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Mirrors > Home > MPE Home > Th. List > nvgf | Structured version Visualization version GIF version |
Description: Mapping for the vector addition operation. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nvgf.1 | β’ π = (BaseSetβπ) |
nvgf.2 | β’ πΊ = ( +π£ βπ) |
Ref | Expression |
---|---|
nvgf | β’ (π β NrmCVec β πΊ:(π Γ π)βΆπ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nvgf.2 | . . 3 β’ πΊ = ( +π£ βπ) | |
2 | 1 | nvgrp 30342 | . 2 β’ (π β NrmCVec β πΊ β GrpOp) |
3 | nvgf.1 | . . . 4 β’ π = (BaseSetβπ) | |
4 | 3, 1 | bafval 30329 | . . 3 β’ π = ran πΊ |
5 | 4 | grpofo 30224 | . 2 β’ (πΊ β GrpOp β πΊ:(π Γ π)βontoβπ) |
6 | fof 6796 | . 2 β’ (πΊ:(π Γ π)βontoβπ β πΊ:(π Γ π)βΆπ) | |
7 | 2, 5, 6 | 3syl 18 | 1 β’ (π β NrmCVec β πΊ:(π Γ π)βΆπ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 Γ cxp 5665 βΆwf 6530 βontoβwfo 6532 βcfv 6534 GrpOpcgr 30214 NrmCVeccnv 30309 +π£ cpv 30310 BaseSetcba 30311 |
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 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-ov 7405 df-oprab 7406 df-1st 7969 df-2nd 7970 df-grpo 30218 df-ablo 30270 df-vc 30284 df-nv 30317 df-va 30320 df-ba 30321 df-sm 30322 df-0v 30323 df-nmcv 30325 |
This theorem is referenced by: vacn 30419 sspg 30453 hladdf 30624 |
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