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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 30646 | . . 3 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺𝑍) = 𝐴) |
| 5 | 1p0e1 12302 | . . . . 5 ⊢ (1 + 0) = 1 | |
| 6 | 5 | oveq1i 7379 | . . . 4 ⊢ ((1 + 0)𝑆𝐴) = (1𝑆𝐴) |
| 7 | 0cn 11138 | . . . . 5 ⊢ 0 ∈ ℂ | |
| 8 | ax-1cn 11098 | . . . . . 6 ⊢ 1 ∈ ℂ | |
| 9 | vc0.2 | . . . . . . 7 ⊢ 𝑆 = (2nd ‘𝑊) | |
| 10 | 1, 9, 2 | vcdir 30639 | . . . . . 6 ⊢ ((𝑊 ∈ CVecOLD ∧ (1 ∈ ℂ ∧ 0 ∈ ℂ ∧ 𝐴 ∈ 𝑋)) → ((1 + 0)𝑆𝐴) = ((1𝑆𝐴)𝐺(0𝑆𝐴))) |
| 11 | 8, 10 | mp3anr1 1461 | . . . . 5 ⊢ ((𝑊 ∈ CVecOLD ∧ (0 ∈ ℂ ∧ 𝐴 ∈ 𝑋)) → ((1 + 0)𝑆𝐴) = ((1𝑆𝐴)𝐺(0𝑆𝐴))) |
| 12 | 7, 11 | mpanr1 704 | . . . 4 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((1 + 0)𝑆𝐴) = ((1𝑆𝐴)𝐺(0𝑆𝐴))) |
| 13 | 1, 9, 2 | vcidOLD 30637 | . . . 4 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (1𝑆𝐴) = 𝐴) |
| 14 | 6, 12, 13 | 3eqtr3a 2796 | . . 3 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((1𝑆𝐴)𝐺(0𝑆𝐴)) = 𝐴) |
| 15 | 13 | oveq1d 7384 | . . 3 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((1𝑆𝐴)𝐺(0𝑆𝐴)) = (𝐴𝐺(0𝑆𝐴))) |
| 16 | 4, 14, 15 | 3eqtr2rd 2779 | . 2 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺(0𝑆𝐴)) = (𝐴𝐺𝑍)) |
| 17 | 1, 9, 2 | vccl 30636 | . . . . 5 ⊢ ((𝑊 ∈ CVecOLD ∧ 0 ∈ ℂ ∧ 𝐴 ∈ 𝑋) → (0𝑆𝐴) ∈ 𝑋) |
| 18 | 7, 17 | mp3an2 1452 | . . . 4 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (0𝑆𝐴) ∈ 𝑋) |
| 19 | 1, 2, 3 | vczcl 30645 | . . . . 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 1129 | . . 3 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((0𝑆𝐴) ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) |
| 23 | 1, 2 | vclcan 30644 | . . 3 ⊢ ((𝑊 ∈ CVecOLD ∧ ((0𝑆𝐴) ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝐴𝐺(0𝑆𝐴)) = (𝐴𝐺𝑍) ↔ (0𝑆𝐴) = 𝑍)) |
| 24 | 22, 23 | syldan 592 | . 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 1087 = wceq 1542 ∈ wcel 2114 ran crn 5633 ‘cfv 6500 (class class class)co 7369 1st c1st 7942 2nd c2nd 7943 ℂcc 11038 0cc0 11040 1c1 11041 + caddc 11043 GIdcgi 30563 CVecOLDcvc 30631 |
| 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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5308 ax-pr 5376 ax-un 7691 ax-resscn 11097 ax-1cn 11098 ax-icn 11099 ax-addcl 11100 ax-addrcl 11101 ax-mulcl 11102 ax-mulrcl 11103 ax-mulcom 11104 ax-addass 11105 ax-mulass 11106 ax-distr 11107 ax-i2m1 11108 ax-1ne0 11109 ax-1rid 11110 ax-rnegex 11111 ax-rrecex 11112 ax-cnre 11113 ax-pre-lttri 11114 ax-pre-lttrn 11115 ax-pre-ltadd 11116 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5527 df-po 5540 df-so 5541 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7326 df-ov 7372 df-1st 7944 df-2nd 7945 df-er 8645 df-en 8896 df-dom 8897 df-sdom 8898 df-pnf 11183 df-mnf 11184 df-ltxr 11186 df-grpo 30566 df-gid 30567 df-ginv 30568 df-ablo 30618 df-vc 30632 |
| This theorem is referenced by: vcz 30648 vcm 30649 nv0 30710 |
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