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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 28343 | . . . . 5 ⊢ (𝑊 ∈ CVecOLD → 𝑍 ∈ 𝑋) |
5 | 4 | anim2i 618 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑊 ∈ CVecOLD) → (𝐴 ∈ ℂ ∧ 𝑍 ∈ 𝑋)) |
6 | 5 | ancoms 461 | . . 3 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ ℂ) → (𝐴 ∈ ℂ ∧ 𝑍 ∈ 𝑋)) |
7 | 0cn 10627 | . . . 4 ⊢ 0 ∈ ℂ | |
8 | vc0.2 | . . . . 5 ⊢ 𝑆 = (2nd ‘𝑊) | |
9 | 1, 8, 2 | vcass 28338 | . . . 4 ⊢ ((𝑊 ∈ CVecOLD ∧ (𝐴 ∈ ℂ ∧ 0 ∈ ℂ ∧ 𝑍 ∈ 𝑋)) → ((𝐴 · 0)𝑆𝑍) = (𝐴𝑆(0𝑆𝑍))) |
10 | 7, 9 | mp3anr2 1455 | . . 3 ⊢ ((𝑊 ∈ CVecOLD ∧ (𝐴 ∈ ℂ ∧ 𝑍 ∈ 𝑋)) → ((𝐴 · 0)𝑆𝑍) = (𝐴𝑆(0𝑆𝑍))) |
11 | 6, 10 | syldan 593 | . 2 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ ℂ) → ((𝐴 · 0)𝑆𝑍) = (𝐴𝑆(0𝑆𝑍))) |
12 | mul01 10813 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝐴 · 0) = 0) | |
13 | 12 | oveq1d 7165 | . . 3 ⊢ (𝐴 ∈ ℂ → ((𝐴 · 0)𝑆𝑍) = (0𝑆𝑍)) |
14 | 1, 8, 2, 3 | vc0 28345 | . . . 4 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝑍 ∈ 𝑋) → (0𝑆𝑍) = 𝑍) |
15 | 4, 14 | mpdan 685 | . . 3 ⊢ (𝑊 ∈ CVecOLD → (0𝑆𝑍) = 𝑍) |
16 | 13, 15 | sylan9eqr 2878 | . 2 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ ℂ) → ((𝐴 · 0)𝑆𝑍) = 𝑍) |
17 | 15 | oveq2d 7166 | . . 3 ⊢ (𝑊 ∈ CVecOLD → (𝐴𝑆(0𝑆𝑍)) = (𝐴𝑆𝑍)) |
18 | 17 | adantr 483 | . 2 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ ℂ) → (𝐴𝑆(0𝑆𝑍)) = (𝐴𝑆𝑍)) |
19 | 11, 16, 18 | 3eqtr3rd 2865 | 1 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ ℂ) → (𝐴𝑆𝑍) = 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ran crn 5550 ‘cfv 6349 (class class class)co 7150 1st c1st 7681 2nd c2nd 7682 ℂcc 10529 0cc0 10531 · cmul 10536 GIdcgi 28261 CVecOLDcvc 28329 |
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 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 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 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 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 |
This theorem is referenced by: nvsz 28409 |
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