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Theorem ghmrn 18888
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.
StepHypRef Expression
1 eqid 2736 . . . 4 (Base‘𝑆) = (Base‘𝑆)
2 eqid 2736 . . . 4 (Base‘𝑇) = (Base‘𝑇)
31, 2ghmf 18879 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
43frnd 6634 . 2 (𝐹 ∈ (𝑆 GrpHom 𝑇) → ran 𝐹 ⊆ (Base‘𝑇))
53fdmd 6637 . . . 4 (𝐹 ∈ (𝑆 GrpHom 𝑇) → dom 𝐹 = (Base‘𝑆))
6 ghmgrp1 18877 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
71grpbn0 18649 . . . . 5 (𝑆 ∈ Grp → (Base‘𝑆) ≠ ∅)
86, 7syl 17 . . . 4 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (Base‘𝑆) ≠ ∅)
95, 8eqnetrd 3009 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) → dom 𝐹 ≠ ∅)
10 dm0rn0 5842 . . . 4 (dom 𝐹 = ∅ ↔ ran 𝐹 = ∅)
1110necon3bii 2994 . . 3 (dom 𝐹 ≠ ∅ ↔ ran 𝐹 ≠ ∅)
129, 11sylib 217 . 2 (𝐹 ∈ (𝑆 GrpHom 𝑇) → ran 𝐹 ≠ ∅)
13 eqid 2736 . . . . . . . . . 10 (+g𝑆) = (+g𝑆)
14 eqid 2736 . . . . . . . . . 10 (+g𝑇) = (+g𝑇)
151, 13, 14ghmlin 18880 . . . . . . . . 9 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆) ∧ 𝑎 ∈ (Base‘𝑆)) → (𝐹‘(𝑐(+g𝑆)𝑎)) = ((𝐹𝑐)(+g𝑇)(𝐹𝑎)))
163ffnd 6627 . . . . . . . . . . 11 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹 Fn (Base‘𝑆))
17163ad2ant1 1133 . . . . . . . . . 10 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆) ∧ 𝑎 ∈ (Base‘𝑆)) → 𝐹 Fn (Base‘𝑆))
181, 13grpcl 18626 . . . . . . . . . . 11 ((𝑆 ∈ Grp ∧ 𝑐 ∈ (Base‘𝑆) ∧ 𝑎 ∈ (Base‘𝑆)) → (𝑐(+g𝑆)𝑎) ∈ (Base‘𝑆))
196, 18syl3an1 1163 . . . . . . . . . 10 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆) ∧ 𝑎 ∈ (Base‘𝑆)) → (𝑐(+g𝑆)𝑎) ∈ (Base‘𝑆))
20 fnfvelrn 6986 . . . . . . . . . 10 ((𝐹 Fn (Base‘𝑆) ∧ (𝑐(+g𝑆)𝑎) ∈ (Base‘𝑆)) → (𝐹‘(𝑐(+g𝑆)𝑎)) ∈ ran 𝐹)
2117, 19, 20syl2anc 585 . . . . . . . . 9 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆) ∧ 𝑎 ∈ (Base‘𝑆)) → (𝐹‘(𝑐(+g𝑆)𝑎)) ∈ ran 𝐹)
2215, 21eqeltrrd 2838 . . . . . . . 8 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆) ∧ 𝑎 ∈ (Base‘𝑆)) → ((𝐹𝑐)(+g𝑇)(𝐹𝑎)) ∈ ran 𝐹)
23223expia 1121 . . . . . . 7 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → (𝑎 ∈ (Base‘𝑆) → ((𝐹𝑐)(+g𝑇)(𝐹𝑎)) ∈ ran 𝐹))
2423ralrimiv 3139 . . . . . 6 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → ∀𝑎 ∈ (Base‘𝑆)((𝐹𝑐)(+g𝑇)(𝐹𝑎)) ∈ ran 𝐹)
25 oveq2 7311 . . . . . . . . . 10 (𝑏 = (𝐹𝑎) → ((𝐹𝑐)(+g𝑇)𝑏) = ((𝐹𝑐)(+g𝑇)(𝐹𝑎)))
2625eleq1d 2821 . . . . . . . . 9 (𝑏 = (𝐹𝑎) → (((𝐹𝑐)(+g𝑇)𝑏) ∈ ran 𝐹 ↔ ((𝐹𝑐)(+g𝑇)(𝐹𝑎)) ∈ ran 𝐹))
2726ralrn 6992 . . . . . . . 8 (𝐹 Fn (Base‘𝑆) → (∀𝑏 ∈ ran 𝐹((𝐹𝑐)(+g𝑇)𝑏) ∈ ran 𝐹 ↔ ∀𝑎 ∈ (Base‘𝑆)((𝐹𝑐)(+g𝑇)(𝐹𝑎)) ∈ ran 𝐹))
2816, 27syl 17 . . . . . . 7 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (∀𝑏 ∈ ran 𝐹((𝐹𝑐)(+g𝑇)𝑏) ∈ ran 𝐹 ↔ ∀𝑎 ∈ (Base‘𝑆)((𝐹𝑐)(+g𝑇)(𝐹𝑎)) ∈ ran 𝐹))
2928adantr 482 . . . . . 6 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → (∀𝑏 ∈ ran 𝐹((𝐹𝑐)(+g𝑇)𝑏) ∈ ran 𝐹 ↔ ∀𝑎 ∈ (Base‘𝑆)((𝐹𝑐)(+g𝑇)(𝐹𝑎)) ∈ ran 𝐹))
3024, 29mpbird 258 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → ∀𝑏 ∈ ran 𝐹((𝐹𝑐)(+g𝑇)𝑏) ∈ ran 𝐹)
31 eqid 2736 . . . . . . 7 (invg𝑆) = (invg𝑆)
32 eqid 2736 . . . . . . 7 (invg𝑇) = (invg𝑇)
331, 31, 32ghminv 18882 . . . . . 6 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → (𝐹‘((invg𝑆)‘𝑐)) = ((invg𝑇)‘(𝐹𝑐)))
3416adantr 482 . . . . . . 7 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → 𝐹 Fn (Base‘𝑆))
351, 31grpinvcl 18668 . . . . . . . 8 ((𝑆 ∈ Grp ∧ 𝑐 ∈ (Base‘𝑆)) → ((invg𝑆)‘𝑐) ∈ (Base‘𝑆))
366, 35sylan 581 . . . . . . 7 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → ((invg𝑆)‘𝑐) ∈ (Base‘𝑆))
37 fnfvelrn 6986 . . . . . . 7 ((𝐹 Fn (Base‘𝑆) ∧ ((invg𝑆)‘𝑐) ∈ (Base‘𝑆)) → (𝐹‘((invg𝑆)‘𝑐)) ∈ ran 𝐹)
3834, 36, 37syl2anc 585 . . . . . 6 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → (𝐹‘((invg𝑆)‘𝑐)) ∈ ran 𝐹)
3933, 38eqeltrrd 2838 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → ((invg𝑇)‘(𝐹𝑐)) ∈ ran 𝐹)
4030, 39jca 513 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → (∀𝑏 ∈ ran 𝐹((𝐹𝑐)(+g𝑇)𝑏) ∈ ran 𝐹 ∧ ((invg𝑇)‘(𝐹𝑐)) ∈ ran 𝐹))
4140ralrimiva 3140 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) → ∀𝑐 ∈ (Base‘𝑆)(∀𝑏 ∈ ran 𝐹((𝐹𝑐)(+g𝑇)𝑏) ∈ ran 𝐹 ∧ ((invg𝑇)‘(𝐹𝑐)) ∈ ran 𝐹))
42 oveq1 7310 . . . . . . . 8 (𝑎 = (𝐹𝑐) → (𝑎(+g𝑇)𝑏) = ((𝐹𝑐)(+g𝑇)𝑏))
4342eleq1d 2821 . . . . . . 7 (𝑎 = (𝐹𝑐) → ((𝑎(+g𝑇)𝑏) ∈ ran 𝐹 ↔ ((𝐹𝑐)(+g𝑇)𝑏) ∈ ran 𝐹))
4443ralbidv 3171 . . . . . 6 (𝑎 = (𝐹𝑐) → (∀𝑏 ∈ ran 𝐹(𝑎(+g𝑇)𝑏) ∈ ran 𝐹 ↔ ∀𝑏 ∈ ran 𝐹((𝐹𝑐)(+g𝑇)𝑏) ∈ ran 𝐹))
45 fveq2 6800 . . . . . . 7 (𝑎 = (𝐹𝑐) → ((invg𝑇)‘𝑎) = ((invg𝑇)‘(𝐹𝑐)))
4645eleq1d 2821 . . . . . 6 (𝑎 = (𝐹𝑐) → (((invg𝑇)‘𝑎) ∈ ran 𝐹 ↔ ((invg𝑇)‘(𝐹𝑐)) ∈ ran 𝐹))
4744, 46anbi12d 632 . . . . 5 (𝑎 = (𝐹𝑐) → ((∀𝑏 ∈ ran 𝐹(𝑎(+g𝑇)𝑏) ∈ ran 𝐹 ∧ ((invg𝑇)‘𝑎) ∈ ran 𝐹) ↔ (∀𝑏 ∈ ran 𝐹((𝐹𝑐)(+g𝑇)𝑏) ∈ ran 𝐹 ∧ ((invg𝑇)‘(𝐹𝑐)) ∈ ran 𝐹)))
4847ralrn 6992 . . . 4 (𝐹 Fn (Base‘𝑆) → (∀𝑎 ∈ ran 𝐹(∀𝑏 ∈ ran 𝐹(𝑎(+g𝑇)𝑏) ∈ ran 𝐹 ∧ ((invg𝑇)‘𝑎) ∈ ran 𝐹) ↔ ∀𝑐 ∈ (Base‘𝑆)(∀𝑏 ∈ ran 𝐹((𝐹𝑐)(+g𝑇)𝑏) ∈ ran 𝐹 ∧ ((invg𝑇)‘(𝐹𝑐)) ∈ ran 𝐹)))
4916, 48syl 17 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (∀𝑎 ∈ ran 𝐹(∀𝑏 ∈ ran 𝐹(𝑎(+g𝑇)𝑏) ∈ ran 𝐹 ∧ ((invg𝑇)‘𝑎) ∈ ran 𝐹) ↔ ∀𝑐 ∈ (Base‘𝑆)(∀𝑏 ∈ ran 𝐹((𝐹𝑐)(+g𝑇)𝑏) ∈ ran 𝐹 ∧ ((invg𝑇)‘(𝐹𝑐)) ∈ ran 𝐹)))
5041, 49mpbird 258 . 2 (𝐹 ∈ (𝑆 GrpHom 𝑇) → ∀𝑎 ∈ ran 𝐹(∀𝑏 ∈ ran 𝐹(𝑎(+g𝑇)𝑏) ∈ ran 𝐹 ∧ ((invg𝑇)‘𝑎) ∈ ran 𝐹))
51 ghmgrp2 18878 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
522, 14, 32issubg2 18811 . . 3 (𝑇 ∈ Grp → (ran 𝐹 ∈ (SubGrp‘𝑇) ↔ (ran 𝐹 ⊆ (Base‘𝑇) ∧ ran 𝐹 ≠ ∅ ∧ ∀𝑎 ∈ ran 𝐹(∀𝑏 ∈ ran 𝐹(𝑎(+g𝑇)𝑏) ∈ ran 𝐹 ∧ ((invg𝑇)‘𝑎) ∈ ran 𝐹))))
5351, 52syl 17 . 2 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (ran 𝐹 ∈ (SubGrp‘𝑇) ↔ (ran 𝐹 ⊆ (Base‘𝑇) ∧ ran 𝐹 ≠ ∅ ∧ ∀𝑎 ∈ ran 𝐹(∀𝑏 ∈ ran 𝐹(𝑎(+g𝑇)𝑏) ∈ ran 𝐹 ∧ ((invg𝑇)‘𝑎) ∈ ran 𝐹))))
544, 12, 50, 53mpbir3and 1342 1 (𝐹 ∈ (𝑆 GrpHom 𝑇) → ran 𝐹 ∈ (SubGrp‘𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1087   = wceq 1539  wcel 2104  wne 2941  wral 3062  wss 3892  c0 4262  dom cdm 5596  ran crn 5597   Fn wfn 6449  cfv 6454  (class class class)co 7303  Basecbs 16953  +gcplusg 17003  Grpcgrp 18618  invgcminusg 18619  SubGrpcsubg 18790   GrpHom cghm 18872
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-rep 5218  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361  ax-un 7616  ax-cnex 10969  ax-resscn 10970  ax-1cn 10971  ax-icn 10972  ax-addcl 10973  ax-addrcl 10974  ax-mulcl 10975  ax-mulrcl 10976  ax-mulcom 10977  ax-addass 10978  ax-mulass 10979  ax-distr 10980  ax-i2m1 10981  ax-1ne0 10982  ax-1rid 10983  ax-rnegex 10984  ax-rrecex 10985  ax-cnre 10986  ax-pre-lttri 10987  ax-pre-lttrn 10988  ax-pre-ltadd 10989  ax-pre-mulgt0 10990
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3285  df-reu 3286  df-rab 3287  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-iun 4933  df-br 5082  df-opab 5144  df-mpt 5165  df-tr 5199  df-id 5496  df-eprel 5502  df-po 5510  df-so 5511  df-fr 5551  df-we 5553  df-xp 5602  df-rel 5603  df-cnv 5604  df-co 5605  df-dm 5606  df-rn 5607  df-res 5608  df-ima 5609  df-pred 6213  df-ord 6280  df-on 6281  df-lim 6282  df-suc 6283  df-iota 6406  df-fun 6456  df-fn 6457  df-f 6458  df-f1 6459  df-fo 6460  df-f1o 6461  df-fv 6462  df-riota 7260  df-ov 7306  df-oprab 7307  df-mpo 7308  df-om 7741  df-2nd 7860  df-frecs 8124  df-wrecs 8155  df-recs 8229  df-rdg 8268  df-er 8525  df-en 8761  df-dom 8762  df-sdom 8763  df-pnf 11053  df-mnf 11054  df-xr 11055  df-ltxr 11056  df-le 11057  df-sub 11249  df-neg 11250  df-nn 12016  df-2 12078  df-sets 16906  df-slot 16924  df-ndx 16936  df-base 16954  df-ress 16983  df-plusg 17016  df-0g 17193  df-mgm 18367  df-sgrp 18416  df-mnd 18427  df-grp 18621  df-minusg 18622  df-subg 18793  df-ghm 18873
This theorem is referenced by:  ghmghmrn  18894  ghmima  18896  cayley  19063
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