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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 23690 together with cvsclm 23722 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 28333 | . . 3 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (1𝑆𝐴) = 𝐴) |
5 | 4, 4 | oveq12d 7166 | . 2 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((1𝑆𝐴)𝐺(1𝑆𝐴)) = (𝐴𝐺𝐴)) |
6 | df-2 11692 | . . . 4 ⊢ 2 = (1 + 1) | |
7 | 6 | oveq1i 7158 | . . 3 ⊢ (2𝑆𝐴) = ((1 + 1)𝑆𝐴) |
8 | ax-1cn 10587 | . . . 4 ⊢ 1 ∈ ℂ | |
9 | 1, 2, 3 | vcdir 28335 | . . . . 5 ⊢ ((𝑊 ∈ CVecOLD ∧ (1 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝐴 ∈ 𝑋)) → ((1 + 1)𝑆𝐴) = ((1𝑆𝐴)𝐺(1𝑆𝐴))) |
10 | 8, 9 | mp3anr1 1452 | . . . 4 ⊢ ((𝑊 ∈ CVecOLD ∧ (1 ∈ ℂ ∧ 𝐴 ∈ 𝑋)) → ((1 + 1)𝑆𝐴) = ((1𝑆𝐴)𝐺(1𝑆𝐴))) |
11 | 8, 10 | mpanr1 701 | . . 3 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((1 + 1)𝑆𝐴) = ((1𝑆𝐴)𝐺(1𝑆𝐴))) |
12 | 7, 11 | syl5req 2867 | . 2 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((1𝑆𝐴)𝐺(1𝑆𝐴)) = (2𝑆𝐴)) |
13 | 5, 12 | eqtr3d 2856 | 1 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺𝐴) = (2𝑆𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1531 ∈ wcel 2108 ran crn 5549 ‘cfv 6348 (class class class)co 7148 1st c1st 7679 2nd c2nd 7680 ℂcc 10527 1c1 10530 + caddc 10532 2c2 11684 CVecOLDcvc 28327 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-1cn 10587 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-sbc 3771 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-ov 7151 df-1st 7681 df-2nd 7682 df-2 11692 df-vc 28328 |
This theorem is referenced by: nv2 28401 ipdirilem 28598 |
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