Theorem ghmrn 19127

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 2729	. . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆)
2		eqid 2729	. . . 4 ⊢ (Base‘𝑇) = (Base‘𝑇)
3	1, 2	ghmf 19118	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
4	3	frnd 6664	. 2 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → ran 𝐹 ⊆ (Base‘𝑇))
5	3	fdmd 6666	. . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → dom 𝐹 = (Base‘𝑆))
6		ghmgrp1 19116	. . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
7	1	grpbn0 18864	. . . . 5 ⊢ (𝑆 ∈ Grp → (Base‘𝑆) ≠ ∅)
8	6, 7	syl 17	. . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (Base‘𝑆) ≠ ∅)
9	5, 8	eqnetrd 2992	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → dom 𝐹 ≠ ∅)
10		dm0rn0 5871	. . . 4 ⊢ (dom 𝐹 = ∅ ↔ ran 𝐹 = ∅)
11	10	necon3bii 2977	. . 3 ⊢ (dom 𝐹 ≠ ∅ ↔ ran 𝐹 ≠ ∅)
12	9, 11	sylib 218	. 2 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → ran 𝐹 ≠ ∅)
13		eqid 2729	. . . . . . . . . 10 ⊢ (+g‘𝑆) = (+g‘𝑆)
14		eqid 2729	. . . . . . . . . 10 ⊢ (+g‘𝑇) = (+g‘𝑇)
15	1, 13, 14	ghmlin 19119	. . . . . . . . 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 6657	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹 Fn (Base‘𝑆))
17	16	3ad2ant1 1133	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆) ∧ 𝑎 ∈ (Base‘𝑆)) → 𝐹 Fn (Base‘𝑆))
18	1, 13	grpcl 18839	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Grp ∧ 𝑐 ∈ (Base‘𝑆) ∧ 𝑎 ∈ (Base‘𝑆)) → (𝑐(+g‘𝑆)𝑎) ∈ (Base‘𝑆))
19	6, 18	syl3an1 1163	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆) ∧ 𝑎 ∈ (Base‘𝑆)) → (𝑐(+g‘𝑆)𝑎) ∈ (Base‘𝑆))
20		fnfvelrn 7018	. . . . . . . . . 10 ⊢ ((𝐹 Fn (Base‘𝑆) ∧ (𝑐(+g‘𝑆)𝑎) ∈ (Base‘𝑆)) → (𝐹‘(𝑐(+g‘𝑆)𝑎)) ∈ ran 𝐹)
21	17, 19, 20	syl2anc 584	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆) ∧ 𝑎 ∈ (Base‘𝑆)) → (𝐹‘(𝑐(+g‘𝑆)𝑎)) ∈ ran 𝐹)
22	15, 21	eqeltrrd 2829	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆) ∧ 𝑎 ∈ (Base‘𝑆)) → ((𝐹‘𝑐)(+g‘𝑇)(𝐹‘𝑎)) ∈ ran 𝐹)
23	22	3expia 1121	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → (𝑎 ∈ (Base‘𝑆) → ((𝐹‘𝑐)(+g‘𝑇)(𝐹‘𝑎)) ∈ ran 𝐹))
24	23	ralrimiv 3120	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → ∀𝑎 ∈ (Base‘𝑆)((𝐹‘𝑐)(+g‘𝑇)(𝐹‘𝑎)) ∈ ran 𝐹)
25		oveq2 7361	. . . . . . . . . 10 ⊢ (𝑏 = (𝐹‘𝑎) → ((𝐹‘𝑐)(+g‘𝑇)𝑏) = ((𝐹‘𝑐)(+g‘𝑇)(𝐹‘𝑎)))
26	25	eleq1d 2813	. . . . . . . . 9 ⊢ (𝑏 = (𝐹‘𝑎) → (((𝐹‘𝑐)(+g‘𝑇)𝑏) ∈ ran 𝐹 ↔ ((𝐹‘𝑐)(+g‘𝑇)(𝐹‘𝑎)) ∈ ran 𝐹))
27	26	ralrn 7026	. . . . . . . 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 17	. . . . . . 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 480	. . . . . 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 257	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → ∀𝑏 ∈ ran 𝐹((𝐹‘𝑐)(+g‘𝑇)𝑏) ∈ ran 𝐹)
31		eqid 2729	. . . . . . 7 ⊢ (invg‘𝑆) = (invg‘𝑆)
32		eqid 2729	. . . . . . 7 ⊢ (invg‘𝑇) = (invg‘𝑇)
33	1, 31, 32	ghminv 19121	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → (𝐹‘((invg‘𝑆)‘𝑐)) = ((invg‘𝑇)‘(𝐹‘𝑐)))
34	16	adantr 480	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → 𝐹 Fn (Base‘𝑆))
35	1, 31	grpinvcl 18885	. . . . . . . 8 ⊢ ((𝑆 ∈ Grp ∧ 𝑐 ∈ (Base‘𝑆)) → ((invg‘𝑆)‘𝑐) ∈ (Base‘𝑆))
36	6, 35	sylan 580	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → ((invg‘𝑆)‘𝑐) ∈ (Base‘𝑆))
37		fnfvelrn 7018	. . . . . . 7 ⊢ ((𝐹 Fn (Base‘𝑆) ∧ ((invg‘𝑆)‘𝑐) ∈ (Base‘𝑆)) → (𝐹‘((invg‘𝑆)‘𝑐)) ∈ ran 𝐹)
38	34, 36, 37	syl2anc 584	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → (𝐹‘((invg‘𝑆)‘𝑐)) ∈ ran 𝐹)
39	33, 38	eqeltrrd 2829	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → ((invg‘𝑇)‘(𝐹‘𝑐)) ∈ ran 𝐹)
40	30, 39	jca 511	. . . 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 3121	. . 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 7360	. . . . . . . 8 ⊢ (𝑎 = (𝐹‘𝑐) → (𝑎(+g‘𝑇)𝑏) = ((𝐹‘𝑐)(+g‘𝑇)𝑏))
43	42	eleq1d 2813	. . . . . . 7 ⊢ (𝑎 = (𝐹‘𝑐) → ((𝑎(+g‘𝑇)𝑏) ∈ ran 𝐹 ↔ ((𝐹‘𝑐)(+g‘𝑇)𝑏) ∈ ran 𝐹))
44	43	ralbidv 3152	. . . . . 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 6826	. . . . . . 7 ⊢ (𝑎 = (𝐹‘𝑐) → ((invg‘𝑇)‘𝑎) = ((invg‘𝑇)‘(𝐹‘𝑐)))
46	45	eleq1d 2813	. . . . . 6 ⊢ (𝑎 = (𝐹‘𝑐) → (((invg‘𝑇)‘𝑎) ∈ ran 𝐹 ↔ ((invg‘𝑇)‘(𝐹‘𝑐)) ∈ ran 𝐹))
47	44, 46	anbi12d 632	. . . . 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 7026	. . . 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 17	. . 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 257	. 2 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → ∀𝑎 ∈ ran 𝐹(∀𝑏 ∈ ran 𝐹(𝑎(+g‘𝑇)𝑏) ∈ ran 𝐹 ∧ ((invg‘𝑇)‘𝑎) ∈ ran 𝐹))
51		ghmgrp2 19117	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
52	2, 14, 32	issubg2 19039	. . 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 17	. 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 1343	1 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → ran 𝐹 ∈ (SubGrp‘𝑇))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044   ⊆ wss 3905  ∅c0 4286  dom cdm 5623  ran crn 5624   Fn wfn 6481  ‘cfv 6486  (class class class)co 7353  Basecbs 17139  +_gcplusg 17180  Grpcgrp 18831  inv_gcminusg 18832  SubGrpcsubg 19018   GrpHom cghm 19110

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 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

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 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-er 8632  df-map 8762  df-en 8880  df-dom 8881  df-sdom 8882  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-nn 12148  df-2 12210  df-sets 17094  df-slot 17112  df-ndx 17124  df-base 17140  df-ress 17161  df-plusg 17193  df-0g 17364  df-mgm 18533  df-sgrp 18612  df-mnd 18628  df-grp 18834  df-minusg 18835  df-subg 19021  df-ghm 19111

This theorem is referenced by:  ghmghmrn  19133  ghmima  19135  ghmqusnsg  19180  ghmquskerlem3  19184  cayley  19312