Theorem ghmcyg 19826

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 2729	. . . 4 ⊢ (.g‘𝐺) = (.g‘𝐺)
3	1, 2	iscyg 19809	. . 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 2729	. . . 4 ⊢ (.g‘𝐻) = (.g‘𝐻)
7		ghmgrp2 19151	. . . . 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 6772	. . . . . 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 7057	. . . 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 6769	. . . . . . . . 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 7079	. . . . . . 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 7420	. . . . . . . 8 ⊢ (𝑚(.g‘𝐺)𝑥) ∈ V
20	19	rgenw 3048	. . . . . . 7 ⊢ ∀𝑚 ∈ ℤ (𝑚(.g‘𝐺)𝑥) ∈ V
21		oveq1 7394	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑛(.g‘𝐺)𝑥) = (𝑚(.g‘𝐺)𝑥))
22	21	cbvmptv 5211	. . . . . . . 8 ⊢ (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = (𝑚 ∈ ℤ ↦ (𝑚(.g‘𝐺)𝑥))
23		fveq2 6858	. . . . . . . . 9 ⊢ (𝑧 = (𝑚(.g‘𝐺)𝑥) → (𝐹‘𝑧) = (𝐹‘(𝑚(.g‘𝐺)𝑥)))
24	23	eqeq2d 2740	. . . . . . . 8 ⊢ (𝑧 = (𝑚(.g‘𝐺)𝑥) → (𝑦 = (𝐹‘𝑧) ↔ 𝑦 = (𝐹‘(𝑚(.g‘𝐺)𝑥))))
25	22, 24	rexrnmptw 7067	. . . . . . 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 19160	. . . . . . . 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 2740	. . . . . 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 19817	. . 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 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  Vcvv 3447   ↦ cmpt 5188  ran crn 5639  ⟶wf 6507  –onto→wfo 6509  ‘cfv 6511  (class class class)co 7387  ℤcz 12529  Basecbs 17179  Grpcgrp 18865  ._gcmg 18999   GrpHom cghm 19144  CycGrpccyg 19807

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-er 8671  df-map 8801  df-en 8919  df-dom 8920  df-sdom 8921  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-n0 12443  df-z 12530  df-uz 12794  df-fz 13469  df-seq 13967  df-0g 17404  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-mhm 18710  df-grp 18868  df-minusg 18869  df-mulg 19000  df-ghm 19145  df-cyg 19808

This theorem is referenced by:  giccyg  19830