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Mirrors > Home > MPE Home > Th. List > clmvs2 | Structured version Visualization version GIF version |
Description: A vector plus itself is two times the vector. (Contributed by NM, 1-Feb-2007.) (Revised by AV, 21-Sep-2021.) |
Ref | Expression |
---|---|
clmvs1.v | ⊢ 𝑉 = (Base‘𝑊) |
clmvs1.s | ⊢ · = ( ·𝑠 ‘𝑊) |
clmvs2.a | ⊢ + = (+g‘𝑊) |
Ref | Expression |
---|---|
clmvs2 | ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → (𝐴 + 𝐴) = (2 · 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-2 12327 | . . . 4 ⊢ 2 = (1 + 1) | |
2 | 1 | oveq1i 7434 | . . 3 ⊢ (2 · 𝐴) = ((1 + 1) · 𝐴) |
3 | 2 | a1i 11 | . 2 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → (2 · 𝐴) = ((1 + 1) · 𝐴)) |
4 | simpl 481 | . . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → 𝑊 ∈ ℂMod) | |
5 | eqid 2726 | . . . . . 6 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
6 | 5 | clm1 25091 | . . . . 5 ⊢ (𝑊 ∈ ℂMod → 1 = (1r‘(Scalar‘𝑊))) |
7 | clmlmod 25085 | . . . . . 6 ⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ LMod) | |
8 | eqid 2726 | . . . . . . 7 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
9 | eqid 2726 | . . . . . . 7 ⊢ (1r‘(Scalar‘𝑊)) = (1r‘(Scalar‘𝑊)) | |
10 | 5, 8, 9 | lmod1cl 20865 | . . . . . 6 ⊢ (𝑊 ∈ LMod → (1r‘(Scalar‘𝑊)) ∈ (Base‘(Scalar‘𝑊))) |
11 | 7, 10 | syl 17 | . . . . 5 ⊢ (𝑊 ∈ ℂMod → (1r‘(Scalar‘𝑊)) ∈ (Base‘(Scalar‘𝑊))) |
12 | 6, 11 | eqeltrd 2826 | . . . 4 ⊢ (𝑊 ∈ ℂMod → 1 ∈ (Base‘(Scalar‘𝑊))) |
13 | 12 | adantr 479 | . . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → 1 ∈ (Base‘(Scalar‘𝑊))) |
14 | simpr 483 | . . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → 𝐴 ∈ 𝑉) | |
15 | clmvs1.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
16 | clmvs1.s | . . . 4 ⊢ · = ( ·𝑠 ‘𝑊) | |
17 | clmvs2.a | . . . 4 ⊢ + = (+g‘𝑊) | |
18 | 15, 5, 16, 8, 17 | clmvsdir 25109 | . . 3 ⊢ ((𝑊 ∈ ℂMod ∧ (1 ∈ (Base‘(Scalar‘𝑊)) ∧ 1 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝐴 ∈ 𝑉)) → ((1 + 1) · 𝐴) = ((1 · 𝐴) + (1 · 𝐴))) |
19 | 4, 13, 13, 14, 18 | syl13anc 1369 | . 2 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → ((1 + 1) · 𝐴) = ((1 · 𝐴) + (1 · 𝐴))) |
20 | 15, 16 | clmvs1 25111 | . . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → (1 · 𝐴) = 𝐴) |
21 | 20, 20 | oveq12d 7442 | . 2 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → ((1 · 𝐴) + (1 · 𝐴)) = (𝐴 + 𝐴)) |
22 | 3, 19, 21 | 3eqtrrd 2771 | 1 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → (𝐴 + 𝐴) = (2 · 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ‘cfv 6554 (class class class)co 7424 1c1 11159 + caddc 11161 2c2 12319 Basecbs 17213 +gcplusg 17266 Scalarcsca 17269 ·𝑠 cvsca 17270 1rcur 20164 LModclmod 20836 ℂModcclm 25080 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-addf 11237 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12611 df-dec 12730 df-uz 12875 df-fz 13539 df-struct 17149 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-ress 17243 df-plusg 17279 df-mulr 17280 df-starv 17281 df-tset 17285 df-ple 17286 df-ds 17288 df-unif 17289 df-0g 17456 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-grp 18931 df-subg 19117 df-cmn 19780 df-mgp 20118 df-ur 20165 df-ring 20218 df-cring 20219 df-subrg 20553 df-lmod 20838 df-cnfld 21344 df-clm 25081 |
This theorem is referenced by: (None) |
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