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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 20624 | . . 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 18247 | . . . . . 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 6880 | . . 3 ⊢ ((𝑅 ∈ Grp ∧ 1 ∈ 𝐵) → 𝐹:ℤ⟶𝐵) |
11 | eqid 2823 | . . . . . . . . 9 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
12 | 4, 5, 11 | mulgdir 18261 | . . . . . . . 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 12025 | . . . . . . 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 7165 | . . . . . . 7 ⊢ (𝑛 = (𝑥 + 𝑦) → (𝑛 · 1 ) = ((𝑥 + 𝑦) · 1 )) | |
19 | ovex 7191 | . . . . . . 7 ⊢ ((𝑥 + 𝑦) · 1 ) ∈ V | |
20 | 18, 9, 19 | fvmpt 6770 | . . . . . 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 7165 | . . . . . . . 8 ⊢ (𝑛 = 𝑥 → (𝑛 · 1 ) = (𝑥 · 1 )) | |
23 | ovex 7191 | . . . . . . . 8 ⊢ (𝑥 · 1 ) ∈ V | |
24 | 22, 9, 23 | fvmpt 6770 | . . . . . . 7 ⊢ (𝑥 ∈ ℤ → (𝐹‘𝑥) = (𝑥 · 1 )) |
25 | oveq1 7165 | . . . . . . . 8 ⊢ (𝑛 = 𝑦 → (𝑛 · 1 ) = (𝑦 · 1 )) | |
26 | ovex 7191 | . . . . . . . 8 ⊢ (𝑦 · 1 ) ∈ V | |
27 | 25, 9, 26 | fvmpt 6770 | . . . . . . 7 ⊢ (𝑦 ∈ ℤ → (𝐹‘𝑦) = (𝑦 · 1 )) |
28 | 24, 27 | oveqan12d 7177 | . . . . . 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 2868 | . . . 4 ⊢ (((𝑅 ∈ Grp ∧ 1 ∈ 𝐵) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥)(+g‘𝑅)(𝐹‘𝑦))) |
31 | 30 | ralrimivva 3193 | . . 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 20625 | . . 3 ⊢ ℤ = (Base‘ℤring) | |
34 | zringplusg 20626 | . . 3 ⊢ + = (+g‘ℤring) | |
35 | 33, 4, 34, 11 | isghm 18360 | . 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 3140 ↦ cmpt 5148 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 + caddc 10542 ℤcz 11984 Basecbs 16485 +gcplusg 16567 Grpcgrp 18105 .gcmg 18226 GrpHom cghm 18357 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-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-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-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-mulg 18227 df-subg 18278 df-ghm 18358 df-cmn 18910 df-mgp 19242 df-ur 19254 df-ring 19301 df-cring 19302 df-subrg 19535 df-cnfld 20548 df-zring 20620 |
This theorem is referenced by: mulgrhm 20647 frgpcyg 20722 |
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