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Mirrors > Home > MPE Home > Th. List > rrxf | Structured version Visualization version GIF version |
Description: Euclidean vectors as functions. (Contributed by Thierry Arnoux, 7-Jul-2019.) |
Ref | Expression |
---|---|
rrxmval.1 | β’ π = {β β (β βm πΌ) β£ β finSupp 0} |
rrxf.1 | β’ (π β πΉ β π) |
Ref | Expression |
---|---|
rrxf | β’ (π β πΉ:πΌβΆβ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rrxmval.1 | . . . 4 β’ π = {β β (β βm πΌ) β£ β finSupp 0} | |
2 | 1 | ssrab3 4078 | . . 3 β’ π β (β βm πΌ) |
3 | rrxf.1 | . . 3 β’ (π β πΉ β π) | |
4 | 2, 3 | sselid 3978 | . 2 β’ (π β πΉ β (β βm πΌ)) |
5 | elmapi 8867 | . 2 β’ (πΉ β (β βm πΌ) β πΉ:πΌβΆβ) | |
6 | 4, 5 | syl 17 | 1 β’ (π β πΉ:πΌβΆβ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1534 β wcel 2099 {crab 3429 class class class wbr 5148 βΆwf 6544 (class class class)co 7420 βm cmap 8844 finSupp cfsupp 9385 βcr 11137 0cc0 11138 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 7993 df-2nd 7994 df-map 8846 |
This theorem is referenced by: rrxsuppss 25330 rrxmval 25332 rrxmetlem 25334 rrxmet 25335 rrxdstprj1 25336 |
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