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Mirrors > Home > MPE Home > Th. List > vcz | Structured version Visualization version GIF version |
Description: Anything times the zero vector is the zero vector. Equation 1b of [Kreyszig] p. 51. (Contributed by NM, 24-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 |
---|---|
vcz | ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ ℂ) → (𝐴𝑆𝑍) = 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vc0.1 | . . . . . 6 ⊢ 𝐺 = (1st ‘𝑊) | |
2 | vc0.3 | . . . . . 6 ⊢ 𝑋 = ran 𝐺 | |
3 | vc0.4 | . . . . . 6 ⊢ 𝑍 = (GId‘𝐺) | |
4 | 1, 2, 3 | vczcl 28355 | . . . . 5 ⊢ (𝑊 ∈ CVecOLD → 𝑍 ∈ 𝑋) |
5 | 4 | anim2i 619 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑊 ∈ CVecOLD) → (𝐴 ∈ ℂ ∧ 𝑍 ∈ 𝑋)) |
6 | 5 | ancoms 462 | . . 3 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ ℂ) → (𝐴 ∈ ℂ ∧ 𝑍 ∈ 𝑋)) |
7 | 0cn 10622 | . . . 4 ⊢ 0 ∈ ℂ | |
8 | vc0.2 | . . . . 5 ⊢ 𝑆 = (2nd ‘𝑊) | |
9 | 1, 8, 2 | vcass 28350 | . . . 4 ⊢ ((𝑊 ∈ CVecOLD ∧ (𝐴 ∈ ℂ ∧ 0 ∈ ℂ ∧ 𝑍 ∈ 𝑋)) → ((𝐴 · 0)𝑆𝑍) = (𝐴𝑆(0𝑆𝑍))) |
10 | 7, 9 | mp3anr2 1456 | . . 3 ⊢ ((𝑊 ∈ CVecOLD ∧ (𝐴 ∈ ℂ ∧ 𝑍 ∈ 𝑋)) → ((𝐴 · 0)𝑆𝑍) = (𝐴𝑆(0𝑆𝑍))) |
11 | 6, 10 | syldan 594 | . 2 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ ℂ) → ((𝐴 · 0)𝑆𝑍) = (𝐴𝑆(0𝑆𝑍))) |
12 | mul01 10808 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝐴 · 0) = 0) | |
13 | 12 | oveq1d 7150 | . . 3 ⊢ (𝐴 ∈ ℂ → ((𝐴 · 0)𝑆𝑍) = (0𝑆𝑍)) |
14 | 1, 8, 2, 3 | vc0 28357 | . . . 4 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝑍 ∈ 𝑋) → (0𝑆𝑍) = 𝑍) |
15 | 4, 14 | mpdan 686 | . . 3 ⊢ (𝑊 ∈ CVecOLD → (0𝑆𝑍) = 𝑍) |
16 | 13, 15 | sylan9eqr 2855 | . 2 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ ℂ) → ((𝐴 · 0)𝑆𝑍) = 𝑍) |
17 | 15 | oveq2d 7151 | . . 3 ⊢ (𝑊 ∈ CVecOLD → (𝐴𝑆(0𝑆𝑍)) = (𝐴𝑆𝑍)) |
18 | 17 | adantr 484 | . 2 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ ℂ) → (𝐴𝑆(0𝑆𝑍)) = (𝐴𝑆𝑍)) |
19 | 11, 16, 18 | 3eqtr3rd 2842 | 1 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ ℂ) → (𝐴𝑆𝑍) = 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ran crn 5520 ‘cfv 6324 (class class class)co 7135 1st c1st 7669 2nd c2nd 7670 ℂcc 10524 0cc0 10526 · cmul 10531 GIdcgi 28273 CVecOLDcvc 28341 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-1st 7671 df-2nd 7672 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-ltxr 10669 df-grpo 28276 df-gid 28277 df-ginv 28278 df-ablo 28328 df-vc 28342 |
This theorem is referenced by: nvsz 28421 |
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