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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 23672 | . . . . 5 ⊢ (𝑊 ∈ ℂMod → 𝐹 = (ℂfld ↾s 𝐾)) |
4 | 3 | fveq2d 6677 | . . . 4 ⊢ (𝑊 ∈ ℂMod → (invg‘𝐹) = (invg‘(ℂfld ↾s 𝐾))) |
5 | 4 | adantr 483 | . . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝐾) → (invg‘𝐹) = (invg‘(ℂfld ↾s 𝐾))) |
6 | 5 | fveq1d 6675 | . 2 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝐾) → ((invg‘𝐹)‘𝐴) = ((invg‘(ℂfld ↾s 𝐾))‘𝐴)) |
7 | 1, 2 | clmsubrg 23673 | . . . 4 ⊢ (𝑊 ∈ ℂMod → 𝐾 ∈ (SubRing‘ℂfld)) |
8 | subrgsubg 19544 | . . . 4 ⊢ (𝐾 ∈ (SubRing‘ℂfld) → 𝐾 ∈ (SubGrp‘ℂfld)) | |
9 | 7, 8 | syl 17 | . . 3 ⊢ (𝑊 ∈ ℂMod → 𝐾 ∈ (SubGrp‘ℂfld)) |
10 | eqid 2824 | . . . 4 ⊢ (ℂfld ↾s 𝐾) = (ℂfld ↾s 𝐾) | |
11 | eqid 2824 | . . . 4 ⊢ (invg‘ℂfld) = (invg‘ℂfld) | |
12 | eqid 2824 | . . . 4 ⊢ (invg‘(ℂfld ↾s 𝐾)) = (invg‘(ℂfld ↾s 𝐾)) | |
13 | 10, 11, 12 | subginv 18289 | . . 3 ⊢ ((𝐾 ∈ (SubGrp‘ℂfld) ∧ 𝐴 ∈ 𝐾) → ((invg‘ℂfld)‘𝐴) = ((invg‘(ℂfld ↾s 𝐾))‘𝐴)) |
14 | 9, 13 | sylan 582 | . 2 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝐾) → ((invg‘ℂfld)‘𝐴) = ((invg‘(ℂfld ↾s 𝐾))‘𝐴)) |
15 | 1, 2 | clmsscn 23686 | . . . 4 ⊢ (𝑊 ∈ ℂMod → 𝐾 ⊆ ℂ) |
16 | 15 | sselda 3970 | . . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝐾) → 𝐴 ∈ ℂ) |
17 | cnfldneg 20574 | . . 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 2866 | 1 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝐾) → -𝐴 = ((invg‘𝐹)‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ‘cfv 6358 (class class class)co 7159 ℂcc 10538 -cneg 10874 Basecbs 16486 ↾s cress 16487 Scalarcsca 16571 invgcminusg 18107 SubGrpcsubg 18276 SubRingcsubrg 19534 ℂfldccnfld 20548 ℂModcclm 23669 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-addf 10619 ax-mulf 10620 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-starv 16583 df-tset 16587 df-ple 16588 df-ds 16590 df-unif 16591 df-0g 16718 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-grp 18109 df-minusg 18110 df-subg 18279 df-cmn 18911 df-mgp 19243 df-ring 19302 df-cring 19303 df-subrg 19536 df-cnfld 20549 df-clm 23670 |
This theorem is referenced by: clmvneg1 23706 clmvsneg 23707 clmvsubval 23716 ncvspi 23763 |
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