Theorem ghmcyg 19927

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 2761	. . . 4 ⊢ (.g‘𝐺) = (.g‘𝐺)
3	1, 2	iscyg 19910	. . 3 ⊢ (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑥 ∈ 𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵))
4	3	simprbi 501	. 2 ⊢ (𝐺 ∈ CycGrp → ∃𝑥 ∈ 𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵)
5		ghmcyg.1	. . . 4 ⊢ 𝐶 = (Base‘𝐻)
6		eqid 2761	. . . 4 ⊢ (.g‘𝐻) = (.g‘𝐻)
7		ghmgrp2 19250	. . . . 5 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐻 ∈ Grp)
8	7	ad2antrr 736	. . . 4 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵–onto→𝐶) ∧ (𝑥 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵)) → 𝐻 ∈ Grp)
9		fof 6773	. . . . . 6 ⊢ (𝐹:𝐵–onto→𝐶 → 𝐹:𝐵⟶𝐶)
10	9	ad2antlr 737	. . . . 5 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵–onto→𝐶) ∧ (𝑥 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵)) → 𝐹:𝐵⟶𝐶)
11		simprl 780	. . . . 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 7061	. . . 4 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵–onto→𝐶) ∧ (𝑥 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵)) → (𝐹‘𝑥) ∈ 𝐶)
13		simplr 778	. . . . . . . 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 6770	. . . . . . . . 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 739	. . . . . . . 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 259	. . . . . . 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 7083	. . . . . . 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 589	. . . . . 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 7424	. . . . . . . 8 ⊢ (𝑚(.g‘𝐺)𝑥) ∈ V
20	19	rgenw 3079	. . . . . . 7 ⊢ ∀𝑚 ∈ ℤ (𝑚(.g‘𝐺)𝑥) ∈ V
21		oveq1 7398	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑛(.g‘𝐺)𝑥) = (𝑚(.g‘𝐺)𝑥))
22	21	cbvmptv 5201	. . . . . . . 8 ⊢ (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = (𝑚 ∈ ℤ ↦ (𝑚(.g‘𝐺)𝑥))
23		fveq2 6862	. . . . . . . . 9 ⊢ (𝑧 = (𝑚(.g‘𝐺)𝑥) → (𝐹‘𝑧) = (𝐹‘(𝑚(.g‘𝐺)𝑥)))
24	23	eqeq2d 2772	. . . . . . . 8 ⊢ (𝑧 = (𝑚(.g‘𝐺)𝑥) → (𝑦 = (𝐹‘𝑧) ↔ 𝑦 = (𝐹‘(𝑚(.g‘𝐺)𝑥))))
25	22, 24	rexrnmptw 7071	. . . . . . 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 220	. . . . 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 792	. . . . . . . 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 488	. . . . . . . 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 736	. . . . . . . 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 19259	. . . . . . . 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 1389	. . . . . . 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 2772	. . . . . 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 3183	. . . . 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 234	. . . 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 19918	. . 3 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵–onto→𝐶) ∧ (𝑥 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵)) → 𝐻 ∈ CycGrp)
37	36	rexlimdvaa 3163	. 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 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∀wral 3075  ∃wrex 3085  Vcvv 3453   ↦ cmpt 5178  ran crn 5644  ⟶wf 6512  –onto→wfo 6514  ‘cfv 6516  (class class class)co 7391  ℤcz 12562  Basecbs 17236  Grpcgrp 18966  ._gcmg 19100   GrpHom cghm 19244  CycGrpccyg 19908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713  ax-cnex 11123  ax-resscn 11124  ax-1cn 11125  ax-icn 11126  ax-addcl 11127  ax-addrcl 11128  ax-mulcl 11129  ax-mulrcl 11130  ax-mulcom 11131  ax-addass 11132  ax-mulass 11133  ax-distr 11134  ax-i2m1 11135  ax-1ne0 11136  ax-1rid 11137  ax-rnegex 11138  ax-rrecex 11139  ax-cnre 11140  ax-pre-lttri 11141  ax-pre-lttrn 11142  ax-pre-ltadd 11143  ax-pre-mulgt0 11144

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6283  df-ord 6344  df-on 6345  df-lim 6346  df-suc 6347  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-riota 7348  df-ov 7394  df-oprab 7395  df-mpo 7396  df-om 7842  df-1st 7965  df-2nd 7966  df-frecs 8256  df-wrecs 8287  df-recs 8336  df-rdg 8375  df-er 8672  df-map 8804  df-en 8922  df-dom 8923  df-sdom 8924  df-pnf 11212  df-mnf 11213  df-xr 11214  df-ltxr 11215  df-le 11216  df-sub 11410  df-neg 11411  df-nn 12205  df-n0 12476  df-z 12563  df-uz 12834  df-fz 13507  df-seq 14009  df-0g 17461  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-mhm 18808  df-grp 18969  df-minusg 18970  df-mulg 19101  df-ghm 19245  df-cyg 19909

This theorem is referenced by:  giccyg  19931