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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 6985 | . 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 19831 | . 2 ⊢ 𝐴 = (𝑥 ∈ 𝐾 ↦ (𝑥 · 1 )) |
8 | ovex 7010 | . 2 ⊢ (𝑋 · 1 ) ∈ V | |
9 | 1, 7, 8 | fvmpt 6597 | 1 ⊢ (𝑋 ∈ 𝐾 → (𝐴‘𝑋) = (𝑋 · 1 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 ∈ wcel 2050 ‘cfv 6190 (class class class)co 6978 Basecbs 16342 Scalarcsca 16427 ·𝑠 cvsca 16428 1rcur 18977 algSccascl 19808 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5050 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-ral 3093 df-rex 3094 df-reu 3095 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-nul 4181 df-if 4352 df-sn 4443 df-pr 4445 df-op 4449 df-uni 4714 df-iun 4795 df-br 4931 df-opab 4993 df-mpt 5010 df-id 5313 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-ov 6981 df-slot 16346 df-base 16348 df-ascl 19811 |
This theorem is referenced by: asclghm 19835 asclmul1 19836 asclmul2 19837 asclrhm 19839 mplascl 19992 ply1scltm 20155 ply1scl0 20164 ply1scl1 20166 lply1binomsc 20181 pmatcollpwscmatlem1 21104 cayhamlem2 21199 ascl0 43799 ascl1 43800 ply1sclrmsm 43805 |
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