Theorem ghmrn 19105

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 2733	. . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆)
2		eqid 2733	. . . 4 ⊢ (Base‘𝑇) = (Base‘𝑇)
3	1, 2	ghmf 19096	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
4	3	frnd 6726	. 2 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → ran 𝐹 ⊆ (Base‘𝑇))
5	3	fdmd 6729	. . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → dom 𝐹 = (Base‘𝑆))
6		ghmgrp1 19094	. . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
7	1	grpbn0 18851	. . . . 5 ⊢ (𝑆 ∈ Grp → (Base‘𝑆) ≠ ∅)
8	6, 7	syl 17	. . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (Base‘𝑆) ≠ ∅)
9	5, 8	eqnetrd 3009	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → dom 𝐹 ≠ ∅)
10		dm0rn0 5925	. . . 4 ⊢ (dom 𝐹 = ∅ ↔ ran 𝐹 = ∅)
11	10	necon3bii 2994	. . 3 ⊢ (dom 𝐹 ≠ ∅ ↔ ran 𝐹 ≠ ∅)
12	9, 11	sylib 217	. 2 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → ran 𝐹 ≠ ∅)
13		eqid 2733	. . . . . . . . . 10 ⊢ (+g‘𝑆) = (+g‘𝑆)
14		eqid 2733	. . . . . . . . . 10 ⊢ (+g‘𝑇) = (+g‘𝑇)
15	1, 13, 14	ghmlin 19097	. . . . . . . . 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 6719	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹 Fn (Base‘𝑆))
17	16	3ad2ant1 1134	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆) ∧ 𝑎 ∈ (Base‘𝑆)) → 𝐹 Fn (Base‘𝑆))
18	1, 13	grpcl 18827	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Grp ∧ 𝑐 ∈ (Base‘𝑆) ∧ 𝑎 ∈ (Base‘𝑆)) → (𝑐(+g‘𝑆)𝑎) ∈ (Base‘𝑆))
19	6, 18	syl3an1 1164	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆) ∧ 𝑎 ∈ (Base‘𝑆)) → (𝑐(+g‘𝑆)𝑎) ∈ (Base‘𝑆))
20		fnfvelrn 7083	. . . . . . . . . 10 ⊢ ((𝐹 Fn (Base‘𝑆) ∧ (𝑐(+g‘𝑆)𝑎) ∈ (Base‘𝑆)) → (𝐹‘(𝑐(+g‘𝑆)𝑎)) ∈ ran 𝐹)
21	17, 19, 20	syl2anc 585	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆) ∧ 𝑎 ∈ (Base‘𝑆)) → (𝐹‘(𝑐(+g‘𝑆)𝑎)) ∈ ran 𝐹)
22	15, 21	eqeltrrd 2835	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆) ∧ 𝑎 ∈ (Base‘𝑆)) → ((𝐹‘𝑐)(+g‘𝑇)(𝐹‘𝑎)) ∈ ran 𝐹)
23	22	3expia 1122	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → (𝑎 ∈ (Base‘𝑆) → ((𝐹‘𝑐)(+g‘𝑇)(𝐹‘𝑎)) ∈ ran 𝐹))
24	23	ralrimiv 3146	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → ∀𝑎 ∈ (Base‘𝑆)((𝐹‘𝑐)(+g‘𝑇)(𝐹‘𝑎)) ∈ ran 𝐹)
25		oveq2 7417	. . . . . . . . . 10 ⊢ (𝑏 = (𝐹‘𝑎) → ((𝐹‘𝑐)(+g‘𝑇)𝑏) = ((𝐹‘𝑐)(+g‘𝑇)(𝐹‘𝑎)))
26	25	eleq1d 2819	. . . . . . . . 9 ⊢ (𝑏 = (𝐹‘𝑎) → (((𝐹‘𝑐)(+g‘𝑇)𝑏) ∈ ran 𝐹 ↔ ((𝐹‘𝑐)(+g‘𝑇)(𝐹‘𝑎)) ∈ ran 𝐹))
27	26	ralrn 7090	. . . . . . . 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 482	. . . . . 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 2733	. . . . . . 7 ⊢ (invg‘𝑆) = (invg‘𝑆)
32		eqid 2733	. . . . . . 7 ⊢ (invg‘𝑇) = (invg‘𝑇)
33	1, 31, 32	ghminv 19099	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → (𝐹‘((invg‘𝑆)‘𝑐)) = ((invg‘𝑇)‘(𝐹‘𝑐)))
34	16	adantr 482	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → 𝐹 Fn (Base‘𝑆))
35	1, 31	grpinvcl 18872	. . . . . . . 8 ⊢ ((𝑆 ∈ Grp ∧ 𝑐 ∈ (Base‘𝑆)) → ((invg‘𝑆)‘𝑐) ∈ (Base‘𝑆))
36	6, 35	sylan 581	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → ((invg‘𝑆)‘𝑐) ∈ (Base‘𝑆))
37		fnfvelrn 7083	. . . . . . 7 ⊢ ((𝐹 Fn (Base‘𝑆) ∧ ((invg‘𝑆)‘𝑐) ∈ (Base‘𝑆)) → (𝐹‘((invg‘𝑆)‘𝑐)) ∈ ran 𝐹)
38	34, 36, 37	syl2anc 585	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → (𝐹‘((invg‘𝑆)‘𝑐)) ∈ ran 𝐹)
39	33, 38	eqeltrrd 2835	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → ((invg‘𝑇)‘(𝐹‘𝑐)) ∈ ran 𝐹)
40	30, 39	jca 513	. . . 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 3147	. . 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 7416	. . . . . . . 8 ⊢ (𝑎 = (𝐹‘𝑐) → (𝑎(+g‘𝑇)𝑏) = ((𝐹‘𝑐)(+g‘𝑇)𝑏))
43	42	eleq1d 2819	. . . . . . 7 ⊢ (𝑎 = (𝐹‘𝑐) → ((𝑎(+g‘𝑇)𝑏) ∈ ran 𝐹 ↔ ((𝐹‘𝑐)(+g‘𝑇)𝑏) ∈ ran 𝐹))
44	43	ralbidv 3178	. . . . . 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 6892	. . . . . . 7 ⊢ (𝑎 = (𝐹‘𝑐) → ((invg‘𝑇)‘𝑎) = ((invg‘𝑇)‘(𝐹‘𝑐)))
46	45	eleq1d 2819	. . . . . 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 7090	. . . 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 19095	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
52	2, 14, 32	issubg2 19021	. . 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 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   ≠ wne 2941  ∀wral 3062   ⊆ wss 3949  ∅c0 4323  dom cdm 5677  ran crn 5678   Fn wfn 6539  ‘cfv 6544  (class class class)co 7409  Basecbs 17144  +_gcplusg 17197  Grpcgrp 18819  inv_gcminusg 18820  SubGrpcsubg 19000   GrpHom cghm 19089

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  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 theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-ress 17174  df-plusg 17210  df-0g 17387  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-grp 18822  df-minusg 18823  df-subg 19003  df-ghm 19090

This theorem is referenced by:  ghmghmrn  19111  ghmima  19113  cayley  19282  ghmquskerlem3  32531