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Mirrors > Home > MPE Home > Th. List > vc2OLD | Structured version Visualization version GIF version |
Description: A vector plus itself is two times the vector. (Contributed by NM, 1-Feb-2007.) Obsolete theorem, use clmvs2 24972 together with cvsclm 25004 instead. (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
vciOLD.1 | ⊢ 𝐺 = (1st ‘𝑊) |
vciOLD.2 | ⊢ 𝑆 = (2nd ‘𝑊) |
vciOLD.3 | ⊢ 𝑋 = ran 𝐺 |
Ref | Expression |
---|---|
vc2OLD | ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺𝐴) = (2𝑆𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vciOLD.1 | . . . 4 ⊢ 𝐺 = (1st ‘𝑊) | |
2 | vciOLD.2 | . . . 4 ⊢ 𝑆 = (2nd ‘𝑊) | |
3 | vciOLD.3 | . . . 4 ⊢ 𝑋 = ran 𝐺 | |
4 | 1, 2, 3 | vcidOLD 30322 | . . 3 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (1𝑆𝐴) = 𝐴) |
5 | 4, 4 | oveq12d 7422 | . 2 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((1𝑆𝐴)𝐺(1𝑆𝐴)) = (𝐴𝐺𝐴)) |
6 | df-2 12276 | . . . 4 ⊢ 2 = (1 + 1) | |
7 | 6 | oveq1i 7414 | . . 3 ⊢ (2𝑆𝐴) = ((1 + 1)𝑆𝐴) |
8 | ax-1cn 11167 | . . . 4 ⊢ 1 ∈ ℂ | |
9 | 1, 2, 3 | vcdir 30324 | . . . . 5 ⊢ ((𝑊 ∈ CVecOLD ∧ (1 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝐴 ∈ 𝑋)) → ((1 + 1)𝑆𝐴) = ((1𝑆𝐴)𝐺(1𝑆𝐴))) |
10 | 8, 9 | mp3anr1 1454 | . . . 4 ⊢ ((𝑊 ∈ CVecOLD ∧ (1 ∈ ℂ ∧ 𝐴 ∈ 𝑋)) → ((1 + 1)𝑆𝐴) = ((1𝑆𝐴)𝐺(1𝑆𝐴))) |
11 | 8, 10 | mpanr1 700 | . . 3 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((1 + 1)𝑆𝐴) = ((1𝑆𝐴)𝐺(1𝑆𝐴))) |
12 | 7, 11 | eqtr2id 2779 | . 2 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((1𝑆𝐴)𝐺(1𝑆𝐴)) = (2𝑆𝐴)) |
13 | 5, 12 | eqtr3d 2768 | 1 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺𝐴) = (2𝑆𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ran crn 5670 ‘cfv 6536 (class class class)co 7404 1st c1st 7969 2nd c2nd 7970 ℂcc 11107 1c1 11110 + caddc 11112 2c2 12268 CVecOLDcvc 30316 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7721 ax-1cn 11167 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-fv 6544 df-ov 7407 df-1st 7971 df-2nd 7972 df-2 12276 df-vc 30317 |
This theorem is referenced by: nv2 30390 ipdirilem 30587 |
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