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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 30602 | . . 3 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺𝑍) = 𝐴) |
5 | 1p0e1 12388 | . . . . 5 ⊢ (1 + 0) = 1 | |
6 | 5 | oveq1i 7441 | . . . 4 ⊢ ((1 + 0)𝑆𝐴) = (1𝑆𝐴) |
7 | 0cn 11251 | . . . . 5 ⊢ 0 ∈ ℂ | |
8 | ax-1cn 11211 | . . . . . 6 ⊢ 1 ∈ ℂ | |
9 | vc0.2 | . . . . . . 7 ⊢ 𝑆 = (2nd ‘𝑊) | |
10 | 1, 9, 2 | vcdir 30595 | . . . . . 6 ⊢ ((𝑊 ∈ CVecOLD ∧ (1 ∈ ℂ ∧ 0 ∈ ℂ ∧ 𝐴 ∈ 𝑋)) → ((1 + 0)𝑆𝐴) = ((1𝑆𝐴)𝐺(0𝑆𝐴))) |
11 | 8, 10 | mp3anr1 1457 | . . . . 5 ⊢ ((𝑊 ∈ CVecOLD ∧ (0 ∈ ℂ ∧ 𝐴 ∈ 𝑋)) → ((1 + 0)𝑆𝐴) = ((1𝑆𝐴)𝐺(0𝑆𝐴))) |
12 | 7, 11 | mpanr1 703 | . . . 4 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((1 + 0)𝑆𝐴) = ((1𝑆𝐴)𝐺(0𝑆𝐴))) |
13 | 1, 9, 2 | vcidOLD 30593 | . . . 4 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (1𝑆𝐴) = 𝐴) |
14 | 6, 12, 13 | 3eqtr3a 2799 | . . 3 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((1𝑆𝐴)𝐺(0𝑆𝐴)) = 𝐴) |
15 | 13 | oveq1d 7446 | . . 3 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((1𝑆𝐴)𝐺(0𝑆𝐴)) = (𝐴𝐺(0𝑆𝐴))) |
16 | 4, 14, 15 | 3eqtr2rd 2782 | . 2 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺(0𝑆𝐴)) = (𝐴𝐺𝑍)) |
17 | 1, 9, 2 | vccl 30592 | . . . . 5 ⊢ ((𝑊 ∈ CVecOLD ∧ 0 ∈ ℂ ∧ 𝐴 ∈ 𝑋) → (0𝑆𝐴) ∈ 𝑋) |
18 | 7, 17 | mp3an2 1448 | . . . 4 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (0𝑆𝐴) ∈ 𝑋) |
19 | 1, 2, 3 | vczcl 30601 | . . . . 5 ⊢ (𝑊 ∈ CVecOLD → 𝑍 ∈ 𝑋) |
20 | 19 | adantr 480 | . . . 4 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → 𝑍 ∈ 𝑋) |
21 | simpr 484 | . . . 4 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋) | |
22 | 18, 20, 21 | 3jca 1127 | . . 3 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((0𝑆𝐴) ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) |
23 | 1, 2 | vclcan 30600 | . . 3 ⊢ ((𝑊 ∈ CVecOLD ∧ ((0𝑆𝐴) ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝐴𝐺(0𝑆𝐴)) = (𝐴𝐺𝑍) ↔ (0𝑆𝐴) = 𝑍)) |
24 | 22, 23 | syldan 591 | . 2 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((𝐴𝐺(0𝑆𝐴)) = (𝐴𝐺𝑍) ↔ (0𝑆𝐴) = 𝑍)) |
25 | 16, 24 | mpbid 232 | 1 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (0𝑆𝐴) = 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ran crn 5690 ‘cfv 6563 (class class class)co 7431 1st c1st 8011 2nd c2nd 8012 ℂcc 11151 0cc0 11153 1c1 11154 + caddc 11156 GIdcgi 30519 CVecOLDcvc 30587 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-1st 8013 df-2nd 8014 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-ltxr 11298 df-grpo 30522 df-gid 30523 df-ginv 30524 df-ablo 30574 df-vc 30588 |
This theorem is referenced by: vcz 30604 vcm 30605 nv0 30666 |
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