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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 6657 | . 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 19334 | . 2 ⊢ 𝐴 = (𝑥 ∈ 𝐾 ↦ (𝑥 · 1 )) |
8 | ovex 6678 | . 2 ⊢ (𝑋 · 1 ) ∈ V | |
9 | 1, 7, 8 | fvmpt 6282 | 1 ⊢ (𝑋 ∈ 𝐾 → (𝐴‘𝑋) = (𝑋 · 1 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 Scalarcsca 15944 ·𝑠 cvsca 15945 1rcur 18501 algSccascl 19311 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-slot 15861 df-base 15863 df-ascl 19314 |
This theorem is referenced by: asclghm 19338 asclmul1 19339 asclmul2 19340 asclrhm 19342 mplascl 19496 ply1scltm 19651 ply1scl0 19660 ply1scl1 19662 lply1binomsc 19677 pmatcollpwscmatlem1 20594 cayhamlem2 20689 ascl0 42165 ascl1 42166 ply1sclrmsm 42171 |
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