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Mirrors > Home > MPE Home > Th. List > asclf | Structured version Visualization version GIF version |
Description: The algebra scalars function is a function into the base set. (Contributed by Mario Carneiro, 4-Jul-2015.) |
Ref | Expression |
---|---|
asclf.a | ⊢ 𝐴 = (algSc‘𝑊) |
asclf.f | ⊢ 𝐹 = (Scalar‘𝑊) |
asclf.r | ⊢ (𝜑 → 𝑊 ∈ Ring) |
asclf.l | ⊢ (𝜑 → 𝑊 ∈ LMod) |
asclf.k | ⊢ 𝐾 = (Base‘𝐹) |
asclf.b | ⊢ 𝐵 = (Base‘𝑊) |
Ref | Expression |
---|---|
asclf | ⊢ (𝜑 → 𝐴:𝐾⟶𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | asclf.l | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
2 | 1 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → 𝑊 ∈ LMod) |
3 | simpr 485 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → 𝑥 ∈ 𝐾) | |
4 | asclf.r | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ Ring) | |
5 | asclf.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑊) | |
6 | eqid 2740 | . . . . . 6 ⊢ (1r‘𝑊) = (1r‘𝑊) | |
7 | 5, 6 | ringidcl 19803 | . . . . 5 ⊢ (𝑊 ∈ Ring → (1r‘𝑊) ∈ 𝐵) |
8 | 4, 7 | syl 17 | . . . 4 ⊢ (𝜑 → (1r‘𝑊) ∈ 𝐵) |
9 | 8 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → (1r‘𝑊) ∈ 𝐵) |
10 | asclf.f | . . . 4 ⊢ 𝐹 = (Scalar‘𝑊) | |
11 | eqid 2740 | . . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
12 | asclf.k | . . . 4 ⊢ 𝐾 = (Base‘𝐹) | |
13 | 5, 10, 11, 12 | lmodvscl 20136 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑥 ∈ 𝐾 ∧ (1r‘𝑊) ∈ 𝐵) → (𝑥( ·𝑠 ‘𝑊)(1r‘𝑊)) ∈ 𝐵) |
14 | 2, 3, 9, 13 | syl3anc 1370 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → (𝑥( ·𝑠 ‘𝑊)(1r‘𝑊)) ∈ 𝐵) |
15 | asclf.a | . . 3 ⊢ 𝐴 = (algSc‘𝑊) | |
16 | 15, 10, 12, 11, 6 | asclfval 21079 | . 2 ⊢ 𝐴 = (𝑥 ∈ 𝐾 ↦ (𝑥( ·𝑠 ‘𝑊)(1r‘𝑊))) |
17 | 14, 16 | fmptd 6983 | 1 ⊢ (𝜑 → 𝐴:𝐾⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 ⟶wf 6427 ‘cfv 6431 (class class class)co 7269 Basecbs 16908 Scalarcsca 16961 ·𝑠 cvsca 16962 1rcur 19733 Ringcrg 19779 LModclmod 20119 algSccascl 21055 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 ax-cnex 10926 ax-resscn 10927 ax-1cn 10928 ax-icn 10929 ax-addcl 10930 ax-addrcl 10931 ax-mulcl 10932 ax-mulrcl 10933 ax-mulcom 10934 ax-addass 10935 ax-mulass 10936 ax-distr 10937 ax-i2m1 10938 ax-1ne0 10939 ax-1rid 10940 ax-rnegex 10941 ax-rrecex 10942 ax-cnre 10943 ax-pre-lttri 10944 ax-pre-lttrn 10945 ax-pre-ltadd 10946 ax-pre-mulgt0 10947 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-om 7705 df-2nd 7823 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-er 8479 df-en 8715 df-dom 8716 df-sdom 8717 df-pnf 11010 df-mnf 11011 df-xr 11012 df-ltxr 11013 df-le 11014 df-sub 11205 df-neg 11206 df-nn 11972 df-2 12034 df-sets 16861 df-slot 16879 df-ndx 16891 df-base 16909 df-plusg 16971 df-0g 17148 df-mgm 18322 df-sgrp 18371 df-mnd 18382 df-mgp 19717 df-ur 19734 df-ring 19781 df-lmod 20121 df-ascl 21058 |
This theorem is referenced by: asclghm 21083 ascldimul 21088 aspval2 21098 mplasclf 21269 subrgasclcl 21271 mpfconst 21307 ply1sclf 21452 cply1coe0bi 21467 lply1binomsc 21474 evls1sca 21485 evl1scvarpw 21525 mat2pmatbas 21871 chpscmat 21987 chpscmatgsumbin 21989 |
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