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Theorem ghmrn 17594
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 2621 . . . 4 (Base‘𝑆) = (Base‘𝑆)
2 eqid 2621 . . . 4 (Base‘𝑇) = (Base‘𝑇)
31, 2ghmf 17585 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
4 frn 6010 . . 3 (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → ran 𝐹 ⊆ (Base‘𝑇))
53, 4syl 17 . 2 (𝐹 ∈ (𝑆 GrpHom 𝑇) → ran 𝐹 ⊆ (Base‘𝑇))
6 fdm 6008 . . . . 5 (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → dom 𝐹 = (Base‘𝑆))
73, 6syl 17 . . . 4 (𝐹 ∈ (𝑆 GrpHom 𝑇) → dom 𝐹 = (Base‘𝑆))
8 ghmgrp1 17583 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
91grpbn0 17372 . . . . 5 (𝑆 ∈ Grp → (Base‘𝑆) ≠ ∅)
108, 9syl 17 . . . 4 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (Base‘𝑆) ≠ ∅)
117, 10eqnetrd 2857 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) → dom 𝐹 ≠ ∅)
12 dm0rn0 5302 . . . 4 (dom 𝐹 = ∅ ↔ ran 𝐹 = ∅)
1312necon3bii 2842 . . 3 (dom 𝐹 ≠ ∅ ↔ ran 𝐹 ≠ ∅)
1411, 13sylib 208 . 2 (𝐹 ∈ (𝑆 GrpHom 𝑇) → ran 𝐹 ≠ ∅)
15 eqid 2621 . . . . . . . . . 10 (+g𝑆) = (+g𝑆)
16 eqid 2621 . . . . . . . . . 10 (+g𝑇) = (+g𝑇)
171, 15, 16ghmlin 17586 . . . . . . . . 9 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆) ∧ 𝑎 ∈ (Base‘𝑆)) → (𝐹‘(𝑐(+g𝑆)𝑎)) = ((𝐹𝑐)(+g𝑇)(𝐹𝑎)))
18 ffn 6002 . . . . . . . . . . . 12 (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → 𝐹 Fn (Base‘𝑆))
193, 18syl 17 . . . . . . . . . . 11 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹 Fn (Base‘𝑆))
20193ad2ant1 1080 . . . . . . . . . 10 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆) ∧ 𝑎 ∈ (Base‘𝑆)) → 𝐹 Fn (Base‘𝑆))
211, 15grpcl 17351 . . . . . . . . . . 11 ((𝑆 ∈ Grp ∧ 𝑐 ∈ (Base‘𝑆) ∧ 𝑎 ∈ (Base‘𝑆)) → (𝑐(+g𝑆)𝑎) ∈ (Base‘𝑆))
228, 21syl3an1 1356 . . . . . . . . . 10 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆) ∧ 𝑎 ∈ (Base‘𝑆)) → (𝑐(+g𝑆)𝑎) ∈ (Base‘𝑆))
23 fnfvelrn 6312 . . . . . . . . . 10 ((𝐹 Fn (Base‘𝑆) ∧ (𝑐(+g𝑆)𝑎) ∈ (Base‘𝑆)) → (𝐹‘(𝑐(+g𝑆)𝑎)) ∈ ran 𝐹)
2420, 22, 23syl2anc 692 . . . . . . . . 9 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆) ∧ 𝑎 ∈ (Base‘𝑆)) → (𝐹‘(𝑐(+g𝑆)𝑎)) ∈ ran 𝐹)
2517, 24eqeltrrd 2699 . . . . . . . 8 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆) ∧ 𝑎 ∈ (Base‘𝑆)) → ((𝐹𝑐)(+g𝑇)(𝐹𝑎)) ∈ ran 𝐹)
26253expia 1264 . . . . . . 7 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → (𝑎 ∈ (Base‘𝑆) → ((𝐹𝑐)(+g𝑇)(𝐹𝑎)) ∈ ran 𝐹))
2726ralrimiv 2959 . . . . . 6 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → ∀𝑎 ∈ (Base‘𝑆)((𝐹𝑐)(+g𝑇)(𝐹𝑎)) ∈ ran 𝐹)
28 oveq2 6612 . . . . . . . . . 10 (𝑏 = (𝐹𝑎) → ((𝐹𝑐)(+g𝑇)𝑏) = ((𝐹𝑐)(+g𝑇)(𝐹𝑎)))
2928eleq1d 2683 . . . . . . . . 9 (𝑏 = (𝐹𝑎) → (((𝐹𝑐)(+g𝑇)𝑏) ∈ ran 𝐹 ↔ ((𝐹𝑐)(+g𝑇)(𝐹𝑎)) ∈ ran 𝐹))
3029ralrn 6318 . . . . . . . 8 (𝐹 Fn (Base‘𝑆) → (∀𝑏 ∈ ran 𝐹((𝐹𝑐)(+g𝑇)𝑏) ∈ ran 𝐹 ↔ ∀𝑎 ∈ (Base‘𝑆)((𝐹𝑐)(+g𝑇)(𝐹𝑎)) ∈ ran 𝐹))
3119, 30syl 17 . . . . . . 7 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (∀𝑏 ∈ ran 𝐹((𝐹𝑐)(+g𝑇)𝑏) ∈ ran 𝐹 ↔ ∀𝑎 ∈ (Base‘𝑆)((𝐹𝑐)(+g𝑇)(𝐹𝑎)) ∈ ran 𝐹))
3231adantr 481 . . . . . 6 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → (∀𝑏 ∈ ran 𝐹((𝐹𝑐)(+g𝑇)𝑏) ∈ ran 𝐹 ↔ ∀𝑎 ∈ (Base‘𝑆)((𝐹𝑐)(+g𝑇)(𝐹𝑎)) ∈ ran 𝐹))
3327, 32mpbird 247 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → ∀𝑏 ∈ ran 𝐹((𝐹𝑐)(+g𝑇)𝑏) ∈ ran 𝐹)
34 eqid 2621 . . . . . . 7 (invg𝑆) = (invg𝑆)
35 eqid 2621 . . . . . . 7 (invg𝑇) = (invg𝑇)
361, 34, 35ghminv 17588 . . . . . 6 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → (𝐹‘((invg𝑆)‘𝑐)) = ((invg𝑇)‘(𝐹𝑐)))
3719adantr 481 . . . . . . 7 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → 𝐹 Fn (Base‘𝑆))
381, 34grpinvcl 17388 . . . . . . . 8 ((𝑆 ∈ Grp ∧ 𝑐 ∈ (Base‘𝑆)) → ((invg𝑆)‘𝑐) ∈ (Base‘𝑆))
398, 38sylan 488 . . . . . . 7 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → ((invg𝑆)‘𝑐) ∈ (Base‘𝑆))
40 fnfvelrn 6312 . . . . . . 7 ((𝐹 Fn (Base‘𝑆) ∧ ((invg𝑆)‘𝑐) ∈ (Base‘𝑆)) → (𝐹‘((invg𝑆)‘𝑐)) ∈ ran 𝐹)
4137, 39, 40syl2anc 692 . . . . . 6 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → (𝐹‘((invg𝑆)‘𝑐)) ∈ ran 𝐹)
4236, 41eqeltrrd 2699 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → ((invg𝑇)‘(𝐹𝑐)) ∈ ran 𝐹)
4333, 42jca 554 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → (∀𝑏 ∈ ran 𝐹((𝐹𝑐)(+g𝑇)𝑏) ∈ ran 𝐹 ∧ ((invg𝑇)‘(𝐹𝑐)) ∈ ran 𝐹))
4443ralrimiva 2960 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) → ∀𝑐 ∈ (Base‘𝑆)(∀𝑏 ∈ ran 𝐹((𝐹𝑐)(+g𝑇)𝑏) ∈ ran 𝐹 ∧ ((invg𝑇)‘(𝐹𝑐)) ∈ ran 𝐹))
45 oveq1 6611 . . . . . . . 8 (𝑎 = (𝐹𝑐) → (𝑎(+g𝑇)𝑏) = ((𝐹𝑐)(+g𝑇)𝑏))
4645eleq1d 2683 . . . . . . 7 (𝑎 = (𝐹𝑐) → ((𝑎(+g𝑇)𝑏) ∈ ran 𝐹 ↔ ((𝐹𝑐)(+g𝑇)𝑏) ∈ ran 𝐹))
4746ralbidv 2980 . . . . . 6 (𝑎 = (𝐹𝑐) → (∀𝑏 ∈ ran 𝐹(𝑎(+g𝑇)𝑏) ∈ ran 𝐹 ↔ ∀𝑏 ∈ ran 𝐹((𝐹𝑐)(+g𝑇)𝑏) ∈ ran 𝐹))
48 fveq2 6148 . . . . . . 7 (𝑎 = (𝐹𝑐) → ((invg𝑇)‘𝑎) = ((invg𝑇)‘(𝐹𝑐)))
4948eleq1d 2683 . . . . . 6 (𝑎 = (𝐹𝑐) → (((invg𝑇)‘𝑎) ∈ ran 𝐹 ↔ ((invg𝑇)‘(𝐹𝑐)) ∈ ran 𝐹))
5047, 49anbi12d 746 . . . . 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)))
5150ralrn 6318 . . . 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)))
5219, 51syl 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)))
5344, 52mpbird 247 . 2 (𝐹 ∈ (𝑆 GrpHom 𝑇) → ∀𝑎 ∈ ran 𝐹(∀𝑏 ∈ ran 𝐹(𝑎(+g𝑇)𝑏) ∈ ran 𝐹 ∧ ((invg𝑇)‘𝑎) ∈ ran 𝐹))
54 ghmgrp2 17584 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
552, 16, 35issubg2 17530 . . 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))))
5654, 55syl 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))))
575, 14, 53, 56mpbir3and 1243 1 (𝐹 ∈ (𝑆 GrpHom 𝑇) → ran 𝐹 ∈ (SubGrp‘𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  wss 3555  c0 3891  dom cdm 5074  ran crn 5075   Fn wfn 5842  wf 5843  cfv 5847  (class class class)co 6604  Basecbs 15781  +gcplusg 15862  Grpcgrp 17343  invgcminusg 17344  SubGrpcsubg 17509   GrpHom cghm 17578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-0g 16023  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-grp 17346  df-minusg 17347  df-subg 17512  df-ghm 17579
This theorem is referenced by:  ghmghmrn  17600  ghmima  17602  cayley  17755
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