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Mirrors > Home > MPE Home > Th. List > expghm | Structured version Visualization version GIF version |
Description: Exponentiation is a group homomorphism from addition to multiplication. (Contributed by Mario Carneiro, 18-Jun-2015.) (Revised by AV, 10-Jun-2019.) |
Ref | Expression |
---|---|
expghm.m | ⊢ 𝑀 = (mulGrp‘ℂfld) |
expghm.u | ⊢ 𝑈 = (𝑀 ↾s (ℂ ∖ {0})) |
Ref | Expression |
---|---|
expghm | ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝑥 ∈ ℤ ↦ (𝐴↑𝑥)) ∈ (ℤring GrpHom 𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | expclzlem 12884 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑥 ∈ ℤ) → (𝐴↑𝑥) ∈ (ℂ ∖ {0})) | |
2 | 1 | 3expa 1265 | . . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝑥 ∈ ℤ) → (𝐴↑𝑥) ∈ (ℂ ∖ {0})) |
3 | eqid 2622 | . . 3 ⊢ (𝑥 ∈ ℤ ↦ (𝐴↑𝑥)) = (𝑥 ∈ ℤ ↦ (𝐴↑𝑥)) | |
4 | 2, 3 | fmptd 6385 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝑥 ∈ ℤ ↦ (𝐴↑𝑥)):ℤ⟶(ℂ ∖ {0})) |
5 | expaddz 12904 | . . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → (𝐴↑(𝑦 + 𝑧)) = ((𝐴↑𝑦) · (𝐴↑𝑧))) | |
6 | zaddcl 11417 | . . . . . 6 ⊢ ((𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ) → (𝑦 + 𝑧) ∈ ℤ) | |
7 | 6 | adantl 482 | . . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → (𝑦 + 𝑧) ∈ ℤ) |
8 | oveq2 6658 | . . . . . 6 ⊢ (𝑥 = (𝑦 + 𝑧) → (𝐴↑𝑥) = (𝐴↑(𝑦 + 𝑧))) | |
9 | ovex 6678 | . . . . . 6 ⊢ (𝐴↑(𝑦 + 𝑧)) ∈ V | |
10 | 8, 3, 9 | fvmpt 6282 | . . . . 5 ⊢ ((𝑦 + 𝑧) ∈ ℤ → ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘(𝑦 + 𝑧)) = (𝐴↑(𝑦 + 𝑧))) |
11 | 7, 10 | syl 17 | . . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘(𝑦 + 𝑧)) = (𝐴↑(𝑦 + 𝑧))) |
12 | oveq2 6658 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴↑𝑥) = (𝐴↑𝑦)) | |
13 | ovex 6678 | . . . . . . 7 ⊢ (𝐴↑𝑦) ∈ V | |
14 | 12, 3, 13 | fvmpt 6282 | . . . . . 6 ⊢ (𝑦 ∈ ℤ → ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑦) = (𝐴↑𝑦)) |
15 | oveq2 6658 | . . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝐴↑𝑥) = (𝐴↑𝑧)) | |
16 | ovex 6678 | . . . . . . 7 ⊢ (𝐴↑𝑧) ∈ V | |
17 | 15, 3, 16 | fvmpt 6282 | . . . . . 6 ⊢ (𝑧 ∈ ℤ → ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑧) = (𝐴↑𝑧)) |
18 | 14, 17 | oveqan12d 6669 | . . . . 5 ⊢ ((𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ) → (((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑦) · ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑧)) = ((𝐴↑𝑦) · (𝐴↑𝑧))) |
19 | 18 | adantl 482 | . . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → (((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑦) · ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑧)) = ((𝐴↑𝑦) · (𝐴↑𝑧))) |
20 | 5, 11, 19 | 3eqtr4d 2666 | . . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘(𝑦 + 𝑧)) = (((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑦) · ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑧))) |
21 | 20 | ralrimivva 2971 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ∀𝑦 ∈ ℤ ∀𝑧 ∈ ℤ ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘(𝑦 + 𝑧)) = (((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑦) · ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑧))) |
22 | zringgrp 19823 | . . . 4 ⊢ ℤring ∈ Grp | |
23 | cnring 19768 | . . . . 5 ⊢ ℂfld ∈ Ring | |
24 | cnfldbas 19750 | . . . . . . 7 ⊢ ℂ = (Base‘ℂfld) | |
25 | cnfld0 19770 | . . . . . . 7 ⊢ 0 = (0g‘ℂfld) | |
26 | cndrng 19775 | . . . . . . 7 ⊢ ℂfld ∈ DivRing | |
27 | 24, 25, 26 | drngui 18753 | . . . . . 6 ⊢ (ℂ ∖ {0}) = (Unit‘ℂfld) |
28 | expghm.u | . . . . . . 7 ⊢ 𝑈 = (𝑀 ↾s (ℂ ∖ {0})) | |
29 | expghm.m | . . . . . . . 8 ⊢ 𝑀 = (mulGrp‘ℂfld) | |
30 | 29 | oveq1i 6660 | . . . . . . 7 ⊢ (𝑀 ↾s (ℂ ∖ {0})) = ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) |
31 | 28, 30 | eqtri 2644 | . . . . . 6 ⊢ 𝑈 = ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) |
32 | 27, 31 | unitgrp 18667 | . . . . 5 ⊢ (ℂfld ∈ Ring → 𝑈 ∈ Grp) |
33 | 23, 32 | ax-mp 5 | . . . 4 ⊢ 𝑈 ∈ Grp |
34 | 22, 33 | pm3.2i 471 | . . 3 ⊢ (ℤring ∈ Grp ∧ 𝑈 ∈ Grp) |
35 | zringbas 19824 | . . . 4 ⊢ ℤ = (Base‘ℤring) | |
36 | difss 3737 | . . . . 5 ⊢ (ℂ ∖ {0}) ⊆ ℂ | |
37 | 29, 24 | mgpbas 18495 | . . . . . 6 ⊢ ℂ = (Base‘𝑀) |
38 | 28, 37 | ressbas2 15931 | . . . . 5 ⊢ ((ℂ ∖ {0}) ⊆ ℂ → (ℂ ∖ {0}) = (Base‘𝑈)) |
39 | 36, 38 | ax-mp 5 | . . . 4 ⊢ (ℂ ∖ {0}) = (Base‘𝑈) |
40 | zringplusg 19825 | . . . 4 ⊢ + = (+g‘ℤring) | |
41 | fvex 6201 | . . . . . 6 ⊢ (Unit‘ℂfld) ∈ V | |
42 | 27, 41 | eqeltri 2697 | . . . . 5 ⊢ (ℂ ∖ {0}) ∈ V |
43 | cnfldmul 19752 | . . . . . . 7 ⊢ · = (.r‘ℂfld) | |
44 | 29, 43 | mgpplusg 18493 | . . . . . 6 ⊢ · = (+g‘𝑀) |
45 | 28, 44 | ressplusg 15993 | . . . . 5 ⊢ ((ℂ ∖ {0}) ∈ V → · = (+g‘𝑈)) |
46 | 42, 45 | ax-mp 5 | . . . 4 ⊢ · = (+g‘𝑈) |
47 | 35, 39, 40, 46 | isghm 17660 | . . 3 ⊢ ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥)) ∈ (ℤring GrpHom 𝑈) ↔ ((ℤring ∈ Grp ∧ 𝑈 ∈ Grp) ∧ ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥)):ℤ⟶(ℂ ∖ {0}) ∧ ∀𝑦 ∈ ℤ ∀𝑧 ∈ ℤ ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘(𝑦 + 𝑧)) = (((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑦) · ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑧))))) |
48 | 34, 47 | mpbiran 953 | . 2 ⊢ ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥)) ∈ (ℤring GrpHom 𝑈) ↔ ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥)):ℤ⟶(ℂ ∖ {0}) ∧ ∀𝑦 ∈ ℤ ∀𝑧 ∈ ℤ ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘(𝑦 + 𝑧)) = (((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑦) · ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑧)))) |
49 | 4, 21, 48 | sylanbrc 698 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝑥 ∈ ℤ ↦ (𝐴↑𝑥)) ∈ (ℤring GrpHom 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 Vcvv 3200 ∖ cdif 3571 ⊆ wss 3574 {csn 4177 ↦ cmpt 4729 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 0cc0 9936 + caddc 9939 · cmul 9941 ℤcz 11377 ↑cexp 12860 Basecbs 15857 ↾s cress 15858 +gcplusg 15941 Grpcgrp 17422 GrpHom cghm 17657 mulGrpcmgp 18489 Ringcrg 18547 Unitcui 18639 ℂfldccnfld 19746 ℤringzring 19818 |
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 ax-addf 10015 ax-mulf 10016 |
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-int 4476 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-1st 7168 df-2nd 7169 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-seq 12802 df-exp 12861 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-subg 17591 df-ghm 17658 df-cmn 18195 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-dvr 18683 df-drng 18749 df-subrg 18778 df-cnfld 19747 df-zring 19819 |
This theorem is referenced by: lgseisenlem4 25103 |
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