Theorem ghmrn 14037

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 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2238	. . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆)
2		eqid 2238	. . . 4 ⊢ (Base‘𝑇) = (Base‘𝑇)
3	1, 2	ghmf 14027	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
4	3	frnd 5538	. 2 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → ran 𝐹 ⊆ (Base‘𝑇))
5		ghmgrp1 14025	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
6		eqid 2238	. . . . . . 7 ⊢ (0g‘𝑆) = (0g‘𝑆)
7	1, 6	grpidcl 13811	. . . . . 6 ⊢ (𝑆 ∈ Grp → (0g‘𝑆) ∈ (Base‘𝑆))
8	5, 7	syl 14	. . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (0g‘𝑆) ∈ (Base‘𝑆))
9	3	fdmd 5535	. . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → dom 𝐹 = (Base‘𝑆))
10	8, 9	eleqtrrd 2318	. . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (0g‘𝑆) ∈ dom 𝐹)
11		elex2 2838	. . . 4 ⊢ ((0g‘𝑆) ∈ dom 𝐹 → ∃𝑗 𝑗 ∈ dom 𝐹)
12	10, 11	syl 14	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → ∃𝑗 𝑗 ∈ dom 𝐹)
13		dmmrnm 4996	. . 3 ⊢ (∃𝑗 𝑗 ∈ dom 𝐹 ↔ ∃𝑗 𝑗 ∈ ran 𝐹)
14	12, 13	sylib 122	. 2 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → ∃𝑗 𝑗 ∈ ran 𝐹)
15		eqid 2238	. . . . . . . . . 10 ⊢ (+g‘𝑆) = (+g‘𝑆)
16		eqid 2238	. . . . . . . . . 10 ⊢ (+g‘𝑇) = (+g‘𝑇)
17	1, 15, 16	ghmlin 14028	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆) ∧ 𝑎 ∈ (Base‘𝑆)) → (𝐹‘(𝑐(+g‘𝑆)𝑎)) = ((𝐹‘𝑐)(+g‘𝑇)(𝐹‘𝑎)))
18	3	ffnd 5529	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹 Fn (Base‘𝑆))
19	18	3ad2ant1 1049	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆) ∧ 𝑎 ∈ (Base‘𝑆)) → 𝐹 Fn (Base‘𝑆))
20	1, 15	grpcl 13790	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Grp ∧ 𝑐 ∈ (Base‘𝑆) ∧ 𝑎 ∈ (Base‘𝑆)) → (𝑐(+g‘𝑆)𝑎) ∈ (Base‘𝑆))
21	5, 20	syl3an1 1311	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆) ∧ 𝑎 ∈ (Base‘𝑆)) → (𝑐(+g‘𝑆)𝑎) ∈ (Base‘𝑆))
22		fnfvelrn 5831	. . . . . . . . . 10 ⊢ ((𝐹 Fn (Base‘𝑆) ∧ (𝑐(+g‘𝑆)𝑎) ∈ (Base‘𝑆)) → (𝐹‘(𝑐(+g‘𝑆)𝑎)) ∈ ran 𝐹)
23	19, 21, 22	syl2anc 415	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆) ∧ 𝑎 ∈ (Base‘𝑆)) → (𝐹‘(𝑐(+g‘𝑆)𝑎)) ∈ ran 𝐹)
24	17, 23	eqeltrrd 2316	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆) ∧ 𝑎 ∈ (Base‘𝑆)) → ((𝐹‘𝑐)(+g‘𝑇)(𝐹‘𝑎)) ∈ ran 𝐹)
25	24	3expia 1236	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → (𝑎 ∈ (Base‘𝑆) → ((𝐹‘𝑐)(+g‘𝑇)(𝐹‘𝑎)) ∈ ran 𝐹))
26	25	ralrimiv 2622	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → ∀𝑎 ∈ (Base‘𝑆)((𝐹‘𝑐)(+g‘𝑇)(𝐹‘𝑎)) ∈ ran 𝐹)
27		oveq2 6083	. . . . . . . . . 10 ⊢ (𝑏 = (𝐹‘𝑎) → ((𝐹‘𝑐)(+g‘𝑇)𝑏) = ((𝐹‘𝑐)(+g‘𝑇)(𝐹‘𝑎)))
28	27	eleq1d 2307	. . . . . . . . 9 ⊢ (𝑏 = (𝐹‘𝑎) → (((𝐹‘𝑐)(+g‘𝑇)𝑏) ∈ ran 𝐹 ↔ ((𝐹‘𝑐)(+g‘𝑇)(𝐹‘𝑎)) ∈ ran 𝐹))
29	28	ralrn 5837	. . . . . . . 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))
30	18, 29	syl 14	. . . . . . 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))
31	30	adantr 276	. . . . . 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))
32	26, 31	mpbird 167	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → ∀𝑏 ∈ ran 𝐹((𝐹‘𝑐)(+g‘𝑇)𝑏) ∈ ran 𝐹)
33		eqid 2238	. . . . . . 7 ⊢ (invg‘𝑆) = (invg‘𝑆)
34		eqid 2238	. . . . . . 7 ⊢ (invg‘𝑇) = (invg‘𝑇)
35	1, 33, 34	ghminv 14030	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → (𝐹‘((invg‘𝑆)‘𝑐)) = ((invg‘𝑇)‘(𝐹‘𝑐)))
36	18	adantr 276	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → 𝐹 Fn (Base‘𝑆))
37	1, 33	grpinvcl 13830	. . . . . . . 8 ⊢ ((𝑆 ∈ Grp ∧ 𝑐 ∈ (Base‘𝑆)) → ((invg‘𝑆)‘𝑐) ∈ (Base‘𝑆))
38	5, 37	sylan 283	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → ((invg‘𝑆)‘𝑐) ∈ (Base‘𝑆))
39		fnfvelrn 5831	. . . . . . 7 ⊢ ((𝐹 Fn (Base‘𝑆) ∧ ((invg‘𝑆)‘𝑐) ∈ (Base‘𝑆)) → (𝐹‘((invg‘𝑆)‘𝑐)) ∈ ran 𝐹)
40	36, 38, 39	syl2anc 415	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → (𝐹‘((invg‘𝑆)‘𝑐)) ∈ ran 𝐹)
41	35, 40	eqeltrrd 2316	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → ((invg‘𝑇)‘(𝐹‘𝑐)) ∈ ran 𝐹)
42	32, 41	jca 306	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → (∀𝑏 ∈ ran 𝐹((𝐹‘𝑐)(+g‘𝑇)𝑏) ∈ ran 𝐹 ∧ ((invg‘𝑇)‘(𝐹‘𝑐)) ∈ ran 𝐹))
43	42	ralrimiva 2623	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → ∀𝑐 ∈ (Base‘𝑆)(∀𝑏 ∈ ran 𝐹((𝐹‘𝑐)(+g‘𝑇)𝑏) ∈ ran 𝐹 ∧ ((invg‘𝑇)‘(𝐹‘𝑐)) ∈ ran 𝐹))
44		oveq1 6082	. . . . . . . 8 ⊢ (𝑎 = (𝐹‘𝑐) → (𝑎(+g‘𝑇)𝑏) = ((𝐹‘𝑐)(+g‘𝑇)𝑏))
45	44	eleq1d 2307	. . . . . . 7 ⊢ (𝑎 = (𝐹‘𝑐) → ((𝑎(+g‘𝑇)𝑏) ∈ ran 𝐹 ↔ ((𝐹‘𝑐)(+g‘𝑇)𝑏) ∈ ran 𝐹))
46	45	ralbidv 2550	. . . . . 6 ⊢ (𝑎 = (𝐹‘𝑐) → (∀𝑏 ∈ ran 𝐹(𝑎(+g‘𝑇)𝑏) ∈ ran 𝐹 ↔ ∀𝑏 ∈ ran 𝐹((𝐹‘𝑐)(+g‘𝑇)𝑏) ∈ ran 𝐹))
47		fveq2 5690	. . . . . . 7 ⊢ (𝑎 = (𝐹‘𝑐) → ((invg‘𝑇)‘𝑎) = ((invg‘𝑇)‘(𝐹‘𝑐)))
48	47	eleq1d 2307	. . . . . 6 ⊢ (𝑎 = (𝐹‘𝑐) → (((invg‘𝑇)‘𝑎) ∈ ran 𝐹 ↔ ((invg‘𝑇)‘(𝐹‘𝑐)) ∈ ran 𝐹))
49	46, 48	anbi12d 477	. . . . 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)))
50	49	ralrn 5837	. . . 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)))
51	18, 50	syl 14	. . 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)))
52	43, 51	mpbird 167	. 2 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → ∀𝑎 ∈ ran 𝐹(∀𝑏 ∈ ran 𝐹(𝑎(+g‘𝑇)𝑏) ∈ ran 𝐹 ∧ ((invg‘𝑇)‘𝑎) ∈ ran 𝐹))
53		ghmgrp2 14026	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
54	2, 16, 34	issubg2m 13969	. . 3 ⊢ (𝑇 ∈ Grp → (ran 𝐹 ∈ (SubGrp‘𝑇) ↔ (ran 𝐹 ⊆ (Base‘𝑇) ∧ ∃𝑗 𝑗 ∈ ran 𝐹 ∧ ∀𝑎 ∈ ran 𝐹(∀𝑏 ∈ ran 𝐹(𝑎(+g‘𝑇)𝑏) ∈ ran 𝐹 ∧ ((invg‘𝑇)‘𝑎) ∈ ran 𝐹))))
55	53, 54	syl 14	. 2 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (ran 𝐹 ∈ (SubGrp‘𝑇) ↔ (ran 𝐹 ⊆ (Base‘𝑇) ∧ ∃𝑗 𝑗 ∈ ran 𝐹 ∧ ∀𝑎 ∈ ran 𝐹(∀𝑏 ∈ ran 𝐹(𝑎(+g‘𝑇)𝑏) ∈ ran 𝐹 ∧ ((invg‘𝑇)‘𝑎) ∈ ran 𝐹))))
56	4, 14, 52, 55	mpbir3and 1211	1 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → ran 𝐹 ∈ (SubGrp‘𝑇))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1009   = wceq 1402  ∃wex 1545   ∈ wcel 2209  ∀wral 2528   ⊆ wss 3220  dom cdm 4769  ran crn 4770   Fn wfn 5367  ‘cfv 5372  (class class class)co 6075  Basecbs 13330  +_gcplusg 13408  0_gc0g 13587  Grpcgrp 13782  inv_gcminusg 13783  SubGrpcsubg 13947   GrpHom cghm 14020

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-pre-ltirr 8281  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-ltxr 8355  df-inn 9284  df-2 9342  df-ndx 13333  df-slot 13334  df-base 13336  df-sets 13337  df-iress 13338  df-plusg 13421  df-0g 13589  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-grp 13785  df-minusg 13786  df-subg 13950  df-ghm 14021

This theorem is referenced by:  ghmghmrn  14043  ghmima  14045