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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dvhvaddval | Structured version Visualization version GIF version |
Description: The vector sum operation for the constructed full vector space H. (Contributed by NM, 26-Oct-2013.) |
Ref | Expression |
---|---|
dvhvaddval.a | âĒ + = (ð â (ð à ðļ), ð â (ð à ðļ) âĶ âĻ((1st âð) â (1st âð)), ((2nd âð) âĻĢ (2nd âð))âĐ) |
Ref | Expression |
---|---|
dvhvaddval | âĒ ((ðđ â (ð à ðļ) â§ ðš â (ð à ðļ)) â (ðđ + ðš) = âĻ((1st âðđ) â (1st âðš)), ((2nd âðđ) âĻĢ (2nd âðš))âĐ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6892 | . . . 4 âĒ (â = ðđ â (1st ââ) = (1st âðđ)) | |
2 | 1 | coeq1d 5858 | . . 3 âĒ (â = ðđ â ((1st ââ) â (1st âð)) = ((1st âðđ) â (1st âð))) |
3 | fveq2 6892 | . . . 4 âĒ (â = ðđ â (2nd ââ) = (2nd âðđ)) | |
4 | 3 | oveq1d 7431 | . . 3 âĒ (â = ðđ â ((2nd ââ) âĻĢ (2nd âð)) = ((2nd âðđ) âĻĢ (2nd âð))) |
5 | 2, 4 | opeq12d 4877 | . 2 âĒ (â = ðđ â âĻ((1st ââ) â (1st âð)), ((2nd ââ) âĻĢ (2nd âð))âĐ = âĻ((1st âðđ) â (1st âð)), ((2nd âðđ) âĻĢ (2nd âð))âĐ) |
6 | fveq2 6892 | . . . 4 âĒ (ð = ðš â (1st âð) = (1st âðš)) | |
7 | 6 | coeq2d 5859 | . . 3 âĒ (ð = ðš â ((1st âðđ) â (1st âð)) = ((1st âðđ) â (1st âðš))) |
8 | fveq2 6892 | . . . 4 âĒ (ð = ðš â (2nd âð) = (2nd âðš)) | |
9 | 8 | oveq2d 7432 | . . 3 âĒ (ð = ðš â ((2nd âðđ) âĻĢ (2nd âð)) = ((2nd âðđ) âĻĢ (2nd âðš))) |
10 | 7, 9 | opeq12d 4877 | . 2 âĒ (ð = ðš â âĻ((1st âðđ) â (1st âð)), ((2nd âðđ) âĻĢ (2nd âð))âĐ = âĻ((1st âðđ) â (1st âðš)), ((2nd âðđ) âĻĢ (2nd âðš))âĐ) |
11 | dvhvaddval.a | . . 3 âĒ + = (ð â (ð à ðļ), ð â (ð à ðļ) âĶ âĻ((1st âð) â (1st âð)), ((2nd âð) âĻĢ (2nd âð))âĐ) | |
12 | 11 | dvhvaddcbv 40618 | . 2 âĒ + = (â â (ð à ðļ), ð â (ð à ðļ) âĶ âĻ((1st ââ) â (1st âð)), ((2nd ââ) âĻĢ (2nd âð))âĐ) |
13 | opex 5460 | . 2 âĒ âĻ((1st âðđ) â (1st âðš)), ((2nd âðđ) âĻĢ (2nd âðš))âĐ â V | |
14 | 5, 10, 12, 13 | ovmpo 7578 | 1 âĒ ((ðđ â (ð à ðļ) â§ ðš â (ð à ðļ)) â (ðđ + ðš) = âĻ((1st âðđ) â (1st âðš)), ((2nd âðđ) âĻĢ (2nd âðš))âĐ) |
Colors of variables: wff setvar class |
Syntax hints: â wi 4 â§ wa 394 = wceq 1533 â wcel 2098 âĻcop 4630 à cxp 5670 â ccom 5676 âcfv 6543 (class class class)co 7416 â cmpo 7418 1st c1st 7989 2nd c2nd 7990 |
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 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-sbc 3769 df-dif 3942 df-un 3944 df-ss 3956 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7419 df-oprab 7420 df-mpo 7421 |
This theorem is referenced by: dvhvadd 40621 dvhopaddN 40643 |
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