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Mirrors > Home > MPE Home > Th. List > clmsub | Structured version Visualization version GIF version |
Description: Subtraction 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 |
---|---|
clmsub | ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) → (𝐴 − 𝐵) = (𝐴(-g‘𝐹)𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clm0.f | . . . . 5 ⊢ 𝐹 = (Scalar‘𝑊) | |
2 | clmsub.k | . . . . 5 ⊢ 𝐾 = (Base‘𝐹) | |
3 | 1, 2 | clmsubrg 23664 | . . . 4 ⊢ (𝑊 ∈ ℂMod → 𝐾 ∈ (SubRing‘ℂfld)) |
4 | subrgsubg 19535 | . . . 4 ⊢ (𝐾 ∈ (SubRing‘ℂfld) → 𝐾 ∈ (SubGrp‘ℂfld)) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ (𝑊 ∈ ℂMod → 𝐾 ∈ (SubGrp‘ℂfld)) |
6 | cnfldsub 20567 | . . . 4 ⊢ − = (-g‘ℂfld) | |
7 | eqid 2821 | . . . 4 ⊢ (ℂfld ↾s 𝐾) = (ℂfld ↾s 𝐾) | |
8 | eqid 2821 | . . . 4 ⊢ (-g‘(ℂfld ↾s 𝐾)) = (-g‘(ℂfld ↾s 𝐾)) | |
9 | 6, 7, 8 | subgsub 18285 | . . 3 ⊢ ((𝐾 ∈ (SubGrp‘ℂfld) ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) → (𝐴 − 𝐵) = (𝐴(-g‘(ℂfld ↾s 𝐾))𝐵)) |
10 | 5, 9 | syl3an1 1159 | . 2 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) → (𝐴 − 𝐵) = (𝐴(-g‘(ℂfld ↾s 𝐾))𝐵)) |
11 | 1, 2 | clmsca 23663 | . . . . 5 ⊢ (𝑊 ∈ ℂMod → 𝐹 = (ℂfld ↾s 𝐾)) |
12 | 11 | fveq2d 6668 | . . . 4 ⊢ (𝑊 ∈ ℂMod → (-g‘𝐹) = (-g‘(ℂfld ↾s 𝐾))) |
13 | 12 | 3ad2ant1 1129 | . . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) → (-g‘𝐹) = (-g‘(ℂfld ↾s 𝐾))) |
14 | 13 | oveqd 7167 | . 2 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) → (𝐴(-g‘𝐹)𝐵) = (𝐴(-g‘(ℂfld ↾s 𝐾))𝐵)) |
15 | 10, 14 | eqtr4d 2859 | 1 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) → (𝐴 − 𝐵) = (𝐴(-g‘𝐹)𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ‘cfv 6349 (class class class)co 7150 − cmin 10864 Basecbs 16477 ↾s cress 16478 Scalarcsca 16562 -gcsg 18099 SubGrpcsubg 18267 SubRingcsubrg 19525 ℂfldccnfld 20539 ℂModcclm 23660 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-addf 10610 ax-mulf 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12887 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-minusg 18101 df-sbg 18102 df-subg 18270 df-cmn 18902 df-mgp 19234 df-ring 19293 df-cring 19294 df-subrg 19527 df-cnfld 20540 df-clm 23661 |
This theorem is referenced by: clmsubdir 23700 cphsubdir 23806 cphsubdi 23807 cph2subdi 23808 ipcau2 23831 tcphcphlem1 23832 ttgcontlem1 26665 |
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