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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 7426 | . 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 21829 | . 2 ⊢ 𝐴 = (𝑥 ∈ 𝐾 ↦ (𝑥 · 1 )) |
8 | ovex 7452 | . 2 ⊢ (𝑋 · 1 ) ∈ V | |
9 | 1, 7, 8 | fvmpt 7004 | 1 ⊢ (𝑋 ∈ 𝐾 → (𝐴‘𝑋) = (𝑋 · 1 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ‘cfv 6549 (class class class)co 7419 Basecbs 17183 Scalarcsca 17239 ·𝑠 cvsca 17240 1rcur 20133 algSccascl 21803 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-1cn 11198 ax-addcl 11200 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-om 7872 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-nn 12246 df-slot 17154 df-ndx 17166 df-base 17184 df-ascl 21806 |
This theorem is referenced by: asclghm 21833 ascl0 21834 ascl1 21835 asclmul1 21836 asclmul2 21837 ascldimul 21838 asclmulg 21852 psrascl 21941 mplascl 22030 psdascl 22115 ply1scltm 22225 ply1scl0OLD 22235 ply1scl1OLD 22238 lply1binomsc 22255 asclply1subcl 22318 pmatcollpwscmatlem1 22735 cayhamlem2 22830 ply1sclrmsm 47637 |
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