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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) ⧠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 6843 | . . . 4 âĒ (â = ðđ â (1st ââ) = (1st âðđ)) | |
2 | 1 | coeq1d 5818 | . . 3 âĒ (â = ðđ â ((1st ââ) â (1st âð)) = ((1st âðđ) â (1st âð))) |
3 | fveq2 6843 | . . . 4 âĒ (â = ðđ â (2nd ââ) = (2nd âðđ)) | |
4 | 3 | oveq1d 7373 | . . 3 âĒ (â = ðđ â ((2nd ââ) âĻĢ (2nd âð)) = ((2nd âðđ) âĻĢ (2nd âð))) |
5 | 2, 4 | opeq12d 4839 | . 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 6843 | . . . 4 âĒ (ð = ðš â (1st âð) = (1st âðš)) | |
7 | 6 | coeq2d 5819 | . . 3 âĒ (ð = ðš â ((1st âðđ) â (1st âð)) = ((1st âðđ) â (1st âðš))) |
8 | fveq2 6843 | . . . 4 âĒ (ð = ðš â (2nd âð) = (2nd âðš)) | |
9 | 8 | oveq2d 7374 | . . 3 âĒ (ð = ðš â ((2nd âðđ) âĻĢ (2nd âð)) = ((2nd âðđ) âĻĢ (2nd âðš))) |
10 | 7, 9 | opeq12d 4839 | . 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 39555 | . 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 5422 | . 2 âĒ âĻ((1st âðđ) â (1st âðš)), ((2nd âðđ) âĻĢ (2nd âðš))âĐ â V | |
14 | 5, 10, 12, 13 | ovmpo 7516 | 1 âĒ ((ðđ â (ð à ðļ) ⧠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 ⧠wa 397 = wceq 1542 â wcel 2107 âĻcop 4593 à cxp 5632 â ccom 5638 âcfv 6497 (class class class)co 7358 â cmpo 7360 1st c1st 7920 2nd c2nd 7921 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-sbc 3741 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-iota 6449 df-fun 6499 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 |
This theorem is referenced by: dvhvadd 39558 dvhopaddN 39580 |
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