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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 28349 | . . 3 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺𝑍) = 𝐴) |
5 | 1p0e1 11760 | . . . . 5 ⊢ (1 + 0) = 1 | |
6 | 5 | oveq1i 7165 | . . . 4 ⊢ ((1 + 0)𝑆𝐴) = (1𝑆𝐴) |
7 | 0cn 10632 | . . . . 5 ⊢ 0 ∈ ℂ | |
8 | ax-1cn 10594 | . . . . . 6 ⊢ 1 ∈ ℂ | |
9 | vc0.2 | . . . . . . 7 ⊢ 𝑆 = (2nd ‘𝑊) | |
10 | 1, 9, 2 | vcdir 28342 | . . . . . 6 ⊢ ((𝑊 ∈ CVecOLD ∧ (1 ∈ ℂ ∧ 0 ∈ ℂ ∧ 𝐴 ∈ 𝑋)) → ((1 + 0)𝑆𝐴) = ((1𝑆𝐴)𝐺(0𝑆𝐴))) |
11 | 8, 10 | mp3anr1 1454 | . . . . 5 ⊢ ((𝑊 ∈ CVecOLD ∧ (0 ∈ ℂ ∧ 𝐴 ∈ 𝑋)) → ((1 + 0)𝑆𝐴) = ((1𝑆𝐴)𝐺(0𝑆𝐴))) |
12 | 7, 11 | mpanr1 701 | . . . 4 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((1 + 0)𝑆𝐴) = ((1𝑆𝐴)𝐺(0𝑆𝐴))) |
13 | 1, 9, 2 | vcidOLD 28340 | . . . 4 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (1𝑆𝐴) = 𝐴) |
14 | 6, 12, 13 | 3eqtr3a 2880 | . . 3 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((1𝑆𝐴)𝐺(0𝑆𝐴)) = 𝐴) |
15 | 13 | oveq1d 7170 | . . 3 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((1𝑆𝐴)𝐺(0𝑆𝐴)) = (𝐴𝐺(0𝑆𝐴))) |
16 | 4, 14, 15 | 3eqtr2rd 2863 | . 2 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺(0𝑆𝐴)) = (𝐴𝐺𝑍)) |
17 | 1, 9, 2 | vccl 28339 | . . . . 5 ⊢ ((𝑊 ∈ CVecOLD ∧ 0 ∈ ℂ ∧ 𝐴 ∈ 𝑋) → (0𝑆𝐴) ∈ 𝑋) |
18 | 7, 17 | mp3an2 1445 | . . . 4 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (0𝑆𝐴) ∈ 𝑋) |
19 | 1, 2, 3 | vczcl 28348 | . . . . 5 ⊢ (𝑊 ∈ CVecOLD → 𝑍 ∈ 𝑋) |
20 | 19 | adantr 483 | . . . 4 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → 𝑍 ∈ 𝑋) |
21 | simpr 487 | . . . 4 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋) | |
22 | 18, 20, 21 | 3jca 1124 | . . 3 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((0𝑆𝐴) ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) |
23 | 1, 2 | vclcan 28347 | . . 3 ⊢ ((𝑊 ∈ CVecOLD ∧ ((0𝑆𝐴) ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝐴𝐺(0𝑆𝐴)) = (𝐴𝐺𝑍) ↔ (0𝑆𝐴) = 𝑍)) |
24 | 22, 23 | syldan 593 | . 2 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((𝐴𝐺(0𝑆𝐴)) = (𝐴𝐺𝑍) ↔ (0𝑆𝐴) = 𝑍)) |
25 | 16, 24 | mpbid 234 | 1 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (0𝑆𝐴) = 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ran crn 5555 ‘cfv 6354 (class class class)co 7155 1st c1st 7686 2nd c2nd 7687 ℂcc 10534 0cc0 10536 1c1 10537 + caddc 10539 GIdcgi 28266 CVecOLDcvc 28334 |
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 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 |
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 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-po 5473 df-so 5474 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-1st 7688 df-2nd 7689 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-pnf 10676 df-mnf 10677 df-ltxr 10679 df-grpo 28269 df-gid 28270 df-ginv 28271 df-ablo 28321 df-vc 28335 |
This theorem is referenced by: vcz 28351 vcm 28352 nv0 28413 |
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