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Mirrors > Home > MPE Home > Th. List > nv0 | Structured version Visualization version GIF version |
Description: Zero times a vector is the zero vector. (Contributed by NM, 27-Nov-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nv0.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
nv0.4 | ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) |
nv0.6 | ⊢ 𝑍 = (0vec‘𝑈) |
Ref | Expression |
---|---|
nv0 | ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (0𝑆𝐴) = 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2728 | . . . 4 ⊢ (1st ‘𝑈) = (1st ‘𝑈) | |
2 | 1 | nvvc 30445 | . . 3 ⊢ (𝑈 ∈ NrmCVec → (1st ‘𝑈) ∈ CVecOLD) |
3 | eqid 2728 | . . . . 5 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
4 | 3 | vafval 30433 | . . . 4 ⊢ ( +𝑣 ‘𝑈) = (1st ‘(1st ‘𝑈)) |
5 | nv0.4 | . . . . 5 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
6 | 5 | smfval 30435 | . . . 4 ⊢ 𝑆 = (2nd ‘(1st ‘𝑈)) |
7 | nv0.1 | . . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈) | |
8 | 7, 3 | bafval 30434 | . . . 4 ⊢ 𝑋 = ran ( +𝑣 ‘𝑈) |
9 | eqid 2728 | . . . 4 ⊢ (GId‘( +𝑣 ‘𝑈)) = (GId‘( +𝑣 ‘𝑈)) | |
10 | 4, 6, 8, 9 | vc0 30404 | . . 3 ⊢ (((1st ‘𝑈) ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (0𝑆𝐴) = (GId‘( +𝑣 ‘𝑈))) |
11 | 2, 10 | sylan 578 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (0𝑆𝐴) = (GId‘( +𝑣 ‘𝑈))) |
12 | nv0.6 | . . . 4 ⊢ 𝑍 = (0vec‘𝑈) | |
13 | 3, 12 | 0vfval 30436 | . . 3 ⊢ (𝑈 ∈ NrmCVec → 𝑍 = (GId‘( +𝑣 ‘𝑈))) |
14 | 13 | adantr 479 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → 𝑍 = (GId‘( +𝑣 ‘𝑈))) |
15 | 11, 14 | eqtr4d 2771 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (0𝑆𝐴) = 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ‘cfv 6553 (class class class)co 7426 1st c1st 7997 0cc0 11146 GIdcgi 30320 CVecOLDcvc 30388 NrmCVeccnv 30414 +𝑣 cpv 30415 BaseSetcba 30416 ·𝑠OLD cns 30417 0veccn0v 30418 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-po 5594 df-so 5595 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-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-1st 7999 df-2nd 8000 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-ltxr 11291 df-grpo 30323 df-gid 30324 df-ginv 30325 df-ablo 30375 df-vc 30389 df-nv 30422 df-va 30425 df-ba 30426 df-sm 30427 df-0v 30428 df-nmcv 30430 |
This theorem is referenced by: nvmul0or 30480 nvz0 30498 nvge0 30503 ipasslem1 30661 hlmul0 30739 |
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