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Mirrors > Home > MPE Home > Th. List > asclghm | Structured version Visualization version GIF version |
Description: The algebra scalars function is a group homomorphism. (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) |
Ref | Expression |
---|---|
asclghm | ⊢ (𝜑 → 𝐴 ∈ (𝐹 GrpHom 𝑊)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2737 | . 2 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
2 | eqid 2737 | . 2 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
3 | eqid 2737 | . 2 ⊢ (+g‘𝐹) = (+g‘𝐹) | |
4 | eqid 2737 | . 2 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
5 | asclf.l | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
6 | asclf.f | . . . . 5 ⊢ 𝐹 = (Scalar‘𝑊) | |
7 | 6 | lmodring 19907 | . . . 4 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Ring) |
8 | 5, 7 | syl 17 | . . 3 ⊢ (𝜑 → 𝐹 ∈ Ring) |
9 | ringgrp 19567 | . . 3 ⊢ (𝐹 ∈ Ring → 𝐹 ∈ Grp) | |
10 | 8, 9 | syl 17 | . 2 ⊢ (𝜑 → 𝐹 ∈ Grp) |
11 | asclf.r | . . 3 ⊢ (𝜑 → 𝑊 ∈ Ring) | |
12 | ringgrp 19567 | . . 3 ⊢ (𝑊 ∈ Ring → 𝑊 ∈ Grp) | |
13 | 11, 12 | syl 17 | . 2 ⊢ (𝜑 → 𝑊 ∈ Grp) |
14 | asclf.a | . . 3 ⊢ 𝐴 = (algSc‘𝑊) | |
15 | 14, 6, 11, 5, 1, 2 | asclf 20841 | . 2 ⊢ (𝜑 → 𝐴:(Base‘𝐹)⟶(Base‘𝑊)) |
16 | 5 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → 𝑊 ∈ LMod) |
17 | simprl 771 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → 𝑥 ∈ (Base‘𝐹)) | |
18 | simprr 773 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → 𝑦 ∈ (Base‘𝐹)) | |
19 | eqid 2737 | . . . . . . 7 ⊢ (1r‘𝑊) = (1r‘𝑊) | |
20 | 2, 19 | ringidcl 19586 | . . . . . 6 ⊢ (𝑊 ∈ Ring → (1r‘𝑊) ∈ (Base‘𝑊)) |
21 | 11, 20 | syl 17 | . . . . 5 ⊢ (𝜑 → (1r‘𝑊) ∈ (Base‘𝑊)) |
22 | 21 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → (1r‘𝑊) ∈ (Base‘𝑊)) |
23 | eqid 2737 | . . . . 5 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
24 | 2, 4, 6, 23, 1, 3 | lmodvsdir 19923 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹) ∧ (1r‘𝑊) ∈ (Base‘𝑊))) → ((𝑥(+g‘𝐹)𝑦)( ·𝑠 ‘𝑊)(1r‘𝑊)) = ((𝑥( ·𝑠 ‘𝑊)(1r‘𝑊))(+g‘𝑊)(𝑦( ·𝑠 ‘𝑊)(1r‘𝑊)))) |
25 | 16, 17, 18, 22, 24 | syl13anc 1374 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → ((𝑥(+g‘𝐹)𝑦)( ·𝑠 ‘𝑊)(1r‘𝑊)) = ((𝑥( ·𝑠 ‘𝑊)(1r‘𝑊))(+g‘𝑊)(𝑦( ·𝑠 ‘𝑊)(1r‘𝑊)))) |
26 | 1, 3 | grpcl 18373 | . . . . . 6 ⊢ ((𝐹 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹)) → (𝑥(+g‘𝐹)𝑦) ∈ (Base‘𝐹)) |
27 | 26 | 3expb 1122 | . . . . 5 ⊢ ((𝐹 ∈ Grp ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → (𝑥(+g‘𝐹)𝑦) ∈ (Base‘𝐹)) |
28 | 10, 27 | sylan 583 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → (𝑥(+g‘𝐹)𝑦) ∈ (Base‘𝐹)) |
29 | 14, 6, 1, 23, 19 | asclval 20839 | . . . 4 ⊢ ((𝑥(+g‘𝐹)𝑦) ∈ (Base‘𝐹) → (𝐴‘(𝑥(+g‘𝐹)𝑦)) = ((𝑥(+g‘𝐹)𝑦)( ·𝑠 ‘𝑊)(1r‘𝑊))) |
30 | 28, 29 | syl 17 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → (𝐴‘(𝑥(+g‘𝐹)𝑦)) = ((𝑥(+g‘𝐹)𝑦)( ·𝑠 ‘𝑊)(1r‘𝑊))) |
31 | 14, 6, 1, 23, 19 | asclval 20839 | . . . . 5 ⊢ (𝑥 ∈ (Base‘𝐹) → (𝐴‘𝑥) = (𝑥( ·𝑠 ‘𝑊)(1r‘𝑊))) |
32 | 14, 6, 1, 23, 19 | asclval 20839 | . . . . 5 ⊢ (𝑦 ∈ (Base‘𝐹) → (𝐴‘𝑦) = (𝑦( ·𝑠 ‘𝑊)(1r‘𝑊))) |
33 | 31, 32 | oveqan12d 7232 | . . . 4 ⊢ ((𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹)) → ((𝐴‘𝑥)(+g‘𝑊)(𝐴‘𝑦)) = ((𝑥( ·𝑠 ‘𝑊)(1r‘𝑊))(+g‘𝑊)(𝑦( ·𝑠 ‘𝑊)(1r‘𝑊)))) |
34 | 33 | adantl 485 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → ((𝐴‘𝑥)(+g‘𝑊)(𝐴‘𝑦)) = ((𝑥( ·𝑠 ‘𝑊)(1r‘𝑊))(+g‘𝑊)(𝑦( ·𝑠 ‘𝑊)(1r‘𝑊)))) |
35 | 25, 30, 34 | 3eqtr4d 2787 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → (𝐴‘(𝑥(+g‘𝐹)𝑦)) = ((𝐴‘𝑥)(+g‘𝑊)(𝐴‘𝑦))) |
36 | 1, 2, 3, 4, 10, 13, 15, 35 | isghmd 18631 | 1 ⊢ (𝜑 → 𝐴 ∈ (𝐹 GrpHom 𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ‘cfv 6380 (class class class)co 7213 Basecbs 16760 +gcplusg 16802 Scalarcsca 16805 ·𝑠 cvsca 16806 Grpcgrp 18365 GrpHom cghm 18619 1rcur 19516 Ringcrg 19562 LModclmod 19899 algSccascl 20814 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-2 11893 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-plusg 16815 df-0g 16946 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-grp 18368 df-ghm 18620 df-mgp 19505 df-ur 19517 df-ring 19564 df-lmod 19901 df-ascl 20817 |
This theorem is referenced by: asclinvg 20849 asclrhm 20850 cpmatacl 21613 cpmatinvcl 21614 mat2pmatghm 21627 mat2pmatmul 21628 |
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