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Mirrors > Home > MPE Home > Th. List > asclval | Structured version Visualization version GIF version |
Description: Value of a mapped algebra scalar. (Contributed by Mario Carneiro, 8-Mar-2015.) |
Ref | Expression |
---|---|
asclfval.a | ⊢ 𝐴 = (algSc‘𝑊) |
asclfval.f | ⊢ 𝐹 = (Scalar‘𝑊) |
asclfval.k | ⊢ 𝐾 = (Base‘𝐹) |
asclfval.s | ⊢ · = ( ·𝑠 ‘𝑊) |
asclfval.o | ⊢ 1 = (1r‘𝑊) |
Ref | Expression |
---|---|
asclval | ⊢ (𝑋 ∈ 𝐾 → (𝐴‘𝑋) = (𝑋 · 1 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 7152 | . 2 ⊢ (𝑥 = 𝑋 → (𝑥 · 1 ) = (𝑋 · 1 )) | |
2 | asclfval.a | . . 3 ⊢ 𝐴 = (algSc‘𝑊) | |
3 | asclfval.f | . . 3 ⊢ 𝐹 = (Scalar‘𝑊) | |
4 | asclfval.k | . . 3 ⊢ 𝐾 = (Base‘𝐹) | |
5 | asclfval.s | . . 3 ⊢ · = ( ·𝑠 ‘𝑊) | |
6 | asclfval.o | . . 3 ⊢ 1 = (1r‘𝑊) | |
7 | 2, 3, 4, 5, 6 | asclfval 20036 | . 2 ⊢ 𝐴 = (𝑥 ∈ 𝐾 ↦ (𝑥 · 1 )) |
8 | ovex 7178 | . 2 ⊢ (𝑋 · 1 ) ∈ V | |
9 | 1, 7, 8 | fvmpt 6761 | 1 ⊢ (𝑋 ∈ 𝐾 → (𝐴‘𝑋) = (𝑋 · 1 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ‘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: asclghm 20040 ascl0 20041 asclmul1 20042 asclmul2 20043 ascldimul 20044 ascldimulOLD 20045 asclrhm 20047 mplascl 20204 ply1scltm 20377 ply1scl0 20386 ply1scl1 20388 lply1binomsc 20403 pmatcollpwscmatlem1 21325 cayhamlem2 21420 ascl1 44360 ply1sclrmsm 44365 |
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