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Mirrors > Home > MPE Home > Th. List > vc0 | Structured version Visualization version GIF version |
Description: Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
vc0.1 | ⊢ 𝐺 = (1st ‘𝑊) |
vc0.2 | ⊢ 𝑆 = (2nd ‘𝑊) |
vc0.3 | ⊢ 𝑋 = ran 𝐺 |
vc0.4 | ⊢ 𝑍 = (GId‘𝐺) |
Ref | Expression |
---|---|
vc0 | ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (0𝑆𝐴) = 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vc0.1 | . . . 4 ⊢ 𝐺 = (1st ‘𝑊) | |
2 | vc0.3 | . . . 4 ⊢ 𝑋 = ran 𝐺 | |
3 | vc0.4 | . . . 4 ⊢ 𝑍 = (GId‘𝐺) | |
4 | 1, 2, 3 | vc0rid 26812 | . . 3 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺𝑍) = 𝐴) |
5 | 1p0e1 11010 | . . . . 5 ⊢ (1 + 0) = 1 | |
6 | 5 | oveq1i 6559 | . . . 4 ⊢ ((1 + 0)𝑆𝐴) = (1𝑆𝐴) |
7 | 0cn 9911 | . . . . 5 ⊢ 0 ∈ ℂ | |
8 | ax-1cn 9873 | . . . . . 6 ⊢ 1 ∈ ℂ | |
9 | vc0.2 | . . . . . . 7 ⊢ 𝑆 = (2nd ‘𝑊) | |
10 | 1, 9, 2 | vcdir 26805 | . . . . . 6 ⊢ ((𝑊 ∈ CVecOLD ∧ (1 ∈ ℂ ∧ 0 ∈ ℂ ∧ 𝐴 ∈ 𝑋)) → ((1 + 0)𝑆𝐴) = ((1𝑆𝐴)𝐺(0𝑆𝐴))) |
11 | 8, 10 | mp3anr1 1413 | . . . . 5 ⊢ ((𝑊 ∈ CVecOLD ∧ (0 ∈ ℂ ∧ 𝐴 ∈ 𝑋)) → ((1 + 0)𝑆𝐴) = ((1𝑆𝐴)𝐺(0𝑆𝐴))) |
12 | 7, 11 | mpanr1 715 | . . . 4 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((1 + 0)𝑆𝐴) = ((1𝑆𝐴)𝐺(0𝑆𝐴))) |
13 | 1, 9, 2 | vcidOLD 26803 | . . . 4 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (1𝑆𝐴) = 𝐴) |
14 | 6, 12, 13 | 3eqtr3a 2668 | . . 3 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((1𝑆𝐴)𝐺(0𝑆𝐴)) = 𝐴) |
15 | 13 | oveq1d 6564 | . . 3 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((1𝑆𝐴)𝐺(0𝑆𝐴)) = (𝐴𝐺(0𝑆𝐴))) |
16 | 4, 14, 15 | 3eqtr2rd 2651 | . 2 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺(0𝑆𝐴)) = (𝐴𝐺𝑍)) |
17 | 1, 9, 2 | vccl 26802 | . . . . 5 ⊢ ((𝑊 ∈ CVecOLD ∧ 0 ∈ ℂ ∧ 𝐴 ∈ 𝑋) → (0𝑆𝐴) ∈ 𝑋) |
18 | 7, 17 | mp3an2 1404 | . . . 4 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (0𝑆𝐴) ∈ 𝑋) |
19 | 1, 2, 3 | vczcl 26811 | . . . . 5 ⊢ (𝑊 ∈ CVecOLD → 𝑍 ∈ 𝑋) |
20 | 19 | adantr 480 | . . . 4 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → 𝑍 ∈ 𝑋) |
21 | simpr 476 | . . . 4 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋) | |
22 | 18, 20, 21 | 3jca 1235 | . . 3 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((0𝑆𝐴) ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) |
23 | 1, 2 | vclcan 26810 | . . 3 ⊢ ((𝑊 ∈ CVecOLD ∧ ((0𝑆𝐴) ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝐴𝐺(0𝑆𝐴)) = (𝐴𝐺𝑍) ↔ (0𝑆𝐴) = 𝑍)) |
24 | 22, 23 | syldan 486 | . 2 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((𝐴𝐺(0𝑆𝐴)) = (𝐴𝐺𝑍) ↔ (0𝑆𝐴) = 𝑍)) |
25 | 16, 24 | mpbid 221 | 1 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (0𝑆𝐴) = 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ran crn 5039 ‘cfv 5804 (class class class)co 6549 1st c1st 7057 2nd c2nd 7058 ℂcc 9813 0cc0 9815 1c1 9816 + caddc 9818 GIdcgi 26728 CVecOLDcvc 26797 |
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-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 |
This theorem is referenced by: vcz 26814 vcm 26815 nv0 26876 |
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