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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 | ⊢ 𝐴 = (algSc‘𝑊) |
asclfn.f | ⊢ 𝐹 = (Scalar‘𝑊) |
asclfn.k | ⊢ 𝐾 = (Base‘𝐹) |
Ref | Expression |
---|---|
asclfn | ⊢ 𝐴 Fn 𝐾 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovex 7178 | . 2 ⊢ (𝑥( ·𝑠 ‘𝑊)(1r‘𝑊)) ∈ V | |
2 | asclfn.a | . . 3 ⊢ 𝐴 = (algSc‘𝑊) | |
3 | asclfn.f | . . 3 ⊢ 𝐹 = (Scalar‘𝑊) | |
4 | asclfn.k | . . 3 ⊢ 𝐾 = (Base‘𝐹) | |
5 | eqid 2818 | . . 3 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
6 | eqid 2818 | . . 3 ⊢ (1r‘𝑊) = (1r‘𝑊) | |
7 | 2, 3, 4, 5, 6 | asclfval 20036 | . 2 ⊢ 𝐴 = (𝑥 ∈ 𝐾 ↦ (𝑥( ·𝑠 ‘𝑊)(1r‘𝑊))) |
8 | 1, 7 | fnmpti 6484 | 1 ⊢ 𝐴 Fn 𝐾 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 Fn wfn 6343 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 Scalarcsca 16556 ·𝑠 cvsca 16557 1rcur 19180 algSccascl 20012 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-slot 16475 df-base 16477 df-ascl 20015 |
This theorem is referenced by: issubassa2 20049 subrgascl 20206 |
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