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Mirrors > Home > MPE Home > Th. List > mulgghm2 | Structured version Visualization version GIF version |
Description: The powers of a group element give a homomorphism from ℤ to a group. (Contributed by Mario Carneiro, 13-Jun-2015.) (Revised by AV, 12-Jun-2019.) |
Ref | Expression |
---|---|
mulgghm2.m | ⊢ · = (.g‘𝑅) |
mulgghm2.f | ⊢ 𝐹 = (𝑛 ∈ ℤ ↦ (𝑛 · 1 )) |
mulgghm2.b | ⊢ 𝐵 = (Base‘𝑅) |
Ref | Expression |
---|---|
mulgghm2 | ⊢ ((𝑅 ∈ Grp ∧ 1 ∈ 𝐵) → 𝐹 ∈ (ℤring GrpHom 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 485 | . . 3 ⊢ ((𝑅 ∈ Grp ∧ 1 ∈ 𝐵) → 𝑅 ∈ Grp) | |
2 | zringgrp 20600 | . . 3 ⊢ ℤring ∈ Grp | |
3 | 1, 2 | jctil 522 | . 2 ⊢ ((𝑅 ∈ Grp ∧ 1 ∈ 𝐵) → (ℤring ∈ Grp ∧ 𝑅 ∈ Grp)) |
4 | mulgghm2.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑅) | |
5 | mulgghm2.m | . . . . . . 7 ⊢ · = (.g‘𝑅) | |
6 | 4, 5 | mulgcl 18223 | . . . . . 6 ⊢ ((𝑅 ∈ Grp ∧ 𝑛 ∈ ℤ ∧ 1 ∈ 𝐵) → (𝑛 · 1 ) ∈ 𝐵) |
7 | 6 | 3expa 1114 | . . . . 5 ⊢ (((𝑅 ∈ Grp ∧ 𝑛 ∈ ℤ) ∧ 1 ∈ 𝐵) → (𝑛 · 1 ) ∈ 𝐵) |
8 | 7 | an32s 650 | . . . 4 ⊢ (((𝑅 ∈ Grp ∧ 1 ∈ 𝐵) ∧ 𝑛 ∈ ℤ) → (𝑛 · 1 ) ∈ 𝐵) |
9 | mulgghm2.f | . . . 4 ⊢ 𝐹 = (𝑛 ∈ ℤ ↦ (𝑛 · 1 )) | |
10 | 8, 9 | fmptd 6859 | . . 3 ⊢ ((𝑅 ∈ Grp ∧ 1 ∈ 𝐵) → 𝐹:ℤ⟶𝐵) |
11 | eqid 2820 | . . . . . . . . 9 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
12 | 4, 5, 11 | mulgdir 18237 | . . . . . . . 8 ⊢ ((𝑅 ∈ Grp ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 1 ∈ 𝐵)) → ((𝑥 + 𝑦) · 1 ) = ((𝑥 · 1 )(+g‘𝑅)(𝑦 · 1 ))) |
13 | 12 | 3exp2 1350 | . . . . . . 7 ⊢ (𝑅 ∈ Grp → (𝑥 ∈ ℤ → (𝑦 ∈ ℤ → ( 1 ∈ 𝐵 → ((𝑥 + 𝑦) · 1 ) = ((𝑥 · 1 )(+g‘𝑅)(𝑦 · 1 )))))) |
14 | 13 | imp42 429 | . . . . . 6 ⊢ (((𝑅 ∈ Grp ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ 1 ∈ 𝐵) → ((𝑥 + 𝑦) · 1 ) = ((𝑥 · 1 )(+g‘𝑅)(𝑦 · 1 ))) |
15 | 14 | an32s 650 | . . . . 5 ⊢ (((𝑅 ∈ Grp ∧ 1 ∈ 𝐵) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → ((𝑥 + 𝑦) · 1 ) = ((𝑥 · 1 )(+g‘𝑅)(𝑦 · 1 ))) |
16 | zaddcl 12004 | . . . . . . 7 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (𝑥 + 𝑦) ∈ ℤ) | |
17 | 16 | adantl 484 | . . . . . 6 ⊢ (((𝑅 ∈ Grp ∧ 1 ∈ 𝐵) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝑥 + 𝑦) ∈ ℤ) |
18 | oveq1 7144 | . . . . . . 7 ⊢ (𝑛 = (𝑥 + 𝑦) → (𝑛 · 1 ) = ((𝑥 + 𝑦) · 1 )) | |
19 | ovex 7170 | . . . . . . 7 ⊢ ((𝑥 + 𝑦) · 1 ) ∈ V | |
20 | 18, 9, 19 | fvmpt 6749 | . . . . . 6 ⊢ ((𝑥 + 𝑦) ∈ ℤ → (𝐹‘(𝑥 + 𝑦)) = ((𝑥 + 𝑦) · 1 )) |
21 | 17, 20 | syl 17 | . . . . 5 ⊢ (((𝑅 ∈ Grp ∧ 1 ∈ 𝐵) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝐹‘(𝑥 + 𝑦)) = ((𝑥 + 𝑦) · 1 )) |
22 | oveq1 7144 | . . . . . . . 8 ⊢ (𝑛 = 𝑥 → (𝑛 · 1 ) = (𝑥 · 1 )) | |
23 | ovex 7170 | . . . . . . . 8 ⊢ (𝑥 · 1 ) ∈ V | |
24 | 22, 9, 23 | fvmpt 6749 | . . . . . . 7 ⊢ (𝑥 ∈ ℤ → (𝐹‘𝑥) = (𝑥 · 1 )) |
25 | oveq1 7144 | . . . . . . . 8 ⊢ (𝑛 = 𝑦 → (𝑛 · 1 ) = (𝑦 · 1 )) | |
26 | ovex 7170 | . . . . . . . 8 ⊢ (𝑦 · 1 ) ∈ V | |
27 | 25, 9, 26 | fvmpt 6749 | . . . . . . 7 ⊢ (𝑦 ∈ ℤ → (𝐹‘𝑦) = (𝑦 · 1 )) |
28 | 24, 27 | oveqan12d 7156 | . . . . . 6 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((𝐹‘𝑥)(+g‘𝑅)(𝐹‘𝑦)) = ((𝑥 · 1 )(+g‘𝑅)(𝑦 · 1 ))) |
29 | 28 | adantl 484 | . . . . 5 ⊢ (((𝑅 ∈ Grp ∧ 1 ∈ 𝐵) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → ((𝐹‘𝑥)(+g‘𝑅)(𝐹‘𝑦)) = ((𝑥 · 1 )(+g‘𝑅)(𝑦 · 1 ))) |
30 | 15, 21, 29 | 3eqtr4d 2865 | . . . 4 ⊢ (((𝑅 ∈ Grp ∧ 1 ∈ 𝐵) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥)(+g‘𝑅)(𝐹‘𝑦))) |
31 | 30 | ralrimivva 3186 | . . 3 ⊢ ((𝑅 ∈ Grp ∧ 1 ∈ 𝐵) → ∀𝑥 ∈ ℤ ∀𝑦 ∈ ℤ (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥)(+g‘𝑅)(𝐹‘𝑦))) |
32 | 10, 31 | jca 514 | . 2 ⊢ ((𝑅 ∈ Grp ∧ 1 ∈ 𝐵) → (𝐹:ℤ⟶𝐵 ∧ ∀𝑥 ∈ ℤ ∀𝑦 ∈ ℤ (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥)(+g‘𝑅)(𝐹‘𝑦)))) |
33 | zringbas 20601 | . . 3 ⊢ ℤ = (Base‘ℤring) | |
34 | zringplusg 20602 | . . 3 ⊢ + = (+g‘ℤring) | |
35 | 33, 4, 34, 11 | isghm 18336 | . 2 ⊢ (𝐹 ∈ (ℤring GrpHom 𝑅) ↔ ((ℤring ∈ Grp ∧ 𝑅 ∈ Grp) ∧ (𝐹:ℤ⟶𝐵 ∧ ∀𝑥 ∈ ℤ ∀𝑦 ∈ ℤ (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥)(+g‘𝑅)(𝐹‘𝑦))))) |
36 | 3, 32, 35 | sylanbrc 585 | 1 ⊢ ((𝑅 ∈ Grp ∧ 1 ∈ 𝐵) → 𝐹 ∈ (ℤring GrpHom 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3133 ↦ cmpt 5127 ⟶wf 6332 ‘cfv 6336 (class class class)co 7137 + caddc 10521 ℤcz 11963 Basecbs 16461 +gcplusg 16543 Grpcgrp 18081 .gcmg 18202 GrpHom cghm 18333 ℤringzring 20595 |
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 2792 ax-rep 5171 ax-sep 5184 ax-nul 5191 ax-pow 5247 ax-pr 5311 ax-un 7442 ax-cnex 10574 ax-resscn 10575 ax-1cn 10576 ax-icn 10577 ax-addcl 10578 ax-addrcl 10579 ax-mulcl 10580 ax-mulrcl 10581 ax-mulcom 10582 ax-addass 10583 ax-mulass 10584 ax-distr 10585 ax-i2m1 10586 ax-1ne0 10587 ax-1rid 10588 ax-rnegex 10589 ax-rrecex 10590 ax-cnre 10591 ax-pre-lttri 10592 ax-pre-lttrn 10593 ax-pre-ltadd 10594 ax-pre-mulgt0 10595 ax-addf 10597 ax-mulf 10598 |
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 2653 df-clab 2799 df-cleq 2813 df-clel 2891 df-nfc 2959 df-ne 3012 df-nel 3119 df-ral 3138 df-rex 3139 df-reu 3140 df-rmo 3141 df-rab 3142 df-v 3483 df-sbc 3759 df-csb 3867 df-dif 3922 df-un 3924 df-in 3926 df-ss 3935 df-pss 3937 df-nul 4275 df-if 4449 df-pw 4522 df-sn 4549 df-pr 4551 df-tp 4553 df-op 4555 df-uni 4820 df-int 4858 df-iun 4902 df-br 5048 df-opab 5110 df-mpt 5128 df-tr 5154 df-id 5441 df-eprel 5446 df-po 5455 df-so 5456 df-fr 5495 df-we 5497 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6295 df-fun 6338 df-fn 6339 df-f 6340 df-f1 6341 df-fo 6342 df-f1o 6343 df-fv 6344 df-riota 7095 df-ov 7140 df-oprab 7141 df-mpo 7142 df-om 7562 df-1st 7670 df-2nd 7671 df-wrecs 7928 df-recs 7989 df-rdg 8027 df-1o 8083 df-oadd 8087 df-er 8270 df-en 8491 df-dom 8492 df-sdom 8493 df-fin 8494 df-pnf 10658 df-mnf 10659 df-xr 10660 df-ltxr 10661 df-le 10662 df-sub 10853 df-neg 10854 df-nn 11620 df-2 11682 df-3 11683 df-4 11684 df-5 11685 df-6 11686 df-7 11687 df-8 11688 df-9 11689 df-n0 11880 df-z 11964 df-dec 12081 df-uz 12226 df-fz 12878 df-seq 13355 df-struct 16463 df-ndx 16464 df-slot 16465 df-base 16467 df-sets 16468 df-ress 16469 df-plusg 16556 df-mulr 16557 df-starv 16558 df-tset 16562 df-ple 16563 df-ds 16565 df-unif 16566 df-0g 16693 df-mgm 17830 df-sgrp 17879 df-mnd 17890 df-grp 18084 df-minusg 18085 df-mulg 18203 df-subg 18254 df-ghm 18334 df-cmn 18886 df-mgp 19218 df-ur 19230 df-ring 19277 df-cring 19278 df-subrg 19511 df-cnfld 20524 df-zring 20596 |
This theorem is referenced by: mulgrhm 20623 frgpcyg 20698 |
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