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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 30669 | . . 3 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺𝑍) = 𝐴) |
| 5 | 1p0e1 12298 | . . . . 5 ⊢ (1 + 0) = 1 | |
| 6 | 5 | oveq1i 7373 | . . . 4 ⊢ ((1 + 0)𝑆𝐴) = (1𝑆𝐴) |
| 7 | 0cn 11134 | . . . . 5 ⊢ 0 ∈ ℂ | |
| 8 | ax-1cn 11094 | . . . . . 6 ⊢ 1 ∈ ℂ | |
| 9 | vc0.2 | . . . . . . 7 ⊢ 𝑆 = (2nd ‘𝑊) | |
| 10 | 1, 9, 2 | vcdir 30662 | . . . . . 6 ⊢ ((𝑊 ∈ CVecOLD ∧ (1 ∈ ℂ ∧ 0 ∈ ℂ ∧ 𝐴 ∈ 𝑋)) → ((1 + 0)𝑆𝐴) = ((1𝑆𝐴)𝐺(0𝑆𝐴))) |
| 11 | 8, 10 | mp3anr1 1466 | . . . . 5 ⊢ ((𝑊 ∈ CVecOLD ∧ (0 ∈ ℂ ∧ 𝐴 ∈ 𝑋)) → ((1 + 0)𝑆𝐴) = ((1𝑆𝐴)𝐺(0𝑆𝐴))) |
| 12 | 7, 11 | mpanr1 709 | . . . 4 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((1 + 0)𝑆𝐴) = ((1𝑆𝐴)𝐺(0𝑆𝐴))) |
| 13 | 1, 9, 2 | vcidOLD 30660 | . . . 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 7378 | . . 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 30659 | . . . . 5 ⊢ ((𝑊 ∈ CVecOLD ∧ 0 ∈ ℂ ∧ 𝐴 ∈ 𝑋) → (0𝑆𝐴) ∈ 𝑋) |
| 18 | 7, 17 | mp3an2 1457 | . . . 4 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (0𝑆𝐴) ∈ 𝑋) |
| 19 | 1, 2, 3 | vczcl 30668 | . . . . 5 ⊢ (𝑊 ∈ CVecOLD → 𝑍 ∈ 𝑋) |
| 20 | 19 | adantr 481 | . . . 4 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → 𝑍 ∈ 𝑋) |
| 21 | simpr 485 | . . . 4 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋) | |
| 22 | 18, 20, 21 | 3jca 1134 | . . 3 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((0𝑆𝐴) ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) |
| 23 | 1, 2 | vclcan 30667 | . . 3 ⊢ ((𝑊 ∈ CVecOLD ∧ ((0𝑆𝐴) ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝐴𝐺(0𝑆𝐴)) = (𝐴𝐺𝑍) ↔ (0𝑆𝐴) = 𝑍)) |
| 24 | 22, 23 | syldan 597 | . 2 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((𝐴𝐺(0𝑆𝐴)) = (𝐴𝐺𝑍) ↔ (0𝑆𝐴) = 𝑍)) |
| 25 | 16, 24 | mpbid 233 | 1 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (0𝑆𝐴) = 𝑍) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 ran crn 5626 ‘cfv 6492 (class class class)co 7363 1st c1st 7936 2nd c2nd 7937 ℂcc 11034 0cc0 11036 1c1 11037 + caddc 11039 GIdcgi 30586 CVecOLDcvc 30654 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 |
| This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4846 df-iun 4930 df-br 5080 df-opab 5142 df-mpt 5161 df-id 5520 df-po 5533 df-so 5534 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7320 df-ov 7366 df-1st 7938 df-2nd 7939 df-er 8640 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11179 df-mnf 11180 df-ltxr 11182 df-grpo 30589 df-gid 30590 df-ginv 30591 df-ablo 30641 df-vc 30655 |
| This theorem is referenced by: vcz 30671 vcm 30672 nv0 30733 |
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