Theorem ghmeql 18381

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)) → dom (𝐹 ∩ 𝐺) ∈ (SubGrp‘𝑆))

Proof of Theorem ghmeql

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ghmmhm 18368	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹 ∈ (𝑆 MndHom 𝑇))
2		ghmmhm 18368	. . 3 ⊢ (𝐺 ∈ (𝑆 GrpHom 𝑇) → 𝐺 ∈ (𝑆 MndHom 𝑇))
3		mhmeql 17990	. . 3 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → dom (𝐹 ∩ 𝐺) ∈ (SubMnd‘𝑆))
4	1, 2, 3	syl2an 597	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → dom (𝐹 ∩ 𝐺) ∈ (SubMnd‘𝑆))
5		fveq2 6670	. . . . . . . 8 ⊢ (𝑦 = ((invg‘𝑆)‘𝑥) → (𝐹‘𝑦) = (𝐹‘((invg‘𝑆)‘𝑥)))
6		fveq2 6670	. . . . . . . 8 ⊢ (𝑦 = ((invg‘𝑆)‘𝑥) → (𝐺‘𝑦) = (𝐺‘((invg‘𝑆)‘𝑥)))
7	5, 6	eqeq12d 2837	. . . . . . 7 ⊢ (𝑦 = ((invg‘𝑆)‘𝑥) → ((𝐹‘𝑦) = (𝐺‘𝑦) ↔ (𝐹‘((invg‘𝑆)‘𝑥)) = (𝐺‘((invg‘𝑆)‘𝑥))))
8		ghmgrp1 18360	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
9	8	adantr 483	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → 𝑆 ∈ Grp)
10	9	adantr 483	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) → 𝑆 ∈ Grp)
11		simprl 769	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) → 𝑥 ∈ (Base‘𝑆))
12		eqid 2821	. . . . . . . . 9 ⊢ (Base‘𝑆) = (Base‘𝑆)
13		eqid 2821	. . . . . . . . 9 ⊢ (invg‘𝑆) = (invg‘𝑆)
14	12, 13	grpinvcl 18151	. . . . . . . 8 ⊢ ((𝑆 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑆)) → ((invg‘𝑆)‘𝑥) ∈ (Base‘𝑆))
15	10, 11, 14	syl2anc 586	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) → ((invg‘𝑆)‘𝑥) ∈ (Base‘𝑆))
16		simprr 771	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) → (𝐹‘𝑥) = (𝐺‘𝑥))
17	16	fveq2d 6674	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) → ((invg‘𝑇)‘(𝐹‘𝑥)) = ((invg‘𝑇)‘(𝐺‘𝑥)))
18		eqid 2821	. . . . . . . . . 10 ⊢ (invg‘𝑇) = (invg‘𝑇)
19	12, 13, 18	ghminv 18365	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆)) → (𝐹‘((invg‘𝑆)‘𝑥)) = ((invg‘𝑇)‘(𝐹‘𝑥)))
20	19	ad2ant2r 745	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) → (𝐹‘((invg‘𝑆)‘𝑥)) = ((invg‘𝑇)‘(𝐹‘𝑥)))
21	12, 13, 18	ghminv 18365	. . . . . . . . 9 ⊢ ((𝐺 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆)) → (𝐺‘((invg‘𝑆)‘𝑥)) = ((invg‘𝑇)‘(𝐺‘𝑥)))
22	21	ad2ant2lr 746	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) → (𝐺‘((invg‘𝑆)‘𝑥)) = ((invg‘𝑇)‘(𝐺‘𝑥)))
23	17, 20, 22	3eqtr4d 2866	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) → (𝐹‘((invg‘𝑆)‘𝑥)) = (𝐺‘((invg‘𝑆)‘𝑥)))
24	7, 15, 23	elrabd 3682	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) → ((invg‘𝑆)‘𝑥) ∈ {𝑦 ∈ (Base‘𝑆) ∣ (𝐹‘𝑦) = (𝐺‘𝑦)})
25	24	expr 459	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑥 ∈ (Base‘𝑆)) → ((𝐹‘𝑥) = (𝐺‘𝑥) → ((invg‘𝑆)‘𝑥) ∈ {𝑦 ∈ (Base‘𝑆) ∣ (𝐹‘𝑦) = (𝐺‘𝑦)}))
26	25	ralrimiva 3182	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → ∀𝑥 ∈ (Base‘𝑆)((𝐹‘𝑥) = (𝐺‘𝑥) → ((invg‘𝑆)‘𝑥) ∈ {𝑦 ∈ (Base‘𝑆) ∣ (𝐹‘𝑦) = (𝐺‘𝑦)}))
27		fveq2 6670	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥))
28		fveq2 6670	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝐺‘𝑦) = (𝐺‘𝑥))
29	27, 28	eqeq12d 2837	. . . . 5 ⊢ (𝑦 = 𝑥 → ((𝐹‘𝑦) = (𝐺‘𝑦) ↔ (𝐹‘𝑥) = (𝐺‘𝑥)))
30	29	ralrab 3685	. . . 4 ⊢ (∀𝑥 ∈ {𝑦 ∈ (Base‘𝑆) ∣ (𝐹‘𝑦) = (𝐺‘𝑦)} ((invg‘𝑆)‘𝑥) ∈ {𝑦 ∈ (Base‘𝑆) ∣ (𝐹‘𝑦) = (𝐺‘𝑦)} ↔ ∀𝑥 ∈ (Base‘𝑆)((𝐹‘𝑥) = (𝐺‘𝑥) → ((invg‘𝑆)‘𝑥) ∈ {𝑦 ∈ (Base‘𝑆) ∣ (𝐹‘𝑦) = (𝐺‘𝑦)}))
31	26, 30	sylibr 236	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → ∀𝑥 ∈ {𝑦 ∈ (Base‘𝑆) ∣ (𝐹‘𝑦) = (𝐺‘𝑦)} ((invg‘𝑆)‘𝑥) ∈ {𝑦 ∈ (Base‘𝑆) ∣ (𝐹‘𝑦) = (𝐺‘𝑦)})
32		eqid 2821	. . . . . . . 8 ⊢ (Base‘𝑇) = (Base‘𝑇)
33	12, 32	ghmf 18362	. . . . . . 7 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
34	33	adantr 483	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
35	34	ffnd 6515	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → 𝐹 Fn (Base‘𝑆))
36	12, 32	ghmf 18362	. . . . . . 7 ⊢ (𝐺 ∈ (𝑆 GrpHom 𝑇) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
37	36	adantl 484	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
38	37	ffnd 6515	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → 𝐺 Fn (Base‘𝑆))
39		fndmin 6815	. . . . 5 ⊢ ((𝐹 Fn (Base‘𝑆) ∧ 𝐺 Fn (Base‘𝑆)) → dom (𝐹 ∩ 𝐺) = {𝑦 ∈ (Base‘𝑆) ∣ (𝐹‘𝑦) = (𝐺‘𝑦)})
40	35, 38, 39	syl2anc 586	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → dom (𝐹 ∩ 𝐺) = {𝑦 ∈ (Base‘𝑆) ∣ (𝐹‘𝑦) = (𝐺‘𝑦)})
41		eleq2 2901	. . . . 5 ⊢ (dom (𝐹 ∩ 𝐺) = {𝑦 ∈ (Base‘𝑆) ∣ (𝐹‘𝑦) = (𝐺‘𝑦)} → (((invg‘𝑆)‘𝑥) ∈ dom (𝐹 ∩ 𝐺) ↔ ((invg‘𝑆)‘𝑥) ∈ {𝑦 ∈ (Base‘𝑆) ∣ (𝐹‘𝑦) = (𝐺‘𝑦)}))
42	41	raleqbi1dv 3403	. . . 4 ⊢ (dom (𝐹 ∩ 𝐺) = {𝑦 ∈ (Base‘𝑆) ∣ (𝐹‘𝑦) = (𝐺‘𝑦)} → (∀𝑥 ∈ dom (𝐹 ∩ 𝐺)((invg‘𝑆)‘𝑥) ∈ dom (𝐹 ∩ 𝐺) ↔ ∀𝑥 ∈ {𝑦 ∈ (Base‘𝑆) ∣ (𝐹‘𝑦) = (𝐺‘𝑦)} ((invg‘𝑆)‘𝑥) ∈ {𝑦 ∈ (Base‘𝑆) ∣ (𝐹‘𝑦) = (𝐺‘𝑦)}))
43	40, 42	syl 17	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → (∀𝑥 ∈ dom (𝐹 ∩ 𝐺)((invg‘𝑆)‘𝑥) ∈ dom (𝐹 ∩ 𝐺) ↔ ∀𝑥 ∈ {𝑦 ∈ (Base‘𝑆) ∣ (𝐹‘𝑦) = (𝐺‘𝑦)} ((invg‘𝑆)‘𝑥) ∈ {𝑦 ∈ (Base‘𝑆) ∣ (𝐹‘𝑦) = (𝐺‘𝑦)}))
44	31, 43	mpbird 259	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → ∀𝑥 ∈ dom (𝐹 ∩ 𝐺)((invg‘𝑆)‘𝑥) ∈ dom (𝐹 ∩ 𝐺))
45	13	issubg3 18297	. . 3 ⊢ (𝑆 ∈ Grp → (dom (𝐹 ∩ 𝐺) ∈ (SubGrp‘𝑆) ↔ (dom (𝐹 ∩ 𝐺) ∈ (SubMnd‘𝑆) ∧ ∀𝑥 ∈ dom (𝐹 ∩ 𝐺)((invg‘𝑆)‘𝑥) ∈ dom (𝐹 ∩ 𝐺))))
46	9, 45	syl 17	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → (dom (𝐹 ∩ 𝐺) ∈ (SubGrp‘𝑆) ↔ (dom (𝐹 ∩ 𝐺) ∈ (SubMnd‘𝑆) ∧ ∀𝑥 ∈ dom (𝐹 ∩ 𝐺)((invg‘𝑆)‘𝑥) ∈ dom (𝐹 ∩ 𝐺))))
47	4, 44, 46	mpbir2and 711	1 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → dom (𝐹 ∩ 𝐺) ∈ (SubGrp‘𝑆))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3138  {crab 3142   ∩ cin 3935  dom cdm 5555   Fn wfn 6350  ⟶wf 6351  ‘cfv 6355  (class class class)co 7156  Basecbs 16483   MndHom cmhm 17954  SubMndcsubmnd 17955  Grpcgrp 18103  inv_gcminusg 18104  SubGrpcsubg 18273   GrpHom cghm 18355

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-er 8289  df-map 8408  df-en 8510  df-dom 8511  df-sdom 8512  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-nn 11639  df-2 11701  df-ndx 16486  df-slot 16487  df-base 16489  df-sets 16490  df-ress 16491  df-plusg 16578  df-0g 16715  df-mgm 17852  df-sgrp 17901  df-mnd 17912  df-mhm 17956  df-submnd 17957  df-grp 18106  df-minusg 18107  df-subg 18276  df-ghm 18356

This theorem is referenced by:  rhmeql  19565  lmhmeql  19827