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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 2797 | . 2 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
2 | eqid 2797 | . 2 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
3 | eqid 2797 | . 2 ⊢ (+g‘𝐹) = (+g‘𝐹) | |
4 | eqid 2797 | . 2 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
5 | asclf.l | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
6 | asclf.f | . . . . 5 ⊢ 𝐹 = (Scalar‘𝑊) | |
7 | 6 | lmodring 19336 | . . . 4 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Ring) |
8 | 5, 7 | syl 17 | . . 3 ⊢ (𝜑 → 𝐹 ∈ Ring) |
9 | ringgrp 18996 | . . 3 ⊢ (𝐹 ∈ Ring → 𝐹 ∈ Grp) | |
10 | 8, 9 | syl 17 | . 2 ⊢ (𝜑 → 𝐹 ∈ Grp) |
11 | asclf.r | . . 3 ⊢ (𝜑 → 𝑊 ∈ Ring) | |
12 | ringgrp 18996 | . . 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 19803 | . 2 ⊢ (𝜑 → 𝐴:(Base‘𝐹)⟶(Base‘𝑊)) |
16 | 5 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → 𝑊 ∈ LMod) |
17 | simprl 767 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → 𝑥 ∈ (Base‘𝐹)) | |
18 | simprr 769 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → 𝑦 ∈ (Base‘𝐹)) | |
19 | eqid 2797 | . . . . . . 7 ⊢ (1r‘𝑊) = (1r‘𝑊) | |
20 | 2, 19 | ringidcl 19012 | . . . . . 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 2797 | . . . . 5 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
24 | 2, 4, 6, 23, 1, 3 | lmodvsdir 19352 | . . . 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 1365 | . . 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 17873 | . . . . . 6 ⊢ ((𝐹 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹)) → (𝑥(+g‘𝐹)𝑦) ∈ (Base‘𝐹)) |
27 | 26 | 3expb 1113 | . . . . 5 ⊢ ((𝐹 ∈ Grp ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → (𝑥(+g‘𝐹)𝑦) ∈ (Base‘𝐹)) |
28 | 10, 27 | sylan 580 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → (𝑥(+g‘𝐹)𝑦) ∈ (Base‘𝐹)) |
29 | 14, 6, 1, 23, 19 | asclval 19801 | . . . 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 19801 | . . . . 5 ⊢ (𝑥 ∈ (Base‘𝐹) → (𝐴‘𝑥) = (𝑥( ·𝑠 ‘𝑊)(1r‘𝑊))) |
32 | 14, 6, 1, 23, 19 | asclval 19801 | . . . . 5 ⊢ (𝑦 ∈ (Base‘𝐹) → (𝐴‘𝑦) = (𝑦( ·𝑠 ‘𝑊)(1r‘𝑊))) |
33 | 31, 32 | oveqan12d 7042 | . . . 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 2843 | . 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 18112 | 1 ⊢ (𝜑 → 𝐴 ∈ (𝐹 GrpHom 𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1525 ∈ wcel 2083 ‘cfv 6232 (class class class)co 7023 Basecbs 16316 +gcplusg 16398 Scalarcsca 16401 ·𝑠 cvsca 16402 Grpcgrp 17865 GrpHom cghm 18100 1rcur 18945 Ringcrg 18991 LModclmod 19328 algSccascl 19777 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-cnex 10446 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 ax-pre-mulgt0 10467 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-iun 4833 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-om 7444 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-er 8146 df-en 8365 df-dom 8366 df-sdom 8367 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-sub 10725 df-neg 10726 df-nn 11493 df-2 11554 df-ndx 16319 df-slot 16320 df-base 16322 df-sets 16323 df-plusg 16411 df-0g 16548 df-mgm 17685 df-sgrp 17727 df-mnd 17738 df-grp 17868 df-ghm 18101 df-mgp 18934 df-ur 18946 df-ring 18993 df-lmod 19330 df-ascl 19780 |
This theorem is referenced by: asclinvg 19810 asclrhm 19811 cpmatacl 21012 cpmatinvcl 21013 mat2pmatghm 21026 mat2pmatmul 21027 |
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