Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2819 | . 2 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
2 | eqid 2819 | . 2 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
3 | eqid 2819 | . 2 ⊢ (+g‘𝐹) = (+g‘𝐹) | |
4 | eqid 2819 | . 2 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
5 | asclf.l | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
6 | asclf.f | . . . . 5 ⊢ 𝐹 = (Scalar‘𝑊) | |
7 | 6 | lmodring 19634 | . . . 4 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Ring) |
8 | 5, 7 | syl 17 | . . 3 ⊢ (𝜑 → 𝐹 ∈ Ring) |
9 | ringgrp 19294 | . . 3 ⊢ (𝐹 ∈ Ring → 𝐹 ∈ Grp) | |
10 | 8, 9 | syl 17 | . 2 ⊢ (𝜑 → 𝐹 ∈ Grp) |
11 | asclf.r | . . 3 ⊢ (𝜑 → 𝑊 ∈ Ring) | |
12 | ringgrp 19294 | . . 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 20103 | . 2 ⊢ (𝜑 → 𝐴:(Base‘𝐹)⟶(Base‘𝑊)) |
16 | 5 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → 𝑊 ∈ LMod) |
17 | simprl 769 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → 𝑥 ∈ (Base‘𝐹)) | |
18 | simprr 771 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → 𝑦 ∈ (Base‘𝐹)) | |
19 | eqid 2819 | . . . . . . 7 ⊢ (1r‘𝑊) = (1r‘𝑊) | |
20 | 2, 19 | ringidcl 19310 | . . . . . 6 ⊢ (𝑊 ∈ Ring → (1r‘𝑊) ∈ (Base‘𝑊)) |
21 | 11, 20 | syl 17 | . . . . 5 ⊢ (𝜑 → (1r‘𝑊) ∈ (Base‘𝑊)) |
22 | 21 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → (1r‘𝑊) ∈ (Base‘𝑊)) |
23 | eqid 2819 | . . . . 5 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
24 | 2, 4, 6, 23, 1, 3 | lmodvsdir 19650 | . . . 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 1367 | . . 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 18103 | . . . . . 6 ⊢ ((𝐹 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹)) → (𝑥(+g‘𝐹)𝑦) ∈ (Base‘𝐹)) |
27 | 26 | 3expb 1115 | . . . . 5 ⊢ ((𝐹 ∈ Grp ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → (𝑥(+g‘𝐹)𝑦) ∈ (Base‘𝐹)) |
28 | 10, 27 | sylan 582 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → (𝑥(+g‘𝐹)𝑦) ∈ (Base‘𝐹)) |
29 | 14, 6, 1, 23, 19 | asclval 20101 | . . . 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 20101 | . . . . 5 ⊢ (𝑥 ∈ (Base‘𝐹) → (𝐴‘𝑥) = (𝑥( ·𝑠 ‘𝑊)(1r‘𝑊))) |
32 | 14, 6, 1, 23, 19 | asclval 20101 | . . . . 5 ⊢ (𝑦 ∈ (Base‘𝐹) → (𝐴‘𝑦) = (𝑦( ·𝑠 ‘𝑊)(1r‘𝑊))) |
33 | 31, 32 | oveqan12d 7167 | . . . 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 484 | . . 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 2864 | . 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 18359 | 1 ⊢ (𝜑 → 𝐴 ∈ (𝐹 GrpHom 𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1531 ∈ wcel 2108 ‘cfv 6348 (class class class)co 7148 Basecbs 16475 +gcplusg 16557 Scalarcsca 16560 ·𝑠 cvsca 16561 Grpcgrp 18095 GrpHom cghm 18347 1rcur 19243 Ringcrg 19289 LModclmod 19626 algSccascl 20076 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-nn 11631 df-2 11692 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-plusg 16570 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-ghm 18348 df-mgp 19232 df-ur 19244 df-ring 19291 df-lmod 19628 df-ascl 20079 |
This theorem is referenced by: asclinvg 20110 asclrhm 20111 cpmatacl 21316 cpmatinvcl 21317 mat2pmatghm 21330 mat2pmatmul 21331 |
Copyright terms: Public domain | W3C validator |