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Mirrors > Home > MPE Home > Th. List > clmneg | Structured version Visualization version GIF version |
Description: Negation in the scalar ring of a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.) |
Ref | Expression |
---|---|
clm0.f | ⊢ 𝐹 = (Scalar‘𝑊) |
clmsub.k | ⊢ 𝐾 = (Base‘𝐹) |
Ref | Expression |
---|---|
clmneg | ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝐾) → -𝐴 = ((invg‘𝐹)‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clm0.f | . . . . . 6 ⊢ 𝐹 = (Scalar‘𝑊) | |
2 | clmsub.k | . . . . . 6 ⊢ 𝐾 = (Base‘𝐹) | |
3 | 1, 2 | clmsca 24238 | . . . . 5 ⊢ (𝑊 ∈ ℂMod → 𝐹 = (ℂfld ↾s 𝐾)) |
4 | 3 | fveq2d 6770 | . . . 4 ⊢ (𝑊 ∈ ℂMod → (invg‘𝐹) = (invg‘(ℂfld ↾s 𝐾))) |
5 | 4 | adantr 481 | . . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝐾) → (invg‘𝐹) = (invg‘(ℂfld ↾s 𝐾))) |
6 | 5 | fveq1d 6768 | . 2 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝐾) → ((invg‘𝐹)‘𝐴) = ((invg‘(ℂfld ↾s 𝐾))‘𝐴)) |
7 | 1, 2 | clmsubrg 24239 | . . . 4 ⊢ (𝑊 ∈ ℂMod → 𝐾 ∈ (SubRing‘ℂfld)) |
8 | subrgsubg 20040 | . . . 4 ⊢ (𝐾 ∈ (SubRing‘ℂfld) → 𝐾 ∈ (SubGrp‘ℂfld)) | |
9 | 7, 8 | syl 17 | . . 3 ⊢ (𝑊 ∈ ℂMod → 𝐾 ∈ (SubGrp‘ℂfld)) |
10 | eqid 2738 | . . . 4 ⊢ (ℂfld ↾s 𝐾) = (ℂfld ↾s 𝐾) | |
11 | eqid 2738 | . . . 4 ⊢ (invg‘ℂfld) = (invg‘ℂfld) | |
12 | eqid 2738 | . . . 4 ⊢ (invg‘(ℂfld ↾s 𝐾)) = (invg‘(ℂfld ↾s 𝐾)) | |
13 | 10, 11, 12 | subginv 18772 | . . 3 ⊢ ((𝐾 ∈ (SubGrp‘ℂfld) ∧ 𝐴 ∈ 𝐾) → ((invg‘ℂfld)‘𝐴) = ((invg‘(ℂfld ↾s 𝐾))‘𝐴)) |
14 | 9, 13 | sylan 580 | . 2 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝐾) → ((invg‘ℂfld)‘𝐴) = ((invg‘(ℂfld ↾s 𝐾))‘𝐴)) |
15 | 1, 2 | clmsscn 24252 | . . . 4 ⊢ (𝑊 ∈ ℂMod → 𝐾 ⊆ ℂ) |
16 | 15 | sselda 3920 | . . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝐾) → 𝐴 ∈ ℂ) |
17 | cnfldneg 20634 | . . 3 ⊢ (𝐴 ∈ ℂ → ((invg‘ℂfld)‘𝐴) = -𝐴) | |
18 | 16, 17 | syl 17 | . 2 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝐾) → ((invg‘ℂfld)‘𝐴) = -𝐴) |
19 | 6, 14, 18 | 3eqtr2rd 2785 | 1 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝐾) → -𝐴 = ((invg‘𝐹)‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ‘cfv 6426 (class class class)co 7267 ℂcc 10879 -cneg 11216 Basecbs 16922 ↾s cress 16951 Scalarcsca 16975 invgcminusg 18588 SubGrpcsubg 18759 SubRingcsubrg 20030 ℂfldccnfld 20607 ℂModcclm 24235 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 ax-cnex 10937 ax-resscn 10938 ax-1cn 10939 ax-icn 10940 ax-addcl 10941 ax-addrcl 10942 ax-mulcl 10943 ax-mulrcl 10944 ax-mulcom 10945 ax-addass 10946 ax-mulass 10947 ax-distr 10948 ax-i2m1 10949 ax-1ne0 10950 ax-1rid 10951 ax-rnegex 10952 ax-rrecex 10953 ax-cnre 10954 ax-pre-lttri 10955 ax-pre-lttrn 10956 ax-pre-ltadd 10957 ax-pre-mulgt0 10958 ax-addf 10960 ax-mulf 10961 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-pss 3905 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-iun 4926 df-br 5074 df-opab 5136 df-mpt 5157 df-tr 5191 df-id 5484 df-eprel 5490 df-po 5498 df-so 5499 df-fr 5539 df-we 5541 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-pred 6195 df-ord 6262 df-on 6263 df-lim 6264 df-suc 6265 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-riota 7224 df-ov 7270 df-oprab 7271 df-mpo 7272 df-om 7703 df-1st 7820 df-2nd 7821 df-frecs 8084 df-wrecs 8115 df-recs 8189 df-rdg 8228 df-1o 8284 df-er 8485 df-en 8721 df-dom 8722 df-sdom 8723 df-fin 8724 df-pnf 11021 df-mnf 11022 df-xr 11023 df-ltxr 11024 df-le 11025 df-sub 11217 df-neg 11218 df-nn 11984 df-2 12046 df-3 12047 df-4 12048 df-5 12049 df-6 12050 df-7 12051 df-8 12052 df-9 12053 df-n0 12244 df-z 12330 df-dec 12448 df-uz 12593 df-fz 13250 df-struct 16858 df-sets 16875 df-slot 16893 df-ndx 16905 df-base 16923 df-ress 16952 df-plusg 16985 df-mulr 16986 df-starv 16987 df-tset 16991 df-ple 16992 df-ds 16994 df-unif 16995 df-0g 17162 df-mgm 18336 df-sgrp 18385 df-mnd 18396 df-grp 18590 df-minusg 18591 df-subg 18762 df-cmn 19398 df-mgp 19731 df-ring 19795 df-cring 19796 df-subrg 20032 df-cnfld 20608 df-clm 24236 |
This theorem is referenced by: clmvneg1 24272 clmvsneg 24273 clmvsubval 24282 ncvspi 24330 |
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