Theorem ghmrn 19309

Description: The range of a homomorphism is a subgroup. (Contributed by Stefan O'Rear, 31-Dec-2014.)

Assertion

Ref	Expression
ghmrn	⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → ran 𝐹 ∈ (SubGrp‘𝑇))

Proof of Theorem ghmrn

Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2766	. . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆)
2		eqid 2766	. . . 4 ⊢ (Base‘𝑇) = (Base‘𝑇)
3	1, 2	ghmf 19300	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
4	3	frnd 6718	. 2 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → ran 𝐹 ⊆ (Base‘𝑇))
5	3	fdmd 6720	. . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → dom 𝐹 = (Base‘𝑆))
6		ghmgrp1 19298	. . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
7	1	grpbn0 19043	. . . . 5 ⊢ (𝑆 ∈ Grp → (Base‘𝑆) ≠ ∅)
8	6, 7	syl 18	. . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (Base‘𝑆) ≠ ∅)
9	5, 8	eqnetrd 3028	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → dom 𝐹 ≠ ∅)
10		dm0rn0 5917	. . . 4 ⊢ (dom 𝐹 = ∅ ↔ ran 𝐹 = ∅)
11	10	necon3bii 3013	. . 3 ⊢ (dom 𝐹 ≠ ∅ ↔ ran 𝐹 ≠ ∅)
12	9, 11	sylib 221	. 2 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → ran 𝐹 ≠ ∅)
13		eqid 2766	. . . . . . . . . 10 ⊢ (+g‘𝑆) = (+g‘𝑆)
14		eqid 2766	. . . . . . . . . 10 ⊢ (+g‘𝑇) = (+g‘𝑇)
15	1, 13, 14	ghmlin 19301	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆) ∧ 𝑎 ∈ (Base‘𝑆)) → (𝐹‘(𝑐(+g‘𝑆)𝑎)) = ((𝐹‘𝑐)(+g‘𝑇)(𝐹‘𝑎)))
16	3	ffnd 6710	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹 Fn (Base‘𝑆))
17	16	3ad2ant1 1151	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆) ∧ 𝑎 ∈ (Base‘𝑆)) → 𝐹 Fn (Base‘𝑆))
18	1, 13	grpcl 19018	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Grp ∧ 𝑐 ∈ (Base‘𝑆) ∧ 𝑎 ∈ (Base‘𝑆)) → (𝑐(+g‘𝑆)𝑎) ∈ (Base‘𝑆))
19	6, 18	syl3an1 1181	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆) ∧ 𝑎 ∈ (Base‘𝑆)) → (𝑐(+g‘𝑆)𝑎) ∈ (Base‘𝑆))
20		fnfvelrn 7079	. . . . . . . . . 10 ⊢ ((𝐹 Fn (Base‘𝑆) ∧ (𝑐(+g‘𝑆)𝑎) ∈ (Base‘𝑆)) → (𝐹‘(𝑐(+g‘𝑆)𝑎)) ∈ ran 𝐹)
21	17, 19, 20	syl2anc 596	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆) ∧ 𝑎 ∈ (Base‘𝑆)) → (𝐹‘(𝑐(+g‘𝑆)𝑎)) ∈ ran 𝐹)
22	15, 21	eqeltrrd 2867	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆) ∧ 𝑎 ∈ (Base‘𝑆)) → ((𝐹‘𝑐)(+g‘𝑇)(𝐹‘𝑎)) ∈ ran 𝐹)
23	22	3expia 1139	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → (𝑎 ∈ (Base‘𝑆) → ((𝐹‘𝑐)(+g‘𝑇)(𝐹‘𝑎)) ∈ ran 𝐹))
24	23	ralrimiv 3159	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → ∀𝑎 ∈ (Base‘𝑆)((𝐹‘𝑐)(+g‘𝑇)(𝐹‘𝑎)) ∈ ran 𝐹)
25		oveq2 7424	. . . . . . . . . 10 ⊢ (𝑏 = (𝐹‘𝑎) → ((𝐹‘𝑐)(+g‘𝑇)𝑏) = ((𝐹‘𝑐)(+g‘𝑇)(𝐹‘𝑎)))
26	25	eleq1d 2851	. . . . . . . . 9 ⊢ (𝑏 = (𝐹‘𝑎) → (((𝐹‘𝑐)(+g‘𝑇)𝑏) ∈ ran 𝐹 ↔ ((𝐹‘𝑐)(+g‘𝑇)(𝐹‘𝑎)) ∈ ran 𝐹))
27	26	ralrn 7087	. . . . . . . 8 ⊢ (𝐹 Fn (Base‘𝑆) → (∀𝑏 ∈ ran 𝐹((𝐹‘𝑐)(+g‘𝑇)𝑏) ∈ ran 𝐹 ↔ ∀𝑎 ∈ (Base‘𝑆)((𝐹‘𝑐)(+g‘𝑇)(𝐹‘𝑎)) ∈ ran 𝐹))
28	16, 27	syl 18	. . . . . . 7 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (∀𝑏 ∈ ran 𝐹((𝐹‘𝑐)(+g‘𝑇)𝑏) ∈ ran 𝐹 ↔ ∀𝑎 ∈ (Base‘𝑆)((𝐹‘𝑐)(+g‘𝑇)(𝐹‘𝑎)) ∈ ran 𝐹))
29	28	adantr 486	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → (∀𝑏 ∈ ran 𝐹((𝐹‘𝑐)(+g‘𝑇)𝑏) ∈ ran 𝐹 ↔ ∀𝑎 ∈ (Base‘𝑆)((𝐹‘𝑐)(+g‘𝑇)(𝐹‘𝑎)) ∈ ran 𝐹))
30	24, 29	mpbird 260	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → ∀𝑏 ∈ ran 𝐹((𝐹‘𝑐)(+g‘𝑇)𝑏) ∈ ran 𝐹)
31		eqid 2766	. . . . . . 7 ⊢ (invg‘𝑆) = (invg‘𝑆)
32		eqid 2766	. . . . . . 7 ⊢ (invg‘𝑇) = (invg‘𝑇)
33	1, 31, 32	ghminv 19303	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → (𝐹‘((invg‘𝑆)‘𝑐)) = ((invg‘𝑇)‘(𝐹‘𝑐)))
34	16	adantr 486	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → 𝐹 Fn (Base‘𝑆))
35	1, 31	grpinvcl 19064	. . . . . . . 8 ⊢ ((𝑆 ∈ Grp ∧ 𝑐 ∈ (Base‘𝑆)) → ((invg‘𝑆)‘𝑐) ∈ (Base‘𝑆))
36	6, 35	sylan 592	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → ((invg‘𝑆)‘𝑐) ∈ (Base‘𝑆))
37		fnfvelrn 7079	. . . . . . 7 ⊢ ((𝐹 Fn (Base‘𝑆) ∧ ((invg‘𝑆)‘𝑐) ∈ (Base‘𝑆)) → (𝐹‘((invg‘𝑆)‘𝑐)) ∈ ran 𝐹)
38	34, 36, 37	syl2anc 596	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → (𝐹‘((invg‘𝑆)‘𝑐)) ∈ ran 𝐹)
39	33, 38	eqeltrrd 2867	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → ((invg‘𝑇)‘(𝐹‘𝑐)) ∈ ran 𝐹)
40	30, 39	jca 521	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → (∀𝑏 ∈ ran 𝐹((𝐹‘𝑐)(+g‘𝑇)𝑏) ∈ ran 𝐹 ∧ ((invg‘𝑇)‘(𝐹‘𝑐)) ∈ ran 𝐹))
41	40	ralrimiva 3160	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → ∀𝑐 ∈ (Base‘𝑆)(∀𝑏 ∈ ran 𝐹((𝐹‘𝑐)(+g‘𝑇)𝑏) ∈ ran 𝐹 ∧ ((invg‘𝑇)‘(𝐹‘𝑐)) ∈ ran 𝐹))
42		oveq1 7423	. . . . . . . 8 ⊢ (𝑎 = (𝐹‘𝑐) → (𝑎(+g‘𝑇)𝑏) = ((𝐹‘𝑐)(+g‘𝑇)𝑏))
43	42	eleq1d 2851	. . . . . . 7 ⊢ (𝑎 = (𝐹‘𝑐) → ((𝑎(+g‘𝑇)𝑏) ∈ ran 𝐹 ↔ ((𝐹‘𝑐)(+g‘𝑇)𝑏) ∈ ran 𝐹))
44	43	ralbidv 3191	. . . . . 6 ⊢ (𝑎 = (𝐹‘𝑐) → (∀𝑏 ∈ ran 𝐹(𝑎(+g‘𝑇)𝑏) ∈ ran 𝐹 ↔ ∀𝑏 ∈ ran 𝐹((𝐹‘𝑐)(+g‘𝑇)𝑏) ∈ ran 𝐹))
45		fveq2 6885	. . . . . . 7 ⊢ (𝑎 = (𝐹‘𝑐) → ((invg‘𝑇)‘𝑎) = ((invg‘𝑇)‘(𝐹‘𝑐)))
46	45	eleq1d 2851	. . . . . 6 ⊢ (𝑎 = (𝐹‘𝑐) → (((invg‘𝑇)‘𝑎) ∈ ran 𝐹 ↔ ((invg‘𝑇)‘(𝐹‘𝑐)) ∈ ran 𝐹))
47	44, 46	anbi12d 644	. . . . 5 ⊢ (𝑎 = (𝐹‘𝑐) → ((∀𝑏 ∈ ran 𝐹(𝑎(+g‘𝑇)𝑏) ∈ ran 𝐹 ∧ ((invg‘𝑇)‘𝑎) ∈ ran 𝐹) ↔ (∀𝑏 ∈ ran 𝐹((𝐹‘𝑐)(+g‘𝑇)𝑏) ∈ ran 𝐹 ∧ ((invg‘𝑇)‘(𝐹‘𝑐)) ∈ ran 𝐹)))
48	47	ralrn 7087	. . . 4 ⊢ (𝐹 Fn (Base‘𝑆) → (∀𝑎 ∈ ran 𝐹(∀𝑏 ∈ ran 𝐹(𝑎(+g‘𝑇)𝑏) ∈ ran 𝐹 ∧ ((invg‘𝑇)‘𝑎) ∈ ran 𝐹) ↔ ∀𝑐 ∈ (Base‘𝑆)(∀𝑏 ∈ ran 𝐹((𝐹‘𝑐)(+g‘𝑇)𝑏) ∈ ran 𝐹 ∧ ((invg‘𝑇)‘(𝐹‘𝑐)) ∈ ran 𝐹)))
49	16, 48	syl 18	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (∀𝑎 ∈ ran 𝐹(∀𝑏 ∈ ran 𝐹(𝑎(+g‘𝑇)𝑏) ∈ ran 𝐹 ∧ ((invg‘𝑇)‘𝑎) ∈ ran 𝐹) ↔ ∀𝑐 ∈ (Base‘𝑆)(∀𝑏 ∈ ran 𝐹((𝐹‘𝑐)(+g‘𝑇)𝑏) ∈ ran 𝐹 ∧ ((invg‘𝑇)‘(𝐹‘𝑐)) ∈ ran 𝐹)))
50	41, 49	mpbird 260	. 2 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → ∀𝑎 ∈ ran 𝐹(∀𝑏 ∈ ran 𝐹(𝑎(+g‘𝑇)𝑏) ∈ ran 𝐹 ∧ ((invg‘𝑇)‘𝑎) ∈ ran 𝐹))
51		ghmgrp2 19299	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
52	2, 14, 32	issubg2 19218	. . 3 ⊢ (𝑇 ∈ Grp → (ran 𝐹 ∈ (SubGrp‘𝑇) ↔ (ran 𝐹 ⊆ (Base‘𝑇) ∧ ran 𝐹 ≠ ∅ ∧ ∀𝑎 ∈ ran 𝐹(∀𝑏 ∈ ran 𝐹(𝑎(+g‘𝑇)𝑏) ∈ ran 𝐹 ∧ ((invg‘𝑇)‘𝑎) ∈ ran 𝐹))))
53	51, 52	syl 18	. 2 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (ran 𝐹 ∈ (SubGrp‘𝑇) ↔ (ran 𝐹 ⊆ (Base‘𝑇) ∧ ran 𝐹 ≠ ∅ ∧ ∀𝑎 ∈ ran 𝐹(∀𝑏 ∈ ran 𝐹(𝑎(+g‘𝑇)𝑏) ∈ ran 𝐹 ∧ ((invg‘𝑇)‘𝑎) ∈ ran 𝐹))))
54	4, 12, 50, 53	mpbir3and 1361	1 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → ran 𝐹 ∈ (SubGrp‘𝑇))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ≠ wne 2961  ∀wral 3082   ⊆ wss 3908  ∅c0 4289  dom cdm 5664  ran crn 5665   Fn wfn 6535  ‘cfv 6540  (class class class)co 7416  Basecbs 17279  +_gcplusg 17320  Grpcgrp 19010  inv_gcminusg 19011  SubGrpcsubg 19196   GrpHom cghm 19293

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5260  ax-nul 5272  ax-pow 5339  ax-pr 5407  ax-un 7738  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-nel 3068  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4876  df-iun 4961  df-br 5113  df-opab 5177  df-mpt 5196  df-tr 5222  df-id 5559  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-xp 5670  df-rel 5671  df-cnv 5672  df-co 5673  df-dm 5674  df-rn 5675  df-res 5676  df-ima 5677  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7865  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-er 8696  df-map 8828  df-en 8946  df-dom 8947  df-sdom 8948  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11453  df-neg 11454  df-nn 12244  df-2 12313  df-sets 17234  df-slot 17252  df-ndx 17264  df-base 17280  df-ress 17301  df-plusg 17333  df-0g 17504  df-mgm 18708  df-sgrp 18787  df-mnd 18803  df-grp 19013  df-minusg 19014  df-subg 19199  df-ghm 19294

This theorem is used by:  ghmghmrn  19315  ghmima  19317  ghmqusnsg  19362  ghmquskerlem3  19366  cayley  19494