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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 7189 | . 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 2758 | . . 3 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
6 | eqid 2758 | . . 3 ⊢ (1r‘𝑊) = (1r‘𝑊) | |
7 | 2, 3, 4, 5, 6 | asclfval 20654 | . 2 ⊢ 𝐴 = (𝑥 ∈ 𝐾 ↦ (𝑥( ·𝑠 ‘𝑊)(1r‘𝑊))) |
8 | 1, 7 | fnmpti 6479 | 1 ⊢ 𝐴 Fn 𝐾 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 Fn wfn 6335 ‘cfv 6340 (class class class)co 7156 Basecbs 16554 Scalarcsca 16639 ·𝑠 cvsca 16640 1rcur 19332 algSccascl 20630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-rep 5160 ax-sep 5173 ax-nul 5180 ax-pr 5302 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-ral 3075 df-rex 3076 df-reu 3077 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-nul 4228 df-if 4424 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4802 df-iun 4888 df-br 5037 df-opab 5099 df-mpt 5117 df-id 5434 df-xp 5534 df-rel 5535 df-cnv 5536 df-co 5537 df-dm 5538 df-rn 5539 df-res 5540 df-ima 5541 df-iota 6299 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-ov 7159 df-slot 16558 df-base 16560 df-ascl 20633 |
This theorem is referenced by: issubassa2 20668 subrgascl 20840 |
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