Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > iscyggen2 | Structured version Visualization version GIF version |
Description: The property of being a cyclic generator for a group. (Contributed by Mario Carneiro, 21-Apr-2016.) |
Ref | Expression |
---|---|
iscyg.1 | ⊢ 𝐵 = (Base‘𝐺) |
iscyg.2 | ⊢ · = (.g‘𝐺) |
iscyg3.e | ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵} |
Ref | Expression |
---|---|
iscyggen2 | ⊢ (𝐺 ∈ Grp → (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑛 ∈ ℤ 𝑦 = (𝑛 · 𝑋)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iscyg.1 | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
2 | iscyg.2 | . . 3 ⊢ · = (.g‘𝐺) | |
3 | iscyg3.e | . . 3 ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵} | |
4 | 1, 2, 3 | iscyggen 18999 | . 2 ⊢ (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵)) |
5 | 1, 2 | mulgcl 18245 | . . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑛 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑛 · 𝑋) ∈ 𝐵) |
6 | 5 | 3expa 1114 | . . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝑛 ∈ ℤ) ∧ 𝑋 ∈ 𝐵) → (𝑛 · 𝑋) ∈ 𝐵) |
7 | 6 | an32s 650 | . . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑛 ∈ ℤ) → (𝑛 · 𝑋) ∈ 𝐵) |
8 | 7 | fmpttd 6879 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)):ℤ⟶𝐵) |
9 | frn 6520 | . . . . 5 ⊢ ((𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)):ℤ⟶𝐵 → ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ⊆ 𝐵) | |
10 | eqss 3982 | . . . . . 6 ⊢ (ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵 ↔ (ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ⊆ 𝐵 ∧ 𝐵 ⊆ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)))) | |
11 | 10 | baib 538 | . . . . 5 ⊢ (ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ⊆ 𝐵 → (ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵 ↔ 𝐵 ⊆ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)))) |
12 | 8, 9, 11 | 3syl 18 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵 ↔ 𝐵 ⊆ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)))) |
13 | dfss3 3956 | . . . . 5 ⊢ (𝐵 ⊆ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ↔ ∀𝑦 ∈ 𝐵 𝑦 ∈ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋))) | |
14 | eqid 2821 | . . . . . . 7 ⊢ (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) | |
15 | ovex 7189 | . . . . . . 7 ⊢ (𝑛 · 𝑋) ∈ V | |
16 | 14, 15 | elrnmpti 5832 | . . . . . 6 ⊢ (𝑦 ∈ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ↔ ∃𝑛 ∈ ℤ 𝑦 = (𝑛 · 𝑋)) |
17 | 16 | ralbii 3165 | . . . . 5 ⊢ (∀𝑦 ∈ 𝐵 𝑦 ∈ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ↔ ∀𝑦 ∈ 𝐵 ∃𝑛 ∈ ℤ 𝑦 = (𝑛 · 𝑋)) |
18 | 13, 17 | bitri 277 | . . . 4 ⊢ (𝐵 ⊆ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ↔ ∀𝑦 ∈ 𝐵 ∃𝑛 ∈ ℤ 𝑦 = (𝑛 · 𝑋)) |
19 | 12, 18 | syl6bb 289 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵 ↔ ∀𝑦 ∈ 𝐵 ∃𝑛 ∈ ℤ 𝑦 = (𝑛 · 𝑋))) |
20 | 19 | pm5.32da 581 | . 2 ⊢ (𝐺 ∈ Grp → ((𝑋 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵) ↔ (𝑋 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑛 ∈ ℤ 𝑦 = (𝑛 · 𝑋)))) |
21 | 4, 20 | syl5bb 285 | 1 ⊢ (𝐺 ∈ Grp → (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑛 ∈ ℤ 𝑦 = (𝑛 · 𝑋)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ∃wrex 3139 {crab 3142 ⊆ wss 3936 ↦ cmpt 5146 ran crn 5556 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ℤcz 11982 Basecbs 16483 Grpcgrp 18103 .gcmg 18224 |
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-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 |
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-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-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-seq 13371 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 df-mulg 18225 |
This theorem is referenced by: cyggeninv 19002 iscygd 19006 cygznlem3 20716 |
Copyright terms: Public domain | W3C validator |