Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > asclmulg | Structured version Visualization version GIF version |
Description: Apply group multiplication to the algebra scalars. (Contributed by Thierry Arnoux, 24-Jul-2024.) |
Ref | Expression |
---|---|
asclmulg.a | ⊢ 𝐴 = (algSc‘𝑊) |
asclmulg.f | ⊢ 𝐹 = (Scalar‘𝑊) |
asclmulg.k | ⊢ 𝐾 = (Base‘𝐹) |
asclmulg.m | ⊢ ↑ = (.g‘𝑊) |
asclmulg.t | ⊢ ∗ = (.g‘𝐹) |
Ref | Expression |
---|---|
asclmulg | ⊢ ((𝑊 ∈ AssAlg ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐾) → (𝐴‘(𝑁 ∗ 𝑋)) = (𝑁 ↑ (𝐴‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | assalmod 21065 | . . . 4 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ LMod) | |
2 | 1 | 3ad2ant1 1132 | . . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐾) → 𝑊 ∈ LMod) |
3 | simp3 1137 | . . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐾) → 𝑋 ∈ 𝐾) | |
4 | simp2 1136 | . . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐾) → 𝑁 ∈ ℕ0) | |
5 | assaring 21066 | . . . . 5 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ Ring) | |
6 | eqid 2740 | . . . . . 6 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
7 | eqid 2740 | . . . . . 6 ⊢ (1r‘𝑊) = (1r‘𝑊) | |
8 | 6, 7 | ringidcl 19805 | . . . . 5 ⊢ (𝑊 ∈ Ring → (1r‘𝑊) ∈ (Base‘𝑊)) |
9 | 5, 8 | syl 17 | . . . 4 ⊢ (𝑊 ∈ AssAlg → (1r‘𝑊) ∈ (Base‘𝑊)) |
10 | 9 | 3ad2ant1 1132 | . . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐾) → (1r‘𝑊) ∈ (Base‘𝑊)) |
11 | asclmulg.f | . . . 4 ⊢ 𝐹 = (Scalar‘𝑊) | |
12 | eqid 2740 | . . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
13 | asclmulg.k | . . . 4 ⊢ 𝐾 = (Base‘𝐹) | |
14 | asclmulg.m | . . . 4 ⊢ ↑ = (.g‘𝑊) | |
15 | asclmulg.t | . . . 4 ⊢ ∗ = (.g‘𝐹) | |
16 | 6, 11, 12, 13, 14, 15 | lmodvsmmulgdi 20156 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑋 ∈ 𝐾 ∧ 𝑁 ∈ ℕ0 ∧ (1r‘𝑊) ∈ (Base‘𝑊))) → (𝑁 ↑ (𝑋( ·𝑠 ‘𝑊)(1r‘𝑊))) = ((𝑁 ∗ 𝑋)( ·𝑠 ‘𝑊)(1r‘𝑊))) |
17 | 2, 3, 4, 10, 16 | syl13anc 1371 | . 2 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐾) → (𝑁 ↑ (𝑋( ·𝑠 ‘𝑊)(1r‘𝑊))) = ((𝑁 ∗ 𝑋)( ·𝑠 ‘𝑊)(1r‘𝑊))) |
18 | asclmulg.a | . . . . 5 ⊢ 𝐴 = (algSc‘𝑊) | |
19 | 18, 11, 13, 12, 7 | asclval 21082 | . . . 4 ⊢ (𝑋 ∈ 𝐾 → (𝐴‘𝑋) = (𝑋( ·𝑠 ‘𝑊)(1r‘𝑊))) |
20 | 3, 19 | syl 17 | . . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐾) → (𝐴‘𝑋) = (𝑋( ·𝑠 ‘𝑊)(1r‘𝑊))) |
21 | 20 | oveq2d 7287 | . 2 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐾) → (𝑁 ↑ (𝐴‘𝑋)) = (𝑁 ↑ (𝑋( ·𝑠 ‘𝑊)(1r‘𝑊)))) |
22 | 11 | assasca 21067 | . . . . . 6 ⊢ (𝑊 ∈ AssAlg → 𝐹 ∈ CRing) |
23 | 22 | 3ad2ant1 1132 | . . . . 5 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐾) → 𝐹 ∈ CRing) |
24 | 23 | crnggrpd 19795 | . . . 4 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐾) → 𝐹 ∈ Grp) |
25 | 4 | nn0zd 12423 | . . . 4 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐾) → 𝑁 ∈ ℤ) |
26 | 13, 15, 24, 25, 3 | mulgcld 18723 | . . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐾) → (𝑁 ∗ 𝑋) ∈ 𝐾) |
27 | 18, 11, 13, 12, 7 | asclval 21082 | . . 3 ⊢ ((𝑁 ∗ 𝑋) ∈ 𝐾 → (𝐴‘(𝑁 ∗ 𝑋)) = ((𝑁 ∗ 𝑋)( ·𝑠 ‘𝑊)(1r‘𝑊))) |
28 | 26, 27 | syl 17 | . 2 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐾) → (𝐴‘(𝑁 ∗ 𝑋)) = ((𝑁 ∗ 𝑋)( ·𝑠 ‘𝑊)(1r‘𝑊))) |
29 | 17, 21, 28 | 3eqtr4rd 2791 | 1 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐾) → (𝐴‘(𝑁 ∗ 𝑋)) = (𝑁 ↑ (𝐴‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1542 ∈ wcel 2110 ‘cfv 6432 (class class class)co 7271 ℕ0cn0 12233 Basecbs 16910 Scalarcsca 16963 ·𝑠 cvsca 16964 .gcmg 18698 1rcur 19735 Ringcrg 19781 CRingccrg 19782 LModclmod 20121 AssAlgcasa 21055 algSccascl 21057 |
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 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 |
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 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-1st 7824 df-2nd 7825 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-er 8481 df-en 8717 df-dom 8718 df-sdom 8719 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-n0 12234 df-z 12320 df-uz 12582 df-fz 13239 df-seq 13720 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-plusg 16973 df-0g 17150 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-grp 18578 df-minusg 18579 df-mulg 18699 df-mgp 19719 df-ur 19736 df-ring 19783 df-cring 19784 df-lmod 20123 df-assa 21058 df-ascl 21060 |
This theorem is referenced by: ply1chr 31665 |
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