Theorem ghmcyg 19803

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 2731	. . . 4 ⊢ (.g‘𝐺) = (.g‘𝐺)
3	1, 2	iscyg 19786	. . 3 ⊢ (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑥 ∈ 𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵))
4	3	simprbi 496	. 2 ⊢ (𝐺 ∈ CycGrp → ∃𝑥 ∈ 𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵)
5		ghmcyg.1	. . . 4 ⊢ 𝐶 = (Base‘𝐻)
6		eqid 2731	. . . 4 ⊢ (.g‘𝐻) = (.g‘𝐻)
7		ghmgrp2 19126	. . . . 5 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐻 ∈ Grp)
8	7	ad2antrr 726	. . . 4 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵–onto→𝐶) ∧ (𝑥 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵)) → 𝐻 ∈ Grp)
9		fof 6730	. . . . . 6 ⊢ (𝐹:𝐵–onto→𝐶 → 𝐹:𝐵⟶𝐶)
10	9	ad2antlr 727	. . . . 5 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵–onto→𝐶) ∧ (𝑥 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵)) → 𝐹:𝐵⟶𝐶)
11		simprl 770	. . . . 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 7013	. . . 4 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵–onto→𝐶) ∧ (𝑥 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵)) → (𝐹‘𝑥) ∈ 𝐶)
13		simplr 768	. . . . . . . 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 6727	. . . . . . . . 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 729	. . . . . . . 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 7035	. . . . . . 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 580	. . . . . 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 7374	. . . . . . . 8 ⊢ (𝑚(.g‘𝐺)𝑥) ∈ V
20	19	rgenw 3051	. . . . . . 7 ⊢ ∀𝑚 ∈ ℤ (𝑚(.g‘𝐺)𝑥) ∈ V
21		oveq1 7348	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑛(.g‘𝐺)𝑥) = (𝑚(.g‘𝐺)𝑥))
22	21	cbvmptv 5190	. . . . . . . 8 ⊢ (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = (𝑚 ∈ ℤ ↦ (𝑚(.g‘𝐺)𝑥))
23		fveq2 6817	. . . . . . . . 9 ⊢ (𝑧 = (𝑚(.g‘𝐺)𝑥) → (𝐹‘𝑧) = (𝐹‘(𝑚(.g‘𝐺)𝑥)))
24	23	eqeq2d 2742	. . . . . . . 8 ⊢ (𝑧 = (𝑚(.g‘𝐺)𝑥) → (𝑦 = (𝐹‘𝑧) ↔ 𝑦 = (𝐹‘(𝑚(.g‘𝐺)𝑥))))
25	22, 24	rexrnmptw 7023	. . . . . . 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 218	. . . . 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 782	. . . . . . . 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 484	. . . . . . . 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 726	. . . . . . . 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 19135	. . . . . . . 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 1373	. . . . . . 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 2742	. . . . . 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 3154	. . . . 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 232	. . . 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 19794	. . 3 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵–onto→𝐶) ∧ (𝑥 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵)) → 𝐻 ∈ CycGrp)
37	36	rexlimdvaa 3134	. 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 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047  ∃wrex 3056  Vcvv 3436   ↦ cmpt 5167  ran crn 5612  ⟶wf 6472  –onto→wfo 6474  ‘cfv 6476  (class class class)co 7341  ℤcz 12463  Basecbs 17115  Grpcgrp 18841  ._gcmg 18975   GrpHom cghm 19119  CycGrpccyg 19784

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663  ax-cnex 11057  ax-resscn 11058  ax-1cn 11059  ax-icn 11060  ax-addcl 11061  ax-addrcl 11062  ax-mulcl 11063  ax-mulrcl 11064  ax-mulcom 11065  ax-addass 11066  ax-mulass 11067  ax-distr 11068  ax-i2m1 11069  ax-1ne0 11070  ax-1rid 11071  ax-rnegex 11072  ax-rrecex 11073  ax-cnre 11074  ax-pre-lttri 11075  ax-pre-lttrn 11076  ax-pre-ltadd 11077  ax-pre-mulgt0 11078

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5506  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-we 5566  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-pred 6243  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7792  df-1st 7916  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-er 8617  df-map 8747  df-en 8865  df-dom 8866  df-sdom 8867  df-pnf 11143  df-mnf 11144  df-xr 11145  df-ltxr 11146  df-le 11147  df-sub 11341  df-neg 11342  df-nn 12121  df-n0 12377  df-z 12464  df-uz 12728  df-fz 13403  df-seq 13904  df-0g 17340  df-mgm 18543  df-sgrp 18622  df-mnd 18638  df-mhm 18686  df-grp 18844  df-minusg 18845  df-mulg 18976  df-ghm 19120  df-cyg 19785

This theorem is referenced by:  giccyg  19807