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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 13989 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑥 ∈ ℤ) → (𝐴↑𝑥) ∈ (ℂ ∖ {0})) | |
2 | 1 | 3expa 1118 | . . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝑥 ∈ ℤ) → (𝐴↑𝑥) ∈ (ℂ ∖ {0})) |
3 | 2 | fmpttd 7063 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝑥 ∈ ℤ ↦ (𝐴↑𝑥)):ℤ⟶(ℂ ∖ {0})) |
4 | expaddz 14012 | . . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → (𝐴↑(𝑦 + 𝑧)) = ((𝐴↑𝑦) · (𝐴↑𝑧))) | |
5 | zaddcl 12543 | . . . . . 6 ⊢ ((𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ) → (𝑦 + 𝑧) ∈ ℤ) | |
6 | 5 | adantl 482 | . . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → (𝑦 + 𝑧) ∈ ℤ) |
7 | oveq2 7365 | . . . . . 6 ⊢ (𝑥 = (𝑦 + 𝑧) → (𝐴↑𝑥) = (𝐴↑(𝑦 + 𝑧))) | |
8 | eqid 2736 | . . . . . 6 ⊢ (𝑥 ∈ ℤ ↦ (𝐴↑𝑥)) = (𝑥 ∈ ℤ ↦ (𝐴↑𝑥)) | |
9 | ovex 7390 | . . . . . 6 ⊢ (𝐴↑(𝑦 + 𝑧)) ∈ V | |
10 | 7, 8, 9 | fvmpt 6948 | . . . . 5 ⊢ ((𝑦 + 𝑧) ∈ ℤ → ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘(𝑦 + 𝑧)) = (𝐴↑(𝑦 + 𝑧))) |
11 | 6, 10 | syl 17 | . . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘(𝑦 + 𝑧)) = (𝐴↑(𝑦 + 𝑧))) |
12 | oveq2 7365 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴↑𝑥) = (𝐴↑𝑦)) | |
13 | ovex 7390 | . . . . . . 7 ⊢ (𝐴↑𝑦) ∈ V | |
14 | 12, 8, 13 | fvmpt 6948 | . . . . . 6 ⊢ (𝑦 ∈ ℤ → ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑦) = (𝐴↑𝑦)) |
15 | oveq2 7365 | . . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝐴↑𝑥) = (𝐴↑𝑧)) | |
16 | ovex 7390 | . . . . . . 7 ⊢ (𝐴↑𝑧) ∈ V | |
17 | 15, 8, 16 | fvmpt 6948 | . . . . . 6 ⊢ (𝑧 ∈ ℤ → ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑧) = (𝐴↑𝑧)) |
18 | 14, 17 | oveqan12d 7376 | . . . . 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 | 4, 11, 19 | 3eqtr4d 2786 | . . 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 3197 | . 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 20874 | . . . 4 ⊢ ℤring ∈ Grp | |
23 | cnring 20819 | . . . . 5 ⊢ ℂfld ∈ Ring | |
24 | cnfldbas 20800 | . . . . . . 7 ⊢ ℂ = (Base‘ℂfld) | |
25 | cnfld0 20821 | . . . . . . 7 ⊢ 0 = (0g‘ℂfld) | |
26 | cndrng 20826 | . . . . . . 7 ⊢ ℂfld ∈ DivRing | |
27 | 24, 25, 26 | drngui 20191 | . . . . . 6 ⊢ (ℂ ∖ {0}) = (Unit‘ℂfld) |
28 | expghm.u | . . . . . . 7 ⊢ 𝑈 = (𝑀 ↾s (ℂ ∖ {0})) | |
29 | expghm.m | . . . . . . . 8 ⊢ 𝑀 = (mulGrp‘ℂfld) | |
30 | 29 | oveq1i 7367 | . . . . . . 7 ⊢ (𝑀 ↾s (ℂ ∖ {0})) = ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) |
31 | 28, 30 | eqtri 2764 | . . . . . 6 ⊢ 𝑈 = ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) |
32 | 27, 31 | unitgrp 20096 | . . . . 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 20875 | . . . 4 ⊢ ℤ = (Base‘ℤring) | |
36 | difss 4091 | . . . . 5 ⊢ (ℂ ∖ {0}) ⊆ ℂ | |
37 | 29, 24 | mgpbas 19902 | . . . . . 6 ⊢ ℂ = (Base‘𝑀) |
38 | 28, 37 | ressbas2 17120 | . . . . 5 ⊢ ((ℂ ∖ {0}) ⊆ ℂ → (ℂ ∖ {0}) = (Base‘𝑈)) |
39 | 36, 38 | ax-mp 5 | . . . 4 ⊢ (ℂ ∖ {0}) = (Base‘𝑈) |
40 | zringplusg 20876 | . . . 4 ⊢ + = (+g‘ℤring) | |
41 | 27 | fvexi 6856 | . . . . 5 ⊢ (ℂ ∖ {0}) ∈ V |
42 | cnfldmul 20802 | . . . . . . 7 ⊢ · = (.r‘ℂfld) | |
43 | 29, 42 | mgpplusg 19900 | . . . . . 6 ⊢ · = (+g‘𝑀) |
44 | 28, 43 | ressplusg 17171 | . . . . 5 ⊢ ((ℂ ∖ {0}) ∈ V → · = (+g‘𝑈)) |
45 | 41, 44 | ax-mp 5 | . . . 4 ⊢ · = (+g‘𝑈) |
46 | 35, 39, 40, 45 | isghm 19008 | . . 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))))) |
47 | 34, 46 | mpbiran 707 | . 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)))) |
48 | 3, 21, 47 | sylanbrc 583 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝑥 ∈ ℤ ↦ (𝐴↑𝑥)) ∈ (ℤring GrpHom 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 Vcvv 3445 ∖ cdif 3907 ⊆ wss 3910 {csn 4586 ↦ cmpt 5188 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 ℂcc 11049 0cc0 11051 + caddc 11054 · cmul 11056 ℤcz 12499 ↑cexp 13967 Basecbs 17083 ↾s cress 17112 +gcplusg 17133 Grpcgrp 18748 GrpHom cghm 19005 mulGrpcmgp 19896 Ringcrg 19964 Unitcui 20068 ℂfldccnfld 20796 ℤringczring 20869 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-addf 11130 ax-mulf 11131 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-tpos 8157 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-fz 13425 df-seq 13907 df-exp 13968 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-0g 17323 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-grp 18751 df-minusg 18752 df-subg 18925 df-ghm 19006 df-cmn 19564 df-mgp 19897 df-ur 19914 df-ring 19966 df-cring 19967 df-oppr 20049 df-dvdsr 20070 df-unit 20071 df-invr 20101 df-dvr 20112 df-drng 20187 df-subrg 20220 df-cnfld 20797 df-zring 20870 |
This theorem is referenced by: lgseisenlem4 26726 |
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