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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 6556 | . 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 19155 | . 2 ⊢ 𝐴 = (𝑥 ∈ 𝐾 ↦ (𝑥 · 1 )) |
8 | ovex 6577 | . 2 ⊢ (𝑋 · 1 ) ∈ V | |
9 | 1, 7, 8 | fvmpt 6191 | 1 ⊢ (𝑋 ∈ 𝐾 → (𝐴‘𝑋) = (𝑋 · 1 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 Scalarcsca 15771 ·𝑠 cvsca 15772 1rcur 18324 algSccascl 19132 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-slot 15699 df-base 15700 df-ascl 19135 |
This theorem is referenced by: asclghm 19159 asclmul1 19160 asclmul2 19161 asclrhm 19163 mplascl 19317 ply1scltm 19472 ply1scl0 19481 ply1scl1 19483 lply1binomsc 19498 pmatcollpwscmatlem1 20413 cayhamlem2 20508 ascl0 41959 ascl1 41960 ply1sclrmsm 41965 |
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