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Mirrors > Home > MPE Home > Th. List > nvsz | Structured version Visualization version GIF version |
Description: Anything times the zero vector is the zero vector. (Contributed by NM, 28-Nov-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nvsz.4 | ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) |
nvsz.6 | ⊢ 𝑍 = (0vec‘𝑈) |
Ref | Expression |
---|---|
nvsz | ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ) → (𝐴𝑆𝑍) = 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2818 | . . . 4 ⊢ (1st ‘𝑈) = (1st ‘𝑈) | |
2 | 1 | nvvc 28319 | . . 3 ⊢ (𝑈 ∈ NrmCVec → (1st ‘𝑈) ∈ CVecOLD) |
3 | eqid 2818 | . . . . 5 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
4 | 3 | vafval 28307 | . . . 4 ⊢ ( +𝑣 ‘𝑈) = (1st ‘(1st ‘𝑈)) |
5 | nvsz.4 | . . . . 5 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
6 | 5 | smfval 28309 | . . . 4 ⊢ 𝑆 = (2nd ‘(1st ‘𝑈)) |
7 | eqid 2818 | . . . . 5 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈) | |
8 | 7, 3 | bafval 28308 | . . . 4 ⊢ (BaseSet‘𝑈) = ran ( +𝑣 ‘𝑈) |
9 | eqid 2818 | . . . 4 ⊢ (GId‘( +𝑣 ‘𝑈)) = (GId‘( +𝑣 ‘𝑈)) | |
10 | 4, 6, 8, 9 | vcz 28279 | . . 3 ⊢ (((1st ‘𝑈) ∈ CVecOLD ∧ 𝐴 ∈ ℂ) → (𝐴𝑆(GId‘( +𝑣 ‘𝑈))) = (GId‘( +𝑣 ‘𝑈))) |
11 | 2, 10 | sylan 580 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ) → (𝐴𝑆(GId‘( +𝑣 ‘𝑈))) = (GId‘( +𝑣 ‘𝑈))) |
12 | nvsz.6 | . . . . 5 ⊢ 𝑍 = (0vec‘𝑈) | |
13 | 3, 12 | 0vfval 28310 | . . . 4 ⊢ (𝑈 ∈ NrmCVec → 𝑍 = (GId‘( +𝑣 ‘𝑈))) |
14 | 13 | adantr 481 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ) → 𝑍 = (GId‘( +𝑣 ‘𝑈))) |
15 | 14 | oveq2d 7161 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ) → (𝐴𝑆𝑍) = (𝐴𝑆(GId‘( +𝑣 ‘𝑈)))) |
16 | 11, 15, 14 | 3eqtr4d 2863 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ) → (𝐴𝑆𝑍) = 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ‘cfv 6348 (class class class)co 7145 1st c1st 7676 ℂcc 10523 GIdcgi 28194 CVecOLDcvc 28262 NrmCVeccnv 28288 +𝑣 cpv 28289 BaseSetcba 28290 ·𝑠OLD cns 28291 0veccn0v 28292 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-1st 7678 df-2nd 7679 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-ltxr 10668 df-grpo 28197 df-gid 28198 df-ginv 28199 df-ablo 28249 df-vc 28263 df-nv 28296 df-va 28299 df-ba 28300 df-sm 28301 df-0v 28302 df-nmcv 28304 |
This theorem is referenced by: nvmul0or 28354 nvnd 28392 dip0r 28421 0lno 28494 |
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