MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ghmcyg Structured version   Visualization version   GIF version

Theorem ghmcyg 18497
Description: The image of a cyclic group under a surjective group homomorphism is cyclic. (Contributed by Mario Carneiro, 21-Apr-2016.)
Hypotheses
Ref Expression
cygctb.1 𝐵 = (Base‘𝐺)
ghmcyg.1 𝐶 = (Base‘𝐻)
Assertion
Ref Expression
ghmcyg ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵onto𝐶) → (𝐺 ∈ CycGrp → 𝐻 ∈ CycGrp))

Proof of Theorem ghmcyg
Dummy variables 𝑚 𝑛 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cygctb.1 . . . 4 𝐵 = (Base‘𝐺)
2 eqid 2760 . . . 4 (.g𝐺) = (.g𝐺)
31, 2iscyg 18481 . . 3 (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑥𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)𝑥)) = 𝐵))
43simprbi 483 . 2 (𝐺 ∈ CycGrp → ∃𝑥𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)𝑥)) = 𝐵)
5 ghmcyg.1 . . . 4 𝐶 = (Base‘𝐻)
6 eqid 2760 . . . 4 (.g𝐻) = (.g𝐻)
7 ghmgrp2 17864 . . . . 5 (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐻 ∈ Grp)
87ad2antrr 764 . . . 4 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵onto𝐶) ∧ (𝑥𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)𝑥)) = 𝐵)) → 𝐻 ∈ Grp)
9 fof 6276 . . . . . 6 (𝐹:𝐵onto𝐶𝐹:𝐵𝐶)
109ad2antlr 765 . . . . 5 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵onto𝐶) ∧ (𝑥𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)𝑥)) = 𝐵)) → 𝐹:𝐵𝐶)
11 simprl 811 . . . . 5 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵onto𝐶) ∧ (𝑥𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)𝑥)) = 𝐵)) → 𝑥𝐵)
1210, 11ffvelrnd 6523 . . . 4 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵onto𝐶) ∧ (𝑥𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)𝑥)) = 𝐵)) → (𝐹𝑥) ∈ 𝐶)
13 simplr 809 . . . . . . . 8 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵onto𝐶) ∧ (𝑥𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)𝑥)) = 𝐵)) → 𝐹:𝐵onto𝐶)
14 foeq2 6273 . . . . . . . . 9 (ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)𝑥)) = 𝐵 → (𝐹:ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)𝑥))–onto𝐶𝐹:𝐵onto𝐶))
1514ad2antll 767 . . . . . . . 8 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵onto𝐶) ∧ (𝑥𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)𝑥)) = 𝐵)) → (𝐹:ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)𝑥))–onto𝐶𝐹:𝐵onto𝐶))
1613, 15mpbird 247 . . . . . . 7 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵onto𝐶) ∧ (𝑥𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)𝑥)) = 𝐵)) → 𝐹:ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)𝑥))–onto𝐶)
17 foelrn 6541 . . . . . . 7 ((𝐹:ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)𝑥))–onto𝐶𝑦𝐶) → ∃𝑧 ∈ ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)𝑥))𝑦 = (𝐹𝑧))
1816, 17sylan 489 . . . . . 6 ((((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵onto𝐶) ∧ (𝑥𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)𝑥)) = 𝐵)) ∧ 𝑦𝐶) → ∃𝑧 ∈ ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)𝑥))𝑦 = (𝐹𝑧))
19 ovex 6841 . . . . . . . 8 (𝑚(.g𝐺)𝑥) ∈ V
2019rgenw 3062 . . . . . . 7 𝑚 ∈ ℤ (𝑚(.g𝐺)𝑥) ∈ V
21 oveq1 6820 . . . . . . . . 9 (𝑛 = 𝑚 → (𝑛(.g𝐺)𝑥) = (𝑚(.g𝐺)𝑥))
2221cbvmptv 4902 . . . . . . . 8 (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)𝑥)) = (𝑚 ∈ ℤ ↦ (𝑚(.g𝐺)𝑥))
23 fveq2 6352 . . . . . . . . 9 (𝑧 = (𝑚(.g𝐺)𝑥) → (𝐹𝑧) = (𝐹‘(𝑚(.g𝐺)𝑥)))
2423eqeq2d 2770 . . . . . . . 8 (𝑧 = (𝑚(.g𝐺)𝑥) → (𝑦 = (𝐹𝑧) ↔ 𝑦 = (𝐹‘(𝑚(.g𝐺)𝑥))))
2522, 24rexrnmpt 6532 . . . . . . 7 (∀𝑚 ∈ ℤ (𝑚(.g𝐺)𝑥) ∈ V → (∃𝑧 ∈ ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)𝑥))𝑦 = (𝐹𝑧) ↔ ∃𝑚 ∈ ℤ 𝑦 = (𝐹‘(𝑚(.g𝐺)𝑥))))
2620, 25ax-mp 5 . . . . . 6 (∃𝑧 ∈ ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)𝑥))𝑦 = (𝐹𝑧) ↔ ∃𝑚 ∈ ℤ 𝑦 = (𝐹‘(𝑚(.g𝐺)𝑥)))
2718, 26sylib 208 . . . . 5 ((((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵onto𝐶) ∧ (𝑥𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)𝑥)) = 𝐵)) ∧ 𝑦𝐶) → ∃𝑚 ∈ ℤ 𝑦 = (𝐹‘(𝑚(.g𝐺)𝑥)))
28 simp-4l 825 . . . . . . . 8 (((((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵onto𝐶) ∧ (𝑥𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)𝑥)) = 𝐵)) ∧ 𝑦𝐶) ∧ 𝑚 ∈ ℤ) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
29 simpr 479 . . . . . . . 8 (((((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵onto𝐶) ∧ (𝑥𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)𝑥)) = 𝐵)) ∧ 𝑦𝐶) ∧ 𝑚 ∈ ℤ) → 𝑚 ∈ ℤ)
3011ad2antrr 764 . . . . . . . 8 (((((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵onto𝐶) ∧ (𝑥𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)𝑥)) = 𝐵)) ∧ 𝑦𝐶) ∧ 𝑚 ∈ ℤ) → 𝑥𝐵)
311, 2, 6ghmmulg 17873 . . . . . . . 8 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑚 ∈ ℤ ∧ 𝑥𝐵) → (𝐹‘(𝑚(.g𝐺)𝑥)) = (𝑚(.g𝐻)(𝐹𝑥)))
3228, 29, 30, 31syl3anc 1477 . . . . . . 7 (((((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵onto𝐶) ∧ (𝑥𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)𝑥)) = 𝐵)) ∧ 𝑦𝐶) ∧ 𝑚 ∈ ℤ) → (𝐹‘(𝑚(.g𝐺)𝑥)) = (𝑚(.g𝐻)(𝐹𝑥)))
3332eqeq2d 2770 . . . . . 6 (((((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵onto𝐶) ∧ (𝑥𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)𝑥)) = 𝐵)) ∧ 𝑦𝐶) ∧ 𝑚 ∈ ℤ) → (𝑦 = (𝐹‘(𝑚(.g𝐺)𝑥)) ↔ 𝑦 = (𝑚(.g𝐻)(𝐹𝑥))))
3433rexbidva 3187 . . . . 5 ((((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵onto𝐶) ∧ (𝑥𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)𝑥)) = 𝐵)) ∧ 𝑦𝐶) → (∃𝑚 ∈ ℤ 𝑦 = (𝐹‘(𝑚(.g𝐺)𝑥)) ↔ ∃𝑚 ∈ ℤ 𝑦 = (𝑚(.g𝐻)(𝐹𝑥))))
3527, 34mpbid 222 . . . 4 ((((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵onto𝐶) ∧ (𝑥𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)𝑥)) = 𝐵)) ∧ 𝑦𝐶) → ∃𝑚 ∈ ℤ 𝑦 = (𝑚(.g𝐻)(𝐹𝑥)))
365, 6, 8, 12, 35iscygd 18489 . . 3 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵onto𝐶) ∧ (𝑥𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)𝑥)) = 𝐵)) → 𝐻 ∈ CycGrp)
3736rexlimdvaa 3170 . 2 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵onto𝐶) → (∃𝑥𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)𝑥)) = 𝐵𝐻 ∈ CycGrp))
384, 37syl5 34 1 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵onto𝐶) → (𝐺 ∈ CycGrp → 𝐻 ∈ CycGrp))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wral 3050  wrex 3051  Vcvv 3340  cmpt 4881  ran crn 5267  wf 6045  ontowfo 6047  cfv 6049  (class class class)co 6813  cz 11569  Basecbs 16059  Grpcgrp 17623  .gcmg 17741   GrpHom cghm 17858  CycGrpccyg 18479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-inf2 8711  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-er 7911  df-map 8025  df-en 8122  df-dom 8123  df-sdom 8124  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-n0 11485  df-z 11570  df-uz 11880  df-fz 12520  df-seq 12996  df-0g 16304  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-mhm 17536  df-grp 17626  df-minusg 17627  df-mulg 17742  df-ghm 17859  df-cyg 18480
This theorem is referenced by:  giccyg  18501
  Copyright terms: Public domain W3C validator