Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 13456 | . . . 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 6881 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝑥 ∈ ℤ ↦ (𝐴↑𝑥)):ℤ⟶(ℂ ∖ {0})) |
4 | expaddz 13476 | . . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → (𝐴↑(𝑦 + 𝑧)) = ((𝐴↑𝑦) · (𝐴↑𝑧))) | |
5 | zaddcl 12025 | . . . . . 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 7166 | . . . . . 6 ⊢ (𝑥 = (𝑦 + 𝑧) → (𝐴↑𝑥) = (𝐴↑(𝑦 + 𝑧))) | |
8 | eqid 2823 | . . . . . 6 ⊢ (𝑥 ∈ ℤ ↦ (𝐴↑𝑥)) = (𝑥 ∈ ℤ ↦ (𝐴↑𝑥)) | |
9 | ovex 7191 | . . . . . 6 ⊢ (𝐴↑(𝑦 + 𝑧)) ∈ V | |
10 | 7, 8, 9 | fvmpt 6770 | . . . . 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 7166 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴↑𝑥) = (𝐴↑𝑦)) | |
13 | ovex 7191 | . . . . . . 7 ⊢ (𝐴↑𝑦) ∈ V | |
14 | 12, 8, 13 | fvmpt 6770 | . . . . . 6 ⊢ (𝑦 ∈ ℤ → ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑦) = (𝐴↑𝑦)) |
15 | oveq2 7166 | . . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝐴↑𝑥) = (𝐴↑𝑧)) | |
16 | ovex 7191 | . . . . . . 7 ⊢ (𝐴↑𝑧) ∈ V | |
17 | 15, 8, 16 | fvmpt 6770 | . . . . . 6 ⊢ (𝑧 ∈ ℤ → ((𝑥 ∈ ℤ ↦ (𝐴↑𝑥))‘𝑧) = (𝐴↑𝑧)) |
18 | 14, 17 | oveqan12d 7177 | . . . . 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 2868 | . . 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 3193 | . 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 20624 | . . . 4 ⊢ ℤring ∈ Grp | |
23 | cnring 20569 | . . . . 5 ⊢ ℂfld ∈ Ring | |
24 | cnfldbas 20551 | . . . . . . 7 ⊢ ℂ = (Base‘ℂfld) | |
25 | cnfld0 20571 | . . . . . . 7 ⊢ 0 = (0g‘ℂfld) | |
26 | cndrng 20576 | . . . . . . 7 ⊢ ℂfld ∈ DivRing | |
27 | 24, 25, 26 | drngui 19510 | . . . . . 6 ⊢ (ℂ ∖ {0}) = (Unit‘ℂfld) |
28 | expghm.u | . . . . . . 7 ⊢ 𝑈 = (𝑀 ↾s (ℂ ∖ {0})) | |
29 | expghm.m | . . . . . . . 8 ⊢ 𝑀 = (mulGrp‘ℂfld) | |
30 | 29 | oveq1i 7168 | . . . . . . 7 ⊢ (𝑀 ↾s (ℂ ∖ {0})) = ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) |
31 | 28, 30 | eqtri 2846 | . . . . . 6 ⊢ 𝑈 = ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) |
32 | 27, 31 | unitgrp 19419 | . . . . 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 20625 | . . . 4 ⊢ ℤ = (Base‘ℤring) | |
36 | difss 4110 | . . . . 5 ⊢ (ℂ ∖ {0}) ⊆ ℂ | |
37 | 29, 24 | mgpbas 19247 | . . . . . 6 ⊢ ℂ = (Base‘𝑀) |
38 | 28, 37 | ressbas2 16557 | . . . . 5 ⊢ ((ℂ ∖ {0}) ⊆ ℂ → (ℂ ∖ {0}) = (Base‘𝑈)) |
39 | 36, 38 | ax-mp 5 | . . . 4 ⊢ (ℂ ∖ {0}) = (Base‘𝑈) |
40 | zringplusg 20626 | . . . 4 ⊢ + = (+g‘ℤring) | |
41 | 27 | fvexi 6686 | . . . . 5 ⊢ (ℂ ∖ {0}) ∈ V |
42 | cnfldmul 20553 | . . . . . . 7 ⊢ · = (.r‘ℂfld) | |
43 | 29, 42 | mgpplusg 19245 | . . . . . 6 ⊢ · = (+g‘𝑀) |
44 | 28, 43 | ressplusg 16614 | . . . . 5 ⊢ ((ℂ ∖ {0}) ∈ V → · = (+g‘𝑈)) |
45 | 41, 44 | ax-mp 5 | . . . 4 ⊢ · = (+g‘𝑈) |
46 | 35, 39, 40, 45 | isghm 18360 | . . 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 3018 ∀wral 3140 Vcvv 3496 ∖ cdif 3935 ⊆ wss 3938 {csn 4569 ↦ cmpt 5148 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 0cc0 10539 + caddc 10542 · cmul 10544 ℤcz 11984 ↑cexp 13432 Basecbs 16485 ↾s cress 16486 +gcplusg 16567 Grpcgrp 18105 GrpHom cghm 18357 mulGrpcmgp 19241 Ringcrg 19299 Unitcui 19391 ℂfldccnfld 20547 ℤringzring 20619 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-addf 10618 ax-mulf 10619 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-tpos 7894 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-seq 13373 df-exp 13433 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-0g 16717 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-minusg 18109 df-subg 18278 df-ghm 18358 df-cmn 18910 df-mgp 19242 df-ur 19254 df-ring 19301 df-cring 19302 df-oppr 19375 df-dvdsr 19393 df-unit 19394 df-invr 19424 df-dvr 19435 df-drng 19506 df-subrg 19535 df-cnfld 20548 df-zring 20620 |
This theorem is referenced by: lgseisenlem4 25956 |
Copyright terms: Public domain | W3C validator |