Theorem ghmcyg 19542

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:𝐵–onto→𝐶) → (𝐺 ∈ CycGrp → 𝐻 ∈ CycGrp))

Proof of Theorem ghmcyg

Dummy variables 𝑚 𝑛 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cygctb.1	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
2		eqid 2736	. . . 4 ⊢ (.g‘𝐺) = (.g‘𝐺)
3	1, 2	iscyg 19524	. . 3 ⊢ (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑥 ∈ 𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵))
4	3	simprbi 498	. 2 ⊢ (𝐺 ∈ CycGrp → ∃𝑥 ∈ 𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵)
5		ghmcyg.1	. . . 4 ⊢ 𝐶 = (Base‘𝐻)
6		eqid 2736	. . . 4 ⊢ (.g‘𝐻) = (.g‘𝐻)
7		ghmgrp2 18882	. . . . 5 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐻 ∈ Grp)
8	7	ad2antrr 724	. . . 4 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵–onto→𝐶) ∧ (𝑥 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵)) → 𝐻 ∈ Grp)
9		fof 6718	. . . . . 6 ⊢ (𝐹:𝐵–onto→𝐶 → 𝐹:𝐵⟶𝐶)
10	9	ad2antlr 725	. . . . 5 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵–onto→𝐶) ∧ (𝑥 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵)) → 𝐹:𝐵⟶𝐶)
11		simprl 769	. . . . 5 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵–onto→𝐶) ∧ (𝑥 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵)) → 𝑥 ∈ 𝐵)
12	10, 11	ffvelcdmd 6994	. . . 4 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵–onto→𝐶) ∧ (𝑥 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵)) → (𝐹‘𝑥) ∈ 𝐶)
13		simplr 767	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵–onto→𝐶) ∧ (𝑥 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵)) → 𝐹:𝐵–onto→𝐶)
14		foeq2 6715	. . . . . . . . 9 ⊢ (ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵 → (𝐹:ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥))–onto→𝐶 ↔ 𝐹:𝐵–onto→𝐶))
15	14	ad2antll 727	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵–onto→𝐶) ∧ (𝑥 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵)) → (𝐹:ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥))–onto→𝐶 ↔ 𝐹:𝐵–onto→𝐶))
16	13, 15	mpbird 257	. . . . . . 7 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵–onto→𝐶) ∧ (𝑥 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵)) → 𝐹:ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥))–onto→𝐶)
17		foelrn 7014	. . . . . . 7 ⊢ ((𝐹:ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥))–onto→𝐶 ∧ 𝑦 ∈ 𝐶) → ∃𝑧 ∈ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥))𝑦 = (𝐹‘𝑧))
18	16, 17	sylan 581	. . . . . 6 ⊢ ((((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵–onto→𝐶) ∧ (𝑥 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵)) ∧ 𝑦 ∈ 𝐶) → ∃𝑧 ∈ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥))𝑦 = (𝐹‘𝑧))
19		ovex 7340	. . . . . . . 8 ⊢ (𝑚(.g‘𝐺)𝑥) ∈ V
20	19	rgenw 3066	. . . . . . 7 ⊢ ∀𝑚 ∈ ℤ (𝑚(.g‘𝐺)𝑥) ∈ V
21		oveq1 7314	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑛(.g‘𝐺)𝑥) = (𝑚(.g‘𝐺)𝑥))
22	21	cbvmptv 5194	. . . . . . . 8 ⊢ (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = (𝑚 ∈ ℤ ↦ (𝑚(.g‘𝐺)𝑥))
23		fveq2 6804	. . . . . . . . 9 ⊢ (𝑧 = (𝑚(.g‘𝐺)𝑥) → (𝐹‘𝑧) = (𝐹‘(𝑚(.g‘𝐺)𝑥)))
24	23	eqeq2d 2747	. . . . . . . 8 ⊢ (𝑧 = (𝑚(.g‘𝐺)𝑥) → (𝑦 = (𝐹‘𝑧) ↔ 𝑦 = (𝐹‘(𝑚(.g‘𝐺)𝑥))))
25	22, 24	rexrnmptw 7003	. . . . . . 7 ⊢ (∀𝑚 ∈ ℤ (𝑚(.g‘𝐺)𝑥) ∈ V → (∃𝑧 ∈ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥))𝑦 = (𝐹‘𝑧) ↔ ∃𝑚 ∈ ℤ 𝑦 = (𝐹‘(𝑚(.g‘𝐺)𝑥))))
26	20, 25	ax-mp 5	. . . . . 6 ⊢ (∃𝑧 ∈ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥))𝑦 = (𝐹‘𝑧) ↔ ∃𝑚 ∈ ℤ 𝑦 = (𝐹‘(𝑚(.g‘𝐺)𝑥)))
27	18, 26	sylib 217	. . . . 5 ⊢ ((((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵–onto→𝐶) ∧ (𝑥 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵)) ∧ 𝑦 ∈ 𝐶) → ∃𝑚 ∈ ℤ 𝑦 = (𝐹‘(𝑚(.g‘𝐺)𝑥)))
28		simp-4l 781	. . . . . . . 8 ⊢ (((((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵–onto→𝐶) ∧ (𝑥 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵)) ∧ 𝑦 ∈ 𝐶) ∧ 𝑚 ∈ ℤ) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
29		simpr 486	. . . . . . . 8 ⊢ (((((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵–onto→𝐶) ∧ (𝑥 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵)) ∧ 𝑦 ∈ 𝐶) ∧ 𝑚 ∈ ℤ) → 𝑚 ∈ ℤ)
30	11	ad2antrr 724	. . . . . . . 8 ⊢ (((((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵–onto→𝐶) ∧ (𝑥 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵)) ∧ 𝑦 ∈ 𝐶) ∧ 𝑚 ∈ ℤ) → 𝑥 ∈ 𝐵)
31	1, 2, 6	ghmmulg 18891	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑚 ∈ ℤ ∧ 𝑥 ∈ 𝐵) → (𝐹‘(𝑚(.g‘𝐺)𝑥)) = (𝑚(.g‘𝐻)(𝐹‘𝑥)))
32	28, 29, 30, 31	syl3anc 1371	. . . . . . 7 ⊢ (((((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵–onto→𝐶) ∧ (𝑥 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵)) ∧ 𝑦 ∈ 𝐶) ∧ 𝑚 ∈ ℤ) → (𝐹‘(𝑚(.g‘𝐺)𝑥)) = (𝑚(.g‘𝐻)(𝐹‘𝑥)))
33	32	eqeq2d 2747	. . . . . 6 ⊢ (((((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵–onto→𝐶) ∧ (𝑥 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵)) ∧ 𝑦 ∈ 𝐶) ∧ 𝑚 ∈ ℤ) → (𝑦 = (𝐹‘(𝑚(.g‘𝐺)𝑥)) ↔ 𝑦 = (𝑚(.g‘𝐻)(𝐹‘𝑥))))
34	33	rexbidva 3170	. . . . 5 ⊢ ((((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵–onto→𝐶) ∧ (𝑥 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵)) ∧ 𝑦 ∈ 𝐶) → (∃𝑚 ∈ ℤ 𝑦 = (𝐹‘(𝑚(.g‘𝐺)𝑥)) ↔ ∃𝑚 ∈ ℤ 𝑦 = (𝑚(.g‘𝐻)(𝐹‘𝑥))))
35	27, 34	mpbid 231	. . . 4 ⊢ ((((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵–onto→𝐶) ∧ (𝑥 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵)) ∧ 𝑦 ∈ 𝐶) → ∃𝑚 ∈ ℤ 𝑦 = (𝑚(.g‘𝐻)(𝐹‘𝑥)))
36	5, 6, 8, 12, 35	iscygd 19532	. . 3 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵–onto→𝐶) ∧ (𝑥 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵)) → 𝐻 ∈ CycGrp)
37	36	rexlimdvaa 3150	. 2 ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵–onto→𝐶) → (∃𝑥 ∈ 𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵 → 𝐻 ∈ CycGrp))
38	4, 37	syl5 34	1 ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵–onto→𝐶) → (𝐺 ∈ CycGrp → 𝐻 ∈ CycGrp))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1539   ∈ wcel 2104  ∀wral 3062  ∃wrex 3071  Vcvv 3437   ↦ cmpt 5164  ran crn 5601  ⟶wf 6454  –onto→wfo 6456  ‘cfv 6458  (class class class)co 7307  ℤcz 12365  Basecbs 16957  Grpcgrp 18622  ._gcmg 18745   GrpHom cghm 18876  CycGrpccyg 19522

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-rep 5218  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361  ax-un 7620  ax-cnex 10973  ax-resscn 10974  ax-1cn 10975  ax-icn 10976  ax-addcl 10977  ax-addrcl 10978  ax-mulcl 10979  ax-mulrcl 10980  ax-mulcom 10981  ax-addass 10982  ax-mulass 10983  ax-distr 10984  ax-i2m1 10985  ax-1ne0 10986  ax-1rid 10987  ax-rnegex 10988  ax-rrecex 10989  ax-cnre 10990  ax-pre-lttri 10991  ax-pre-lttrn 10992  ax-pre-ltadd 10993  ax-pre-mulgt0 10994

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3285  df-reu 3286  df-rab 3287  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-iun 4933  df-br 5082  df-opab 5144  df-mpt 5165  df-tr 5199  df-id 5500  df-eprel 5506  df-po 5514  df-so 5515  df-fr 5555  df-we 5557  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-pred 6217  df-ord 6284  df-on 6285  df-lim 6286  df-suc 6287  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-f1 6463  df-fo 6464  df-f1o 6465  df-fv 6466  df-riota 7264  df-ov 7310  df-oprab 7311  df-mpo 7312  df-om 7745  df-1st 7863  df-2nd 7864  df-frecs 8128  df-wrecs 8159  df-recs 8233  df-rdg 8272  df-er 8529  df-map 8648  df-en 8765  df-dom 8766  df-sdom 8767  df-pnf 11057  df-mnf 11058  df-xr 11059  df-ltxr 11060  df-le 11061  df-sub 11253  df-neg 11254  df-nn 12020  df-n0 12280  df-z 12366  df-uz 12629  df-fz 13286  df-seq 13768  df-0g 17197  df-mgm 18371  df-sgrp 18420  df-mnd 18431  df-mhm 18475  df-grp 18625  df-minusg 18626  df-mulg 18746  df-ghm 18877  df-cyg 19523

This theorem is referenced by:  giccyg  19546