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Mirrors > Home > MPE Home > Th. List > vcdir | Structured version Visualization version Unicode version |
Description: Distributive law for the scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
vciOLD.1 | |
vciOLD.2 | |
vciOLD.3 |
Ref | Expression |
---|---|
vcdir |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vciOLD.1 | . . . . . 6 | |
2 | vciOLD.2 | . . . . . 6 | |
3 | vciOLD.3 | . . . . . 6 | |
4 | 1, 2, 3 | vciOLD 27416 | . . . . 5 |
5 | simpl 473 | . . . . . . . . . . 11 | |
6 | 5 | ralimi 2952 | . . . . . . . . . 10 |
7 | 6 | adantl 482 | . . . . . . . . 9 |
8 | 7 | ralimi 2952 | . . . . . . . 8 |
9 | 8 | adantl 482 | . . . . . . 7 |
10 | 9 | ralimi 2952 | . . . . . 6 |
11 | 10 | 3ad2ant3 1084 | . . . . 5 |
12 | 4, 11 | syl 17 | . . . 4 |
13 | oveq2 6658 | . . . . . 6 | |
14 | oveq2 6658 | . . . . . . 7 | |
15 | oveq2 6658 | . . . . . . 7 | |
16 | 14, 15 | oveq12d 6668 | . . . . . 6 |
17 | 13, 16 | eqeq12d 2637 | . . . . 5 |
18 | oveq1 6657 | . . . . . . 7 | |
19 | 18 | oveq1d 6665 | . . . . . 6 |
20 | oveq1 6657 | . . . . . . 7 | |
21 | 20 | oveq1d 6665 | . . . . . 6 |
22 | 19, 21 | eqeq12d 2637 | . . . . 5 |
23 | oveq2 6658 | . . . . . . 7 | |
24 | 23 | oveq1d 6665 | . . . . . 6 |
25 | oveq1 6657 | . . . . . . 7 | |
26 | 25 | oveq2d 6666 | . . . . . 6 |
27 | 24, 26 | eqeq12d 2637 | . . . . 5 |
28 | 17, 22, 27 | rspc3v 3325 | . . . 4 |
29 | 12, 28 | syl5 34 | . . 3 |
30 | 29 | 3coml 1272 | . 2 |
31 | 30 | impcom 446 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 w3a 1037 wceq 1483 wcel 1990 wral 2912 cxp 5112 crn 5115 wf 5884 cfv 5888 (class class class)co 6650 c1st 7166 c2nd 7167 cc 9934 c1 9937 caddc 9939 cmul 9941 cablo 27398 cvc 27413 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-1st 7168 df-2nd 7169 df-vc 27414 |
This theorem is referenced by: vc2OLD 27423 vc0 27429 vcm 27431 nvdir 27486 |
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