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Mirrors > Home > MPE Home > Th. List > asclfn | Structured version Visualization version GIF version |
Description: Unconditional functionality of the algebra scalars function. (Contributed by Mario Carneiro, 9-Mar-2015.) |
Ref | Expression |
---|---|
asclfn.a | β’ π΄ = (algScβπ) |
asclfn.f | β’ πΉ = (Scalarβπ) |
asclfn.k | β’ πΎ = (BaseβπΉ) |
Ref | Expression |
---|---|
asclfn | β’ π΄ Fn πΎ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovex 7437 | . 2 β’ (π₯( Β·π βπ)(1rβπ)) β V | |
2 | asclfn.a | . . 3 β’ π΄ = (algScβπ) | |
3 | asclfn.f | . . 3 β’ πΉ = (Scalarβπ) | |
4 | asclfn.k | . . 3 β’ πΎ = (BaseβπΉ) | |
5 | eqid 2726 | . . 3 β’ ( Β·π βπ) = ( Β·π βπ) | |
6 | eqid 2726 | . . 3 β’ (1rβπ) = (1rβπ) | |
7 | 2, 3, 4, 5, 6 | asclfval 21769 | . 2 β’ π΄ = (π₯ β πΎ β¦ (π₯( Β·π βπ)(1rβπ))) |
8 | 1, 7 | fnmpti 6686 | 1 β’ π΄ Fn πΎ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 Fn wfn 6531 βcfv 6536 (class class class)co 7404 Basecbs 17151 Scalarcsca 17207 Β·π cvsca 17208 1rcur 20084 algSccascl 21743 |
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 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-1cn 11167 ax-addcl 11169 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-om 7852 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-nn 12214 df-slot 17122 df-ndx 17134 df-base 17152 df-ascl 21746 |
This theorem is referenced by: issubassa2 21782 subrgascl 21965 |
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