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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  –onto→wfo 6047  ‘cfv 6049  (class class class)co 6813  ℤcz 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
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