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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 13454 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑥 ∈ ℤ) → (𝐴↑𝑥) ∈ (ℂ ∖ {0})) | |
2 | 1 | 3expa 1114 | . . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝑥 ∈ ℤ) → (𝐴↑𝑥) ∈ (ℂ ∖ {0})) |
3 | 2 | fmpttd 6879 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝑥 ∈ ℤ ↦ (𝐴↑𝑥)):ℤ⟶(ℂ ∖ {0})) |
4 | expaddz 13474 | . . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → (𝐴↑(𝑦 + 𝑧)) = ((𝐴↑𝑦) · (𝐴↑𝑧))) | |
5 | zaddcl 12023 | . . . . . 6 ⊢ ((𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ) → (𝑦 + 𝑧) ∈ ℤ) | |
6 | 5 | adantl 484 | . . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → (𝑦 + 𝑧) ∈ ℤ) |
7 | oveq2 7164 | . . . . . 6 ⊢ (𝑥 = (𝑦 + 𝑧) → (𝐴↑𝑥) = (𝐴↑(𝑦 + 𝑧))) | |
8 | eqid 2821 | . . . . . 6 ⊢ (𝑥 ∈ ℤ ↦ (𝐴↑𝑥)) = (𝑥 ∈ ℤ ↦ (𝐴↑𝑥)) | |
9 | ovex 7189 | . . . . . 6 ⊢ (𝐴↑(𝑦 + 𝑧)) ∈ V | |
10 | 7, 8, 9 | fvmpt 6768 | . . . . 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 7164 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴↑𝑥) = (𝐴↑𝑦)) | |
13 | ovex 7189 | . . . . . . 7 ⊢ (𝐴↑𝑦) ∈ V | |
14 | 12, 8, 13 | fvmpt 6768 | . . . . . 6 ⊢ (𝑦 ∈ ℤ → ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑦) = (𝐴↑𝑦)) |
15 | oveq2 7164 | . . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝐴↑𝑥) = (𝐴↑𝑧)) | |
16 | ovex 7189 | . . . . . . 7 ⊢ (𝐴↑𝑧) ∈ V | |
17 | 15, 8, 16 | fvmpt 6768 | . . . . . 6 ⊢ (𝑧 ∈ ℤ → ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑧) = (𝐴↑𝑧)) |
18 | 14, 17 | oveqan12d 7175 | . . . . 5 ⊢ ((𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ) → (((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑦) · ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑧)) = ((𝐴↑𝑦) · (𝐴↑𝑧))) |
19 | 18 | adantl 484 | . . . 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 2866 | . . 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 3191 | . 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 20622 | . . . 4 ⊢ ℤring ∈ Grp | |
23 | cnring 20567 | . . . . 5 ⊢ ℂfld ∈ Ring | |
24 | cnfldbas 20549 | . . . . . . 7 ⊢ ℂ = (Base‘ℂfld) | |
25 | cnfld0 20569 | . . . . . . 7 ⊢ 0 = (0g‘ℂfld) | |
26 | cndrng 20574 | . . . . . . 7 ⊢ ℂfld ∈ DivRing | |
27 | 24, 25, 26 | drngui 19508 | . . . . . 6 ⊢ (ℂ ∖ {0}) = (Unit‘ℂfld) |
28 | expghm.u | . . . . . . 7 ⊢ 𝑈 = (𝑀 ↾s (ℂ ∖ {0})) | |
29 | expghm.m | . . . . . . . 8 ⊢ 𝑀 = (mulGrp‘ℂfld) | |
30 | 29 | oveq1i 7166 | . . . . . . 7 ⊢ (𝑀 ↾s (ℂ ∖ {0})) = ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) |
31 | 28, 30 | eqtri 2844 | . . . . . 6 ⊢ 𝑈 = ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) |
32 | 27, 31 | unitgrp 19417 | . . . . 5 ⊢ (ℂfld ∈ Ring → 𝑈 ∈ Grp) |
33 | 23, 32 | ax-mp 5 | . . . 4 ⊢ 𝑈 ∈ Grp |
34 | 22, 33 | pm3.2i 473 | . . 3 ⊢ (ℤring ∈ Grp ∧ 𝑈 ∈ Grp) |
35 | zringbas 20623 | . . . 4 ⊢ ℤ = (Base‘ℤring) | |
36 | difss 4108 | . . . . 5 ⊢ (ℂ ∖ {0}) ⊆ ℂ | |
37 | 29, 24 | mgpbas 19245 | . . . . . 6 ⊢ ℂ = (Base‘𝑀) |
38 | 28, 37 | ressbas2 16555 | . . . . 5 ⊢ ((ℂ ∖ {0}) ⊆ ℂ → (ℂ ∖ {0}) = (Base‘𝑈)) |
39 | 36, 38 | ax-mp 5 | . . . 4 ⊢ (ℂ ∖ {0}) = (Base‘𝑈) |
40 | zringplusg 20624 | . . . 4 ⊢ + = (+g‘ℤring) | |
41 | 27 | fvexi 6684 | . . . . 5 ⊢ (ℂ ∖ {0}) ∈ V |
42 | cnfldmul 20551 | . . . . . . 7 ⊢ · = (.r‘ℂfld) | |
43 | 29, 42 | mgpplusg 19243 | . . . . . 6 ⊢ · = (+g‘𝑀) |
44 | 28, 43 | ressplusg 16612 | . . . . 5 ⊢ ((ℂ ∖ {0}) ∈ V → · = (+g‘𝑈)) |
45 | 41, 44 | ax-mp 5 | . . . 4 ⊢ · = (+g‘𝑈) |
46 | 35, 39, 40, 45 | isghm 18358 | . . 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 585 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝑥 ∈ ℤ ↦ (𝐴↑𝑥)) ∈ (ℤring GrpHom 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∀wral 3138 Vcvv 3494 ∖ cdif 3933 ⊆ wss 3936 {csn 4567 ↦ cmpt 5146 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 0cc0 10537 + caddc 10540 · cmul 10542 ℤcz 11982 ↑cexp 13430 Basecbs 16483 ↾s cress 16484 +gcplusg 16565 Grpcgrp 18103 GrpHom cghm 18355 mulGrpcmgp 19239 Ringcrg 19297 Unitcui 19389 ℂfldccnfld 20545 ℤringzring 20617 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-addf 10616 ax-mulf 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-tpos 7892 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-fz 12894 df-seq 13371 df-exp 13431 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 df-subg 18276 df-ghm 18356 df-cmn 18908 df-mgp 19240 df-ur 19252 df-ring 19299 df-cring 19300 df-oppr 19373 df-dvdsr 19391 df-unit 19392 df-invr 19422 df-dvr 19433 df-drng 19504 df-subrg 19533 df-cnfld 20546 df-zring 20618 |
This theorem is referenced by: lgseisenlem4 25954 |
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