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Mirrors > Home > MPE Home > Th. List > vcablo | Structured version Visualization version GIF version |
Description: Vector addition is an Abelian group operation. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
vcabl.1 | β’ πΊ = (1st βπ) |
Ref | Expression |
---|---|
vcablo | β’ (π β CVecOLD β πΊ β AbelOp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vcabl.1 | . . 3 β’ πΊ = (1st βπ) | |
2 | eqid 2733 | . . 3 β’ (2nd βπ) = (2nd βπ) | |
3 | eqid 2733 | . . 3 β’ ran πΊ = ran πΊ | |
4 | 1, 2, 3 | vciOLD 29545 | . 2 β’ (π β CVecOLD β (πΊ β AbelOp β§ (2nd βπ):(β Γ ran πΊ)βΆran πΊ β§ βπ₯ β ran πΊ((1(2nd βπ)π₯) = π₯ β§ βπ¦ β β (βπ§ β ran πΊ(π¦(2nd βπ)(π₯πΊπ§)) = ((π¦(2nd βπ)π₯)πΊ(π¦(2nd βπ)π§)) β§ βπ§ β β (((π¦ + π§)(2nd βπ)π₯) = ((π¦(2nd βπ)π₯)πΊ(π§(2nd βπ)π₯)) β§ ((π¦ Β· π§)(2nd βπ)π₯) = (π¦(2nd βπ)(π§(2nd βπ)π₯))))))) |
5 | 4 | simp1d 1143 | 1 β’ (π β CVecOLD β πΊ β AbelOp) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 βwral 3061 Γ cxp 5632 ran crn 5635 βΆwf 6493 βcfv 6497 (class class class)co 7358 1st c1st 7920 2nd c2nd 7921 βcc 11054 1c1 11057 + caddc 11059 Β· cmul 11061 AbelOpcablo 29528 CVecOLDcvc 29542 |
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 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 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-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-fv 6505 df-ov 7361 df-1st 7922 df-2nd 7923 df-vc 29543 |
This theorem is referenced by: vcgrp 29554 nvablo 29600 ip0i 29809 ipdirilem 29813 |
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