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Mirrors > Home > MPE Home > Th. List > cmodscmulexp | Structured version Visualization version GIF version |
Description: The scalar product of a vector with powers of i belongs to the base set of a subcomplex module if the scalar subring of th subcomplex module contains i. (Contributed by AV, 18-Oct-2021.) |
Ref | Expression |
---|---|
cmodscexp.f | ⊢ 𝐹 = (Scalar‘𝑊) |
cmodscexp.k | ⊢ 𝐾 = (Base‘𝐹) |
cmodscmulexp.x | ⊢ 𝑋 = (Base‘𝑊) |
cmodscmulexp.s | ⊢ · = ( ·𝑠 ‘𝑊) |
Ref | Expression |
---|---|
cmodscmulexp | ⊢ ((𝑊 ∈ ℂMod ∧ (i ∈ 𝐾 ∧ 𝐵 ∈ 𝑋 ∧ 𝑁 ∈ ℕ)) → ((i↑𝑁) · 𝐵) ∈ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clmlmod 25112 | . . 3 ⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ LMod) | |
2 | 1 | adantr 480 | . 2 ⊢ ((𝑊 ∈ ℂMod ∧ (i ∈ 𝐾 ∧ 𝐵 ∈ 𝑋 ∧ 𝑁 ∈ ℕ)) → 𝑊 ∈ LMod) |
3 | simp1 1136 | . . . 4 ⊢ ((i ∈ 𝐾 ∧ 𝐵 ∈ 𝑋 ∧ 𝑁 ∈ ℕ) → i ∈ 𝐾) | |
4 | 3 | anim2i 616 | . . 3 ⊢ ((𝑊 ∈ ℂMod ∧ (i ∈ 𝐾 ∧ 𝐵 ∈ 𝑋 ∧ 𝑁 ∈ ℕ)) → (𝑊 ∈ ℂMod ∧ i ∈ 𝐾)) |
5 | simpr3 1196 | . . 3 ⊢ ((𝑊 ∈ ℂMod ∧ (i ∈ 𝐾 ∧ 𝐵 ∈ 𝑋 ∧ 𝑁 ∈ ℕ)) → 𝑁 ∈ ℕ) | |
6 | cmodscexp.f | . . . 4 ⊢ 𝐹 = (Scalar‘𝑊) | |
7 | cmodscexp.k | . . . 4 ⊢ 𝐾 = (Base‘𝐹) | |
8 | 6, 7 | cmodscexp 25166 | . . 3 ⊢ (((𝑊 ∈ ℂMod ∧ i ∈ 𝐾) ∧ 𝑁 ∈ ℕ) → (i↑𝑁) ∈ 𝐾) |
9 | 4, 5, 8 | syl2anc 583 | . 2 ⊢ ((𝑊 ∈ ℂMod ∧ (i ∈ 𝐾 ∧ 𝐵 ∈ 𝑋 ∧ 𝑁 ∈ ℕ)) → (i↑𝑁) ∈ 𝐾) |
10 | simpr2 1195 | . 2 ⊢ ((𝑊 ∈ ℂMod ∧ (i ∈ 𝐾 ∧ 𝐵 ∈ 𝑋 ∧ 𝑁 ∈ ℕ)) → 𝐵 ∈ 𝑋) | |
11 | cmodscmulexp.x | . . 3 ⊢ 𝑋 = (Base‘𝑊) | |
12 | cmodscmulexp.s | . . 3 ⊢ · = ( ·𝑠 ‘𝑊) | |
13 | 11, 6, 12, 7 | lmodvscl 20893 | . 2 ⊢ ((𝑊 ∈ LMod ∧ (i↑𝑁) ∈ 𝐾 ∧ 𝐵 ∈ 𝑋) → ((i↑𝑁) · 𝐵) ∈ 𝑋) |
14 | 2, 9, 10, 13 | syl3anc 1371 | 1 ⊢ ((𝑊 ∈ ℂMod ∧ (i ∈ 𝐾 ∧ 𝐵 ∈ 𝑋 ∧ 𝑁 ∈ ℕ)) → ((i↑𝑁) · 𝐵) ∈ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2103 ‘cfv 6572 (class class class)co 7445 ici 11182 ℕcn 12289 ↑cexp 14108 Basecbs 17253 Scalarcsca 17309 ·𝑠 cvsca 17310 LModclmod 20875 ℂModcclm 25107 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-cnex 11236 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 ax-pre-mulgt0 11257 ax-addf 11259 ax-mulf 11260 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3383 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-om 7900 df-1st 8026 df-2nd 8027 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-1o 8518 df-er 8759 df-en 9000 df-dom 9001 df-sdom 9002 df-fin 9003 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-sub 11518 df-neg 11519 df-nn 12290 df-2 12352 df-3 12353 df-4 12354 df-5 12355 df-6 12356 df-7 12357 df-8 12358 df-9 12359 df-n0 12550 df-z 12636 df-dec 12755 df-uz 12900 df-fz 13564 df-seq 14049 df-exp 14109 df-struct 17189 df-sets 17206 df-slot 17224 df-ndx 17236 df-base 17254 df-ress 17283 df-plusg 17319 df-mulr 17320 df-starv 17321 df-tset 17325 df-ple 17326 df-ds 17328 df-unif 17329 df-0g 17496 df-mgm 18673 df-sgrp 18752 df-mnd 18768 df-submnd 18814 df-grp 18971 df-mulg 19103 df-cmn 19819 df-mgp 20157 df-ur 20204 df-ring 20257 df-cring 20258 df-subrg 20592 df-lmod 20877 df-cnfld 21383 df-clm 25108 |
This theorem is referenced by: (None) |
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