Theorem ghmcyg 19816

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 2733	. . . 4 ⊢ (.g‘𝐺) = (.g‘𝐺)
3	1, 2	iscyg 19799	. . 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 2733	. . . 4 ⊢ (.g‘𝐻) = (.g‘𝐻)
7		ghmgrp2 19139	. . . . 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 6743	. . . . . 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 7027	. . . 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 6740	. . . . . . . . 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 7049	. . . . . . 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 7388	. . . . . . . 8 ⊢ (𝑚(.g‘𝐺)𝑥) ∈ V
20	19	rgenw 3052	. . . . . . 7 ⊢ ∀𝑚 ∈ ℤ (𝑚(.g‘𝐺)𝑥) ∈ V
21		oveq1 7362	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑛(.g‘𝐺)𝑥) = (𝑚(.g‘𝐺)𝑥))
22	21	cbvmptv 5199	. . . . . . . 8 ⊢ (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = (𝑚 ∈ ℤ ↦ (𝑚(.g‘𝐺)𝑥))
23		fveq2 6831	. . . . . . . . 9 ⊢ (𝑧 = (𝑚(.g‘𝐺)𝑥) → (𝐹‘𝑧) = (𝐹‘(𝑚(.g‘𝐺)𝑥)))
24	23	eqeq2d 2744	. . . . . . . 8 ⊢ (𝑧 = (𝑚(.g‘𝐺)𝑥) → (𝑦 = (𝐹‘𝑧) ↔ 𝑦 = (𝐹‘(𝑚(.g‘𝐺)𝑥))))
25	22, 24	rexrnmptw 7037	. . . . . . 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 19148	. . . . . . . 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 2744	. . . . . 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 3155	. . . . 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 19807	. . 3 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵–onto→𝐶) ∧ (𝑥 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵)) → 𝐻 ∈ CycGrp)
37	36	rexlimdvaa 3135	. 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 2113  ∀wral 3048  ∃wrex 3057  Vcvv 3437   ↦ cmpt 5176  ran crn 5622  ⟶wf 6485  –onto→wfo 6487  ‘cfv 6489  (class class class)co 7355  ℤcz 12479  Basecbs 17127  Grpcgrp 18854  ._gcmg 18988   GrpHom cghm 19132  CycGrpccyg 19797

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677  ax-cnex 11073  ax-resscn 11074  ax-1cn 11075  ax-icn 11076  ax-addcl 11077  ax-addrcl 11078  ax-mulcl 11079  ax-mulrcl 11080  ax-mulcom 11081  ax-addass 11082  ax-mulass 11083  ax-distr 11084  ax-i2m1 11085  ax-1ne0 11086  ax-1rid 11087  ax-rnegex 11088  ax-rrecex 11089  ax-cnre 11090  ax-pre-lttri 11091  ax-pre-lttrn 11092  ax-pre-ltadd 11093  ax-pre-mulgt0 11094

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7312  df-ov 7358  df-oprab 7359  df-mpo 7360  df-om 7806  df-1st 7930  df-2nd 7931  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-er 8631  df-map 8761  df-en 8880  df-dom 8881  df-sdom 8882  df-pnf 11159  df-mnf 11160  df-xr 11161  df-ltxr 11162  df-le 11163  df-sub 11357  df-neg 11358  df-nn 12137  df-n0 12393  df-z 12480  df-uz 12743  df-fz 13415  df-seq 13916  df-0g 17352  df-mgm 18556  df-sgrp 18635  df-mnd 18651  df-mhm 18699  df-grp 18857  df-minusg 18858  df-mulg 18989  df-ghm 19133  df-cyg 19798

This theorem is referenced by:  giccyg  19820