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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 2737 | . . . 4 ⊢ (1st ‘𝑈) = (1st ‘𝑈) | |
2 | 1 | nvvc 29385 | . . 3 ⊢ (𝑈 ∈ NrmCVec → (1st ‘𝑈) ∈ CVecOLD) |
3 | eqid 2737 | . . . . 5 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
4 | 3 | vafval 29373 | . . . 4 ⊢ ( +𝑣 ‘𝑈) = (1st ‘(1st ‘𝑈)) |
5 | nvsz.4 | . . . . 5 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
6 | 5 | smfval 29375 | . . . 4 ⊢ 𝑆 = (2nd ‘(1st ‘𝑈)) |
7 | eqid 2737 | . . . . 5 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈) | |
8 | 7, 3 | bafval 29374 | . . . 4 ⊢ (BaseSet‘𝑈) = ran ( +𝑣 ‘𝑈) |
9 | eqid 2737 | . . . 4 ⊢ (GId‘( +𝑣 ‘𝑈)) = (GId‘( +𝑣 ‘𝑈)) | |
10 | 4, 6, 8, 9 | vcz 29345 | . . 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 29376 | . . . 4 ⊢ (𝑈 ∈ NrmCVec → 𝑍 = (GId‘( +𝑣 ‘𝑈))) |
14 | 13 | adantr 481 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ) → 𝑍 = (GId‘( +𝑣 ‘𝑈))) |
15 | 14 | oveq2d 7367 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ) → (𝐴𝑆𝑍) = (𝐴𝑆(GId‘( +𝑣 ‘𝑈)))) |
16 | 11, 15, 14 | 3eqtr4d 2787 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ) → (𝐴𝑆𝑍) = 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ‘cfv 6493 (class class class)co 7351 1st c1st 7911 ℂcc 11007 GIdcgi 29260 CVecOLDcvc 29328 NrmCVeccnv 29354 +𝑣 cpv 29355 BaseSetcba 29356 ·𝑠OLD cns 29357 0veccn0v 29358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-po 5543 df-so 5544 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-1st 7913 df-2nd 7914 df-er 8606 df-en 8842 df-dom 8843 df-sdom 8844 df-pnf 11149 df-mnf 11150 df-ltxr 11152 df-grpo 29263 df-gid 29264 df-ginv 29265 df-ablo 29315 df-vc 29329 df-nv 29362 df-va 29365 df-ba 29366 df-sm 29367 df-0v 29368 df-nmcv 29370 |
This theorem is referenced by: nvmul0or 29420 nvnd 29458 dip0r 29487 0lno 29560 |
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