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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 2610 | . 2 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
2 | eqid 2610 | . 2 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
3 | eqid 2610 | . 2 ⊢ (+g‘𝐹) = (+g‘𝐹) | |
4 | eqid 2610 | . 2 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
5 | asclf.l | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
6 | asclf.f | . . . . 5 ⊢ 𝐹 = (Scalar‘𝑊) | |
7 | 6 | lmodring 18694 | . . . 4 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Ring) |
8 | 5, 7 | syl 17 | . . 3 ⊢ (𝜑 → 𝐹 ∈ Ring) |
9 | ringgrp 18375 | . . 3 ⊢ (𝐹 ∈ Ring → 𝐹 ∈ Grp) | |
10 | 8, 9 | syl 17 | . 2 ⊢ (𝜑 → 𝐹 ∈ Grp) |
11 | asclf.r | . . 3 ⊢ (𝜑 → 𝑊 ∈ Ring) | |
12 | ringgrp 18375 | . . 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 19158 | . 2 ⊢ (𝜑 → 𝐴:(Base‘𝐹)⟶(Base‘𝑊)) |
16 | 5 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → 𝑊 ∈ LMod) |
17 | simprl 790 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → 𝑥 ∈ (Base‘𝐹)) | |
18 | simprr 792 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → 𝑦 ∈ (Base‘𝐹)) | |
19 | eqid 2610 | . . . . . . 7 ⊢ (1r‘𝑊) = (1r‘𝑊) | |
20 | 2, 19 | ringidcl 18391 | . . . . . 6 ⊢ (𝑊 ∈ Ring → (1r‘𝑊) ∈ (Base‘𝑊)) |
21 | 11, 20 | syl 17 | . . . . 5 ⊢ (𝜑 → (1r‘𝑊) ∈ (Base‘𝑊)) |
22 | 21 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → (1r‘𝑊) ∈ (Base‘𝑊)) |
23 | eqid 2610 | . . . . 5 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
24 | 2, 4, 6, 23, 1, 3 | lmodvsdir 18710 | . . . 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 1320 | . . 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 17253 | . . . . . 6 ⊢ ((𝐹 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹)) → (𝑥(+g‘𝐹)𝑦) ∈ (Base‘𝐹)) |
27 | 26 | 3expb 1258 | . . . . 5 ⊢ ((𝐹 ∈ Grp ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → (𝑥(+g‘𝐹)𝑦) ∈ (Base‘𝐹)) |
28 | 10, 27 | sylan 487 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → (𝑥(+g‘𝐹)𝑦) ∈ (Base‘𝐹)) |
29 | 14, 6, 1, 23, 19 | asclval 19156 | . . . 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 19156 | . . . . 5 ⊢ (𝑥 ∈ (Base‘𝐹) → (𝐴‘𝑥) = (𝑥( ·𝑠 ‘𝑊)(1r‘𝑊))) |
32 | 14, 6, 1, 23, 19 | asclval 19156 | . . . . 5 ⊢ (𝑦 ∈ (Base‘𝐹) → (𝐴‘𝑦) = (𝑦( ·𝑠 ‘𝑊)(1r‘𝑊))) |
33 | 31, 32 | oveqan12d 6568 | . . . 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 481 | . . 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 2654 | . 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 17492 | 1 ⊢ (𝜑 → 𝐴 ∈ (𝐹 GrpHom 𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 Scalarcsca 15771 ·𝑠 cvsca 15772 Grpcgrp 17245 GrpHom cghm 17480 1rcur 18324 Ringcrg 18370 LModclmod 18686 algSccascl 19132 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-plusg 15781 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-ghm 17481 df-mgp 18313 df-ur 18325 df-ring 18372 df-lmod 18688 df-ascl 19135 |
This theorem is referenced by: asclinvg 19162 asclrhm 19163 cpmatacl 20340 cpmatinvcl 20341 mat2pmatghm 20354 mat2pmatmul 20355 |
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