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Theorem ghmcyg 19723
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 2731 . . . 4 (.g𝐺) = (.g𝐺)
31, 2iscyg 19706 . . 3 (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑥𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)𝑥)) = 𝐵))
43simprbi 497 . 2 (𝐺 ∈ CycGrp → ∃𝑥𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)𝑥)) = 𝐵)
5 ghmcyg.1 . . . 4 𝐶 = (Base‘𝐻)
6 eqid 2731 . . . 4 (.g𝐻) = (.g𝐻)
7 ghmgrp2 19061 . . . . 5 (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐻 ∈ Grp)
87ad2antrr 724 . . . 4 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵onto𝐶) ∧ (𝑥𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)𝑥)) = 𝐵)) → 𝐻 ∈ Grp)
9 fof 6792 . . . . . 6 (𝐹:𝐵onto𝐶𝐹:𝐵𝐶)
109ad2antlr 725 . . . . 5 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵onto𝐶) ∧ (𝑥𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)𝑥)) = 𝐵)) → 𝐹:𝐵𝐶)
11 simprl 769 . . . . 5 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵onto𝐶) ∧ (𝑥𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)𝑥)) = 𝐵)) → 𝑥𝐵)
1210, 11ffvelcdmd 7072 . . . 4 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵onto𝐶) ∧ (𝑥𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)𝑥)) = 𝐵)) → (𝐹𝑥) ∈ 𝐶)
13 simplr 767 . . . . . . . 8 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵onto𝐶) ∧ (𝑥𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)𝑥)) = 𝐵)) → 𝐹:𝐵onto𝐶)
14 foeq2 6789 . . . . . . . . 9 (ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)𝑥)) = 𝐵 → (𝐹:ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)𝑥))–onto𝐶𝐹:𝐵onto𝐶))
1514ad2antll 727 . . . . . . . 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 256 . . . . . . 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 7092 . . . . . . 7 ((𝐹:ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)𝑥))–onto𝐶𝑦𝐶) → ∃𝑧 ∈ ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)𝑥))𝑦 = (𝐹𝑧))
1816, 17sylan 580 . . . . . 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 7426 . . . . . . . 8 (𝑚(.g𝐺)𝑥) ∈ V
2019rgenw 3064 . . . . . . 7 𝑚 ∈ ℤ (𝑚(.g𝐺)𝑥) ∈ V
21 oveq1 7400 . . . . . . . . 9 (𝑛 = 𝑚 → (𝑛(.g𝐺)𝑥) = (𝑚(.g𝐺)𝑥))
2221cbvmptv 5254 . . . . . . . 8 (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)𝑥)) = (𝑚 ∈ ℤ ↦ (𝑚(.g𝐺)𝑥))
23 fveq2 6878 . . . . . . . . 9 (𝑧 = (𝑚(.g𝐺)𝑥) → (𝐹𝑧) = (𝐹‘(𝑚(.g𝐺)𝑥)))
2423eqeq2d 2742 . . . . . . . 8 (𝑧 = (𝑚(.g𝐺)𝑥) → (𝑦 = (𝐹𝑧) ↔ 𝑦 = (𝐹‘(𝑚(.g𝐺)𝑥))))
2522, 24rexrnmptw 7081 . . . . . . 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 217 . . . . 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 781 . . . . . . . 8 (((((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵onto𝐶) ∧ (𝑥𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)𝑥)) = 𝐵)) ∧ 𝑦𝐶) ∧ 𝑚 ∈ ℤ) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
29 simpr 485 . . . . . . . 8 (((((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵onto𝐶) ∧ (𝑥𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)𝑥)) = 𝐵)) ∧ 𝑦𝐶) ∧ 𝑚 ∈ ℤ) → 𝑚 ∈ ℤ)
3011ad2antrr 724 . . . . . . . 8 (((((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵onto𝐶) ∧ (𝑥𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)𝑥)) = 𝐵)) ∧ 𝑦𝐶) ∧ 𝑚 ∈ ℤ) → 𝑥𝐵)
311, 2, 6ghmmulg 19070 . . . . . . . 8 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑚 ∈ ℤ ∧ 𝑥𝐵) → (𝐹‘(𝑚(.g𝐺)𝑥)) = (𝑚(.g𝐻)(𝐹𝑥)))
3228, 29, 30, 31syl3anc 1371 . . . . . . 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 2742 . . . . . 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 3175 . . . . 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 231 . . . 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 19714 . . 3 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵onto𝐶) ∧ (𝑥𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)𝑥)) = 𝐵)) → 𝐻 ∈ CycGrp)
3736rexlimdvaa 3155 . 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 205  wa 396   = wceq 1541  wcel 2106  wral 3060  wrex 3069  Vcvv 3473  cmpt 5224  ran crn 5670  wf 6528  ontowfo 6530  cfv 6532  (class class class)co 7393  cz 12540  Basecbs 17126  Grpcgrp 18794  .gcmg 18922   GrpHom cghm 19055  CycGrpccyg 19704
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708  ax-cnex 11148  ax-resscn 11149  ax-1cn 11150  ax-icn 11151  ax-addcl 11152  ax-addrcl 11153  ax-mulcl 11154  ax-mulrcl 11155  ax-mulcom 11156  ax-addass 11157  ax-mulass 11158  ax-distr 11159  ax-i2m1 11160  ax-1ne0 11161  ax-1rid 11162  ax-rnegex 11163  ax-rrecex 11164  ax-cnre 11165  ax-pre-lttri 11166  ax-pre-lttrn 11167  ax-pre-ltadd 11168  ax-pre-mulgt0 11169
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6289  df-ord 6356  df-on 6357  df-lim 6358  df-suc 6359  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-riota 7349  df-ov 7396  df-oprab 7397  df-mpo 7398  df-om 7839  df-1st 7957  df-2nd 7958  df-frecs 8248  df-wrecs 8279  df-recs 8353  df-rdg 8392  df-er 8686  df-map 8805  df-en 8923  df-dom 8924  df-sdom 8925  df-pnf 11232  df-mnf 11233  df-xr 11234  df-ltxr 11235  df-le 11236  df-sub 11428  df-neg 11429  df-nn 12195  df-n0 12455  df-z 12541  df-uz 12805  df-fz 13467  df-seq 13949  df-0g 17369  df-mgm 18543  df-sgrp 18592  df-mnd 18603  df-mhm 18647  df-grp 18797  df-minusg 18798  df-mulg 18923  df-ghm 19056  df-cyg 19705
This theorem is referenced by:  giccyg  19727
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