (Base‘𝐹)) |
6 | 1, 2 | clmneg 23833 | . . . . . 6 ⊢ ((𝑊 ∈ ℂMod ∧ 1 ∈ (Base‘𝐹)) → -1 = ((invg‘𝐹)‘1)) |
7 | 5, 6 | mpdan 687 | . . . . 5 ⊢ (𝑊 ∈ ℂMod → -1 = ((invg‘𝐹)‘1)) |
8 | 1 | clm1 23825 | . . . . . 6 ⊢ (𝑊 ∈ ℂMod → 1 = (1r‘𝐹)) |
9 | 8 | fveq2d 6678 | . . . . 5 ⊢ (𝑊 ∈ ℂMod → ((invg‘𝐹)‘1) = ((invg‘𝐹)‘(1r‘𝐹))) |
10 | 7, 9 | eqtrd 2773 | . . . 4 ⊢ (𝑊 ∈ ℂMod → -1 = ((invg‘𝐹)‘(1r‘𝐹))) |
11 | 10 | adantr 484 | . . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝑋 ∈ 𝑉) → -1 = ((invg‘𝐹)‘(1r‘𝐹))) |
12 | 11 | oveq1d 7185 | . 2 ⊢ ((𝑊 ∈ ℂMod ∧ 𝑋 ∈ 𝑉) → (-1 · 𝑋) = (((invg‘𝐹)‘(1r‘𝐹)) · 𝑋)) |
13 | clmlmod 23819 | . . 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 2738 | . . . 4 ⊢ (1r‘𝐹) = (1r‘𝐹) | |
18 | eqid 2738 | . . . 4 ⊢ (invg‘𝐹) = (invg‘𝐹) | |
19 | 14, 15, 1, 16, 17, 18 | lmodvneg1 19796 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (((invg‘𝐹)‘(1r‘𝐹)) · 𝑋) = (𝑁‘𝑋)) |
20 | 13, 19 | sylan 583 | . 2 ⊢ ((𝑊 ∈ ℂMod ∧ 𝑋 ∈ 𝑉) → (((invg‘𝐹)‘(1r‘𝐹)) · 𝑋) = (𝑁‘𝑋)) |
21 | 12, 20 | eqtrd 2773 | 1 ⊢ ((𝑊 ∈ ℂMod ∧ 𝑋 ∈ 𝑉) → (-1 · 𝑋) = (𝑁‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ‘cfv 6339 (class class class)co 7170 1c1 10616 -cneg 10949 ℤcz 12062 Basecbs 16586 Scalarcsca 16671 ·𝑠 cvsca 16672 invgcminusg 18220 1rcur 19370 LModclmod 19753 ℂModcclm 23814 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 ax-addf 10694 ax-mulf 10695 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-1st 7714 df-2nd 7715 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-nn 11717 df-2 11779 df-3 11780 df-4 11781 df-5 11782 df-6 11783 df-7 11784 df-8 11785 df-9 11786 df-n0 11977 df-z 12063 df-dec 12180 df-uz 12325 df-fz 12982 df-seq 13461 df-struct 16588 df-ndx 16589 df-slot 16590 df-base 16592 df-sets 16593 df-ress 16594 df-plusg 16681 df-mulr 16682 df-starv 16683 df-tset 16687 df-ple 16688 df-ds 16690 df-unif 16691 df-0g 16818 df-mgm 17968 df-sgrp 18017 df-mnd 18028 df-grp 18222 df-minusg 18223 df-mulg 18343 df-subg 18394 df-cmn 19026 df-mgp 19359 df-ur 19371 df-ring 19418 df-cring 19419 df-subrg 19652 df-lmod 19755 df-cnfld 20218 df-clm 23815 |
This theorem is referenced by: clmpm1dir 23855 clmvsrinv 23859 clmvslinv 23860 |
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