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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 13449 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑥 ∈ ℤ) → (𝐴↑𝑥) ∈ (ℂ ∖ {0})) | |
2 | 1 | 3expa 1115 | . . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝑥 ∈ ℤ) → (𝐴↑𝑥) ∈ (ℂ ∖ {0})) |
3 | 2 | fmpttd 6856 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝑥 ∈ ℤ ↦ (𝐴↑𝑥)):ℤ⟶(ℂ ∖ {0})) |
4 | expaddz 13469 | . . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → (𝐴↑(𝑦 + 𝑧)) = ((𝐴↑𝑦) · (𝐴↑𝑧))) | |
5 | zaddcl 12010 | . . . . . 6 ⊢ ((𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ) → (𝑦 + 𝑧) ∈ ℤ) | |
6 | 5 | adantl 485 | . . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → (𝑦 + 𝑧) ∈ ℤ) |
7 | oveq2 7143 | . . . . . 6 ⊢ (𝑥 = (𝑦 + 𝑧) → (𝐴↑𝑥) = (𝐴↑(𝑦 + 𝑧))) | |
8 | eqid 2798 | . . . . . 6 ⊢ (𝑥 ∈ ℤ ↦ (𝐴↑𝑥)) = (𝑥 ∈ ℤ ↦ (𝐴↑𝑥)) | |
9 | ovex 7168 | . . . . . 6 ⊢ (𝐴↑(𝑦 + 𝑧)) ∈ V | |
10 | 7, 8, 9 | fvmpt 6745 | . . . . 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 7143 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴↑𝑥) = (𝐴↑𝑦)) | |
13 | ovex 7168 | . . . . . . 7 ⊢ (𝐴↑𝑦) ∈ V | |
14 | 12, 8, 13 | fvmpt 6745 | . . . . . 6 ⊢ (𝑦 ∈ ℤ → ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑦) = (𝐴↑𝑦)) |
15 | oveq2 7143 | . . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝐴↑𝑥) = (𝐴↑𝑧)) | |
16 | ovex 7168 | . . . . . . 7 ⊢ (𝐴↑𝑧) ∈ V | |
17 | 15, 8, 16 | fvmpt 6745 | . . . . . 6 ⊢ (𝑧 ∈ ℤ → ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑧) = (𝐴↑𝑧)) |
18 | 14, 17 | oveqan12d 7154 | . . . . 5 ⊢ ((𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ) → (((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑦) · ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑧)) = ((𝐴↑𝑦) · (𝐴↑𝑧))) |
19 | 18 | adantl 485 | . . . 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 2843 | . . 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 3156 | . 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 20168 | . . . 4 ⊢ ℤring ∈ Grp | |
23 | cnring 20113 | . . . . 5 ⊢ ℂfld ∈ Ring | |
24 | cnfldbas 20095 | . . . . . . 7 ⊢ ℂ = (Base‘ℂfld) | |
25 | cnfld0 20115 | . . . . . . 7 ⊢ 0 = (0g‘ℂfld) | |
26 | cndrng 20120 | . . . . . . 7 ⊢ ℂfld ∈ DivRing | |
27 | 24, 25, 26 | drngui 19501 | . . . . . 6 ⊢ (ℂ ∖ {0}) = (Unit‘ℂfld) |
28 | expghm.u | . . . . . . 7 ⊢ 𝑈 = (𝑀 ↾s (ℂ ∖ {0})) | |
29 | expghm.m | . . . . . . . 8 ⊢ 𝑀 = (mulGrp‘ℂfld) | |
30 | 29 | oveq1i 7145 | . . . . . . 7 ⊢ (𝑀 ↾s (ℂ ∖ {0})) = ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) |
31 | 28, 30 | eqtri 2821 | . . . . . 6 ⊢ 𝑈 = ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) |
32 | 27, 31 | unitgrp 19413 | . . . . 5 ⊢ (ℂfld ∈ Ring → 𝑈 ∈ Grp) |
33 | 23, 32 | ax-mp 5 | . . . 4 ⊢ 𝑈 ∈ Grp |
34 | 22, 33 | pm3.2i 474 | . . 3 ⊢ (ℤring ∈ Grp ∧ 𝑈 ∈ Grp) |
35 | zringbas 20169 | . . . 4 ⊢ ℤ = (Base‘ℤring) | |
36 | difss 4059 | . . . . 5 ⊢ (ℂ ∖ {0}) ⊆ ℂ | |
37 | 29, 24 | mgpbas 19238 | . . . . . 6 ⊢ ℂ = (Base‘𝑀) |
38 | 28, 37 | ressbas2 16547 | . . . . 5 ⊢ ((ℂ ∖ {0}) ⊆ ℂ → (ℂ ∖ {0}) = (Base‘𝑈)) |
39 | 36, 38 | ax-mp 5 | . . . 4 ⊢ (ℂ ∖ {0}) = (Base‘𝑈) |
40 | zringplusg 20170 | . . . 4 ⊢ + = (+g‘ℤring) | |
41 | 27 | fvexi 6659 | . . . . 5 ⊢ (ℂ ∖ {0}) ∈ V |
42 | cnfldmul 20097 | . . . . . . 7 ⊢ · = (.r‘ℂfld) | |
43 | 29, 42 | mgpplusg 19236 | . . . . . 6 ⊢ · = (+g‘𝑀) |
44 | 28, 43 | ressplusg 16604 | . . . . 5 ⊢ ((ℂ ∖ {0}) ∈ V → · = (+g‘𝑈)) |
45 | 41, 44 | ax-mp 5 | . . . 4 ⊢ · = (+g‘𝑈) |
46 | 35, 39, 40, 45 | isghm 18350 | . . 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 708 | . 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 586 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝑥 ∈ ℤ ↦ (𝐴↑𝑥)) ∈ (ℤring GrpHom 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 Vcvv 3441 ∖ cdif 3878 ⊆ wss 3881 {csn 4525 ↦ cmpt 5110 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 0cc0 10526 + caddc 10529 · cmul 10531 ℤcz 11969 ↑cexp 13425 Basecbs 16475 ↾s cress 16476 +gcplusg 16557 Grpcgrp 18095 GrpHom cghm 18347 mulGrpcmgp 19232 Ringcrg 19290 Unitcui 19385 ℂfldccnfld 20091 ℤringzring 20163 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12886 df-seq 13365 df-exp 13426 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 df-subg 18268 df-ghm 18348 df-cmn 18900 df-mgp 19233 df-ur 19245 df-ring 19292 df-cring 19293 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 df-dvr 19429 df-drng 19497 df-subrg 19526 df-cnfld 20092 df-zring 20164 |
This theorem is referenced by: lgseisenlem4 25962 |
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