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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 2821 | . . . 4 ⊢ (1st ‘𝑈) = (1st ‘𝑈) | |
2 | 1 | nvvc 28386 | . . 3 ⊢ (𝑈 ∈ NrmCVec → (1st ‘𝑈) ∈ CVecOLD) |
3 | eqid 2821 | . . . . 5 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
4 | 3 | vafval 28374 | . . . 4 ⊢ ( +𝑣 ‘𝑈) = (1st ‘(1st ‘𝑈)) |
5 | nvsz.4 | . . . . 5 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
6 | 5 | smfval 28376 | . . . 4 ⊢ 𝑆 = (2nd ‘(1st ‘𝑈)) |
7 | eqid 2821 | . . . . 5 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈) | |
8 | 7, 3 | bafval 28375 | . . . 4 ⊢ (BaseSet‘𝑈) = ran ( +𝑣 ‘𝑈) |
9 | eqid 2821 | . . . 4 ⊢ (GId‘( +𝑣 ‘𝑈)) = (GId‘( +𝑣 ‘𝑈)) | |
10 | 4, 6, 8, 9 | vcz 28346 | . . 3 ⊢ (((1st ‘𝑈) ∈ CVecOLD ∧ 𝐴 ∈ ℂ) → (𝐴𝑆(GId‘( +𝑣 ‘𝑈))) = (GId‘( +𝑣 ‘𝑈))) |
11 | 2, 10 | sylan 582 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ) → (𝐴𝑆(GId‘( +𝑣 ‘𝑈))) = (GId‘( +𝑣 ‘𝑈))) |
12 | nvsz.6 | . . . . 5 ⊢ 𝑍 = (0vec‘𝑈) | |
13 | 3, 12 | 0vfval 28377 | . . . 4 ⊢ (𝑈 ∈ NrmCVec → 𝑍 = (GId‘( +𝑣 ‘𝑈))) |
14 | 13 | adantr 483 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ) → 𝑍 = (GId‘( +𝑣 ‘𝑈))) |
15 | 14 | oveq2d 7166 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ) → (𝐴𝑆𝑍) = (𝐴𝑆(GId‘( +𝑣 ‘𝑈)))) |
16 | 11, 15, 14 | 3eqtr4d 2866 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ) → (𝐴𝑆𝑍) = 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ‘cfv 6350 (class class class)co 7150 1st c1st 7681 ℂcc 10529 GIdcgi 28261 CVecOLDcvc 28329 NrmCVeccnv 28355 +𝑣 cpv 28356 BaseSetcba 28357 ·𝑠OLD cns 28358 0veccn0v 28359 |
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 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 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 |
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-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-po 5469 df-so 5470 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-1st 7683 df-2nd 7684 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-ltxr 10674 df-grpo 28264 df-gid 28265 df-ginv 28266 df-ablo 28316 df-vc 28330 df-nv 28363 df-va 28366 df-ba 28367 df-sm 28368 df-0v 28369 df-nmcv 28371 |
This theorem is referenced by: nvmul0or 28421 nvnd 28459 dip0r 28488 0lno 28561 |
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