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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 2622 | . 2 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
2 | eqid 2622 | . 2 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
3 | eqid 2622 | . 2 ⊢ (+g‘𝐹) = (+g‘𝐹) | |
4 | eqid 2622 | . 2 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
5 | asclf.l | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
6 | asclf.f | . . . . 5 ⊢ 𝐹 = (Scalar‘𝑊) | |
7 | 6 | lmodring 18871 | . . . 4 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Ring) |
8 | 5, 7 | syl 17 | . . 3 ⊢ (𝜑 → 𝐹 ∈ Ring) |
9 | ringgrp 18552 | . . 3 ⊢ (𝐹 ∈ Ring → 𝐹 ∈ Grp) | |
10 | 8, 9 | syl 17 | . 2 ⊢ (𝜑 → 𝐹 ∈ Grp) |
11 | asclf.r | . . 3 ⊢ (𝜑 → 𝑊 ∈ Ring) | |
12 | ringgrp 18552 | . . 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 19337 | . 2 ⊢ (𝜑 → 𝐴:(Base‘𝐹)⟶(Base‘𝑊)) |
16 | 5 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → 𝑊 ∈ LMod) |
17 | simprl 794 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → 𝑥 ∈ (Base‘𝐹)) | |
18 | simprr 796 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → 𝑦 ∈ (Base‘𝐹)) | |
19 | eqid 2622 | . . . . . . 7 ⊢ (1r‘𝑊) = (1r‘𝑊) | |
20 | 2, 19 | ringidcl 18568 | . . . . . 6 ⊢ (𝑊 ∈ Ring → (1r‘𝑊) ∈ (Base‘𝑊)) |
21 | 11, 20 | syl 17 | . . . . 5 ⊢ (𝜑 → (1r‘𝑊) ∈ (Base‘𝑊)) |
22 | 21 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → (1r‘𝑊) ∈ (Base‘𝑊)) |
23 | eqid 2622 | . . . . 5 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
24 | 2, 4, 6, 23, 1, 3 | lmodvsdir 18887 | . . . 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 1328 | . . 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 17430 | . . . . . 6 ⊢ ((𝐹 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹)) → (𝑥(+g‘𝐹)𝑦) ∈ (Base‘𝐹)) |
27 | 26 | 3expb 1266 | . . . . 5 ⊢ ((𝐹 ∈ Grp ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → (𝑥(+g‘𝐹)𝑦) ∈ (Base‘𝐹)) |
28 | 10, 27 | sylan 488 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → (𝑥(+g‘𝐹)𝑦) ∈ (Base‘𝐹)) |
29 | 14, 6, 1, 23, 19 | asclval 19335 | . . . 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 19335 | . . . . 5 ⊢ (𝑥 ∈ (Base‘𝐹) → (𝐴‘𝑥) = (𝑥( ·𝑠 ‘𝑊)(1r‘𝑊))) |
32 | 14, 6, 1, 23, 19 | asclval 19335 | . . . . 5 ⊢ (𝑦 ∈ (Base‘𝐹) → (𝐴‘𝑦) = (𝑦( ·𝑠 ‘𝑊)(1r‘𝑊))) |
33 | 31, 32 | oveqan12d 6669 | . . . 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 482 | . . 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 2666 | . 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 17669 | 1 ⊢ (𝜑 → 𝐴 ∈ (𝐹 GrpHom 𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 +gcplusg 15941 Scalarcsca 15944 ·𝑠 cvsca 15945 Grpcgrp 17422 GrpHom cghm 17657 1rcur 18501 Ringcrg 18547 LModclmod 18863 algSccascl 19311 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-plusg 15954 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-ghm 17658 df-mgp 18490 df-ur 18502 df-ring 18549 df-lmod 18865 df-ascl 19314 |
This theorem is referenced by: asclinvg 19341 asclrhm 19342 cpmatacl 20521 cpmatinvcl 20522 mat2pmatghm 20535 mat2pmatmul 20536 |
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