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Theorem ghmeql 19155
Description: The equalizer of two group homomorphisms is a subgroup. (Contributed by Stefan O'Rear, 7-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
Assertion
Ref Expression
ghmeql ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) β†’ dom (𝐹 ∩ 𝐺) ∈ (SubGrpβ€˜π‘†))
Proof of Theorem ghmeql
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ghmmhm 19142 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) β†’ 𝐹 ∈ (𝑆 MndHom 𝑇))
2 ghmmhm 19142 . . 3 (𝐺 ∈ (𝑆 GrpHom 𝑇) β†’ 𝐺 ∈ (𝑆 MndHom 𝑇))
3 mhmeql 18745 . . 3 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) β†’ dom (𝐹 ∩ 𝐺) ∈ (SubMndβ€˜π‘†))
41, 2, 3syl2an 594 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) β†’ dom (𝐹 ∩ 𝐺) ∈ (SubMndβ€˜π‘†))
5 fveq2 6892 . . . . . . . 8 (𝑦 = ((invgβ€˜π‘†)β€˜π‘₯) β†’ (πΉβ€˜π‘¦) = (πΉβ€˜((invgβ€˜π‘†)β€˜π‘₯)))
6 fveq2 6892 . . . . . . . 8 (𝑦 = ((invgβ€˜π‘†)β€˜π‘₯) β†’ (πΊβ€˜π‘¦) = (πΊβ€˜((invgβ€˜π‘†)β€˜π‘₯)))
75, 6eqeq12d 2746 . . . . . . 7 (𝑦 = ((invgβ€˜π‘†)β€˜π‘₯) β†’ ((πΉβ€˜π‘¦) = (πΊβ€˜π‘¦) ↔ (πΉβ€˜((invgβ€˜π‘†)β€˜π‘₯)) = (πΊβ€˜((invgβ€˜π‘†)β€˜π‘₯))))
8 ghmgrp1 19134 . . . . . . . . . 10 (𝐹 ∈ (𝑆 GrpHom 𝑇) β†’ 𝑆 ∈ Grp)
98adantr 479 . . . . . . . . 9 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) β†’ 𝑆 ∈ Grp)
109adantr 479 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ (πΉβ€˜π‘₯) = (πΊβ€˜π‘₯))) β†’ 𝑆 ∈ Grp)
11 simprl 767 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ (πΉβ€˜π‘₯) = (πΊβ€˜π‘₯))) β†’ π‘₯ ∈ (Baseβ€˜π‘†))
12 eqid 2730 . . . . . . . . 9 (Baseβ€˜π‘†) = (Baseβ€˜π‘†)
13 eqid 2730 . . . . . . . . 9 (invgβ€˜π‘†) = (invgβ€˜π‘†)
1412, 13grpinvcl 18910 . . . . . . . 8 ((𝑆 ∈ Grp ∧ π‘₯ ∈ (Baseβ€˜π‘†)) β†’ ((invgβ€˜π‘†)β€˜π‘₯) ∈ (Baseβ€˜π‘†))
1510, 11, 14syl2anc 582 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ (πΉβ€˜π‘₯) = (πΊβ€˜π‘₯))) β†’ ((invgβ€˜π‘†)β€˜π‘₯) ∈ (Baseβ€˜π‘†))
16 simprr 769 . . . . . . . . 9 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ (πΉβ€˜π‘₯) = (πΊβ€˜π‘₯))) β†’ (πΉβ€˜π‘₯) = (πΊβ€˜π‘₯))
1716fveq2d 6896 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ (πΉβ€˜π‘₯) = (πΊβ€˜π‘₯))) β†’ ((invgβ€˜π‘‡)β€˜(πΉβ€˜π‘₯)) = ((invgβ€˜π‘‡)β€˜(πΊβ€˜π‘₯)))
18 eqid 2730 . . . . . . . . . 10 (invgβ€˜π‘‡) = (invgβ€˜π‘‡)
1912, 13, 18ghminv 19139 . . . . . . . . 9 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ π‘₯ ∈ (Baseβ€˜π‘†)) β†’ (πΉβ€˜((invgβ€˜π‘†)β€˜π‘₯)) = ((invgβ€˜π‘‡)β€˜(πΉβ€˜π‘₯)))
2019ad2ant2r 743 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ (πΉβ€˜π‘₯) = (πΊβ€˜π‘₯))) β†’ (πΉβ€˜((invgβ€˜π‘†)β€˜π‘₯)) = ((invgβ€˜π‘‡)β€˜(πΉβ€˜π‘₯)))
2112, 13, 18ghminv 19139 . . . . . . . . 9 ((𝐺 ∈ (𝑆 GrpHom 𝑇) ∧ π‘₯ ∈ (Baseβ€˜π‘†)) β†’ (πΊβ€˜((invgβ€˜π‘†)β€˜π‘₯)) = ((invgβ€˜π‘‡)β€˜(πΊβ€˜π‘₯)))
2221ad2ant2lr 744 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ (πΉβ€˜π‘₯) = (πΊβ€˜π‘₯))) β†’ (πΊβ€˜((invgβ€˜π‘†)β€˜π‘₯)) = ((invgβ€˜π‘‡)β€˜(πΊβ€˜π‘₯)))
2317, 20, 223eqtr4d 2780 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ (πΉβ€˜π‘₯) = (πΊβ€˜π‘₯))) β†’ (πΉβ€˜((invgβ€˜π‘†)β€˜π‘₯)) = (πΊβ€˜((invgβ€˜π‘†)β€˜π‘₯)))
247, 15, 23elrabd 3686 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ (πΉβ€˜π‘₯) = (πΊβ€˜π‘₯))) β†’ ((invgβ€˜π‘†)β€˜π‘₯) ∈ {𝑦 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘¦) = (πΊβ€˜π‘¦)})
2524expr 455 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ π‘₯ ∈ (Baseβ€˜π‘†)) β†’ ((πΉβ€˜π‘₯) = (πΊβ€˜π‘₯) β†’ ((invgβ€˜π‘†)β€˜π‘₯) ∈ {𝑦 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘¦) = (πΊβ€˜π‘¦)}))
2625ralrimiva 3144 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) β†’ βˆ€π‘₯ ∈ (Baseβ€˜π‘†)((πΉβ€˜π‘₯) = (πΊβ€˜π‘₯) β†’ ((invgβ€˜π‘†)β€˜π‘₯) ∈ {𝑦 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘¦) = (πΊβ€˜π‘¦)}))
27 fveq2 6892 . . . . . 6 (𝑦 = π‘₯ β†’ (πΉβ€˜π‘¦) = (πΉβ€˜π‘₯))
28 fveq2 6892 . . . . . 6 (𝑦 = π‘₯ β†’ (πΊβ€˜π‘¦) = (πΊβ€˜π‘₯))
2927, 28eqeq12d 2746 . . . . 5 (𝑦 = π‘₯ β†’ ((πΉβ€˜π‘¦) = (πΊβ€˜π‘¦) ↔ (πΉβ€˜π‘₯) = (πΊβ€˜π‘₯)))
3029ralrab 3690 . . . 4 (βˆ€π‘₯ ∈ {𝑦 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘¦) = (πΊβ€˜π‘¦)} ((invgβ€˜π‘†)β€˜π‘₯) ∈ {𝑦 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘¦) = (πΊβ€˜π‘¦)} ↔ βˆ€π‘₯ ∈ (Baseβ€˜π‘†)((πΉβ€˜π‘₯) = (πΊβ€˜π‘₯) β†’ ((invgβ€˜π‘†)β€˜π‘₯) ∈ {𝑦 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘¦) = (πΊβ€˜π‘¦)}))
3126, 30sylibr 233 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) β†’ βˆ€π‘₯ ∈ {𝑦 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘¦) = (πΊβ€˜π‘¦)} ((invgβ€˜π‘†)β€˜π‘₯) ∈ {𝑦 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘¦) = (πΊβ€˜π‘¦)})
32 eqid 2730 . . . . . . . 8 (Baseβ€˜π‘‡) = (Baseβ€˜π‘‡)
3312, 32ghmf 19136 . . . . . . 7 (𝐹 ∈ (𝑆 GrpHom 𝑇) β†’ 𝐹:(Baseβ€˜π‘†)⟢(Baseβ€˜π‘‡))
3433adantr 479 . . . . . 6 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) β†’ 𝐹:(Baseβ€˜π‘†)⟢(Baseβ€˜π‘‡))
3534ffnd 6719 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) β†’ 𝐹 Fn (Baseβ€˜π‘†))
3612, 32ghmf 19136 . . . . . . 7 (𝐺 ∈ (𝑆 GrpHom 𝑇) β†’ 𝐺:(Baseβ€˜π‘†)⟢(Baseβ€˜π‘‡))
3736adantl 480 . . . . . 6 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) β†’ 𝐺:(Baseβ€˜π‘†)⟢(Baseβ€˜π‘‡))
3837ffnd 6719 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) β†’ 𝐺 Fn (Baseβ€˜π‘†))
39 fndmin 7047 . . . . 5 ((𝐹 Fn (Baseβ€˜π‘†) ∧ 𝐺 Fn (Baseβ€˜π‘†)) β†’ dom (𝐹 ∩ 𝐺) = {𝑦 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘¦) = (πΊβ€˜π‘¦)})
4035, 38, 39syl2anc 582 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) β†’ dom (𝐹 ∩ 𝐺) = {𝑦 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘¦) = (πΊβ€˜π‘¦)})
41 eleq2 2820 . . . . 5 (dom (𝐹 ∩ 𝐺) = {𝑦 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘¦) = (πΊβ€˜π‘¦)} β†’ (((invgβ€˜π‘†)β€˜π‘₯) ∈ dom (𝐹 ∩ 𝐺) ↔ ((invgβ€˜π‘†)β€˜π‘₯) ∈ {𝑦 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘¦) = (πΊβ€˜π‘¦)}))
4241raleqbi1dv 3331 . . . 4 (dom (𝐹 ∩ 𝐺) = {𝑦 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘¦) = (πΊβ€˜π‘¦)} β†’ (βˆ€π‘₯ ∈ dom (𝐹 ∩ 𝐺)((invgβ€˜π‘†)β€˜π‘₯) ∈ dom (𝐹 ∩ 𝐺) ↔ βˆ€π‘₯ ∈ {𝑦 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘¦) = (πΊβ€˜π‘¦)} ((invgβ€˜π‘†)β€˜π‘₯) ∈ {𝑦 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘¦) = (πΊβ€˜π‘¦)}))
4340, 42syl 17 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) β†’ (βˆ€π‘₯ ∈ dom (𝐹 ∩ 𝐺)((invgβ€˜π‘†)β€˜π‘₯) ∈ dom (𝐹 ∩ 𝐺) ↔ βˆ€π‘₯ ∈ {𝑦 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘¦) = (πΊβ€˜π‘¦)} ((invgβ€˜π‘†)β€˜π‘₯) ∈ {𝑦 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘¦) = (πΊβ€˜π‘¦)}))
4431, 43mpbird 256 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) β†’ βˆ€π‘₯ ∈ dom (𝐹 ∩ 𝐺)((invgβ€˜π‘†)β€˜π‘₯) ∈ dom (𝐹 ∩ 𝐺))
4513issubg3 19062 . . 3 (𝑆 ∈ Grp β†’ (dom (𝐹 ∩ 𝐺) ∈ (SubGrpβ€˜π‘†) ↔ (dom (𝐹 ∩ 𝐺) ∈ (SubMndβ€˜π‘†) ∧ βˆ€π‘₯ ∈ dom (𝐹 ∩ 𝐺)((invgβ€˜π‘†)β€˜π‘₯) ∈ dom (𝐹 ∩ 𝐺))))
469, 45syl 17 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) β†’ (dom (𝐹 ∩ 𝐺) ∈ (SubGrpβ€˜π‘†) ↔ (dom (𝐹 ∩ 𝐺) ∈ (SubMndβ€˜π‘†) ∧ βˆ€π‘₯ ∈ dom (𝐹 ∩ 𝐺)((invgβ€˜π‘†)β€˜π‘₯) ∈ dom (𝐹 ∩ 𝐺))))
474, 44, 46mpbir2and 709 1 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) β†’ dom (𝐹 ∩ 𝐺) ∈ (SubGrpβ€˜π‘†))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059  {crab 3430   ∩ cin 3948  dom cdm 5677   Fn wfn 6539  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7413  Basecbs 17150   MndHom cmhm 18705  SubMndcsubmnd 18706  Grpcgrp 18857  invgcminusg 18858  SubGrpcsubg 19038   GrpHom cghm 19129
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 2701  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7729  ax-cnex 11170  ax-resscn 11171  ax-1cn 11172  ax-icn 11173  ax-addcl 11174  ax-addrcl 11175  ax-mulcl 11176  ax-mulrcl 11177  ax-mulcom 11178  ax-addass 11179  ax-mulass 11180  ax-distr 11181  ax-i2m1 11182  ax-1ne0 11183  ax-1rid 11184  ax-rnegex 11185  ax-rrecex 11186  ax-cnre 11187  ax-pre-lttri 11188  ax-pre-lttrn 11189  ax-pre-ltadd 11190  ax-pre-mulgt0 11191
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  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 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7860  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-er 8707  df-map 8826  df-en 8944  df-dom 8945  df-sdom 8946  df-pnf 11256  df-mnf 11257  df-xr 11258  df-ltxr 11259  df-le 11260  df-sub 11452  df-neg 11453  df-nn 12219  df-2 12281  df-sets 17103  df-slot 17121  df-ndx 17133  df-base 17151  df-ress 17180  df-plusg 17216  df-0g 17393  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-mhm 18707  df-submnd 18708  df-grp 18860  df-minusg 18861  df-subg 19041  df-ghm 19130
This theorem is referenced by:  rhmeql  20495  lmhmeql  20812
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