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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 2610 | . . . 4 ⊢ (1st ‘𝑈) = (1st ‘𝑈) | |
2 | 1 | nvvc 26854 | . . 3 ⊢ (𝑈 ∈ NrmCVec → (1st ‘𝑈) ∈ CVecOLD) |
3 | eqid 2610 | . . . . 5 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
4 | 3 | vafval 26842 | . . . 4 ⊢ ( +𝑣 ‘𝑈) = (1st ‘(1st ‘𝑈)) |
5 | nvsz.4 | . . . . 5 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
6 | 5 | smfval 26844 | . . . 4 ⊢ 𝑆 = (2nd ‘(1st ‘𝑈)) |
7 | eqid 2610 | . . . . 5 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈) | |
8 | 7, 3 | bafval 26843 | . . . 4 ⊢ (BaseSet‘𝑈) = ran ( +𝑣 ‘𝑈) |
9 | eqid 2610 | . . . 4 ⊢ (GId‘( +𝑣 ‘𝑈)) = (GId‘( +𝑣 ‘𝑈)) | |
10 | 4, 6, 8, 9 | vcz 26814 | . . 3 ⊢ (((1st ‘𝑈) ∈ CVecOLD ∧ 𝐴 ∈ ℂ) → (𝐴𝑆(GId‘( +𝑣 ‘𝑈))) = (GId‘( +𝑣 ‘𝑈))) |
11 | 2, 10 | sylan 487 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ) → (𝐴𝑆(GId‘( +𝑣 ‘𝑈))) = (GId‘( +𝑣 ‘𝑈))) |
12 | nvsz.6 | . . . . 5 ⊢ 𝑍 = (0vec‘𝑈) | |
13 | 3, 12 | 0vfval 26845 | . . . 4 ⊢ (𝑈 ∈ NrmCVec → 𝑍 = (GId‘( +𝑣 ‘𝑈))) |
14 | 13 | adantr 480 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ) → 𝑍 = (GId‘( +𝑣 ‘𝑈))) |
15 | 14 | oveq2d 6565 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ) → (𝐴𝑆𝑍) = (𝐴𝑆(GId‘( +𝑣 ‘𝑈)))) |
16 | 11, 15, 14 | 3eqtr4d 2654 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ) → (𝐴𝑆𝑍) = 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 1st c1st 7057 ℂcc 9813 GIdcgi 26728 CVecOLDcvc 26797 NrmCVeccnv 26823 +𝑣 cpv 26824 BaseSetcba 26825 ·𝑠OLD cns 26826 0veccn0v 26827 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-1st 7059 df-2nd 7060 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-ltxr 9958 df-grpo 26731 df-gid 26732 df-ginv 26733 df-ablo 26783 df-vc 26798 df-nv 26831 df-va 26834 df-ba 26835 df-sm 26836 df-0v 26837 df-nmcv 26839 |
This theorem is referenced by: nvmul0or 26889 nvnd 26927 dip0r 26956 0lno 27029 |
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