Theorem ghmeqker 19361

Description: Two source points map to the same destination point under a group homomorphism iff their difference belongs to the kernel. (Contributed by Stefan O'Rear, 31-Dec-2014.)

Hypotheses

Ref	Expression
ghmeqker.b	⊢ 𝐵 = (Base‘𝑆)
ghmeqker.z	⊢ 0 = (0g‘𝑇)
ghmeqker.k	⊢ 𝐾 = (◡𝐹 “ { 0 })
ghmeqker.m	⊢ − = (-g‘𝑆)

Assertion

Ref	Expression
ghmeqker	⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵) → ((𝐹‘𝑈) = (𝐹‘𝑉) ↔ (𝑈 − 𝑉) ∈ 𝐾))

Proof of Theorem ghmeqker

Step	Hyp	Ref	Expression
1		ghmeqker.k	. . . . 5 ⊢ 𝐾 = (◡𝐹 “ { 0 })
2		ghmeqker.z	. . . . . . 7 ⊢ 0 = (0g‘𝑇)
3	2	sneqi 4602	. . . . . 6 ⊢ { 0 } = {(0g‘𝑇)}
4	3	imaeq2i 6062	. . . . 5 ⊢ (◡𝐹 “ { 0 }) = (◡𝐹 “ {(0g‘𝑇)})
5	1, 4	eqtri 2788	. . . 4 ⊢ 𝐾 = (◡𝐹 “ {(0g‘𝑇)})
6	5	eleq2i 2857	. . 3 ⊢ ((𝑈 − 𝑉) ∈ 𝐾 ↔ (𝑈 − 𝑉) ∈ (◡𝐹 “ {(0g‘𝑇)}))
7		ghmeqker.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑆)
8		eqid 2765	. . . . . . 7 ⊢ (Base‘𝑇) = (Base‘𝑇)
9	7, 8	ghmf 19338	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:𝐵⟶(Base‘𝑇))
10	9	ffnd 6710	. . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹 Fn 𝐵)
11	10	3ad2ant1 1151	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵) → 𝐹 Fn 𝐵)
12		fniniseg 7059	. . . 4 ⊢ (𝐹 Fn 𝐵 → ((𝑈 − 𝑉) ∈ (◡𝐹 “ {(0g‘𝑇)}) ↔ ((𝑈 − 𝑉) ∈ 𝐵 ∧ (𝐹‘(𝑈 − 𝑉)) = (0g‘𝑇))))
13	11, 12	syl 18	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵) → ((𝑈 − 𝑉) ∈ (◡𝐹 “ {(0g‘𝑇)}) ↔ ((𝑈 − 𝑉) ∈ 𝐵 ∧ (𝐹‘(𝑈 − 𝑉)) = (0g‘𝑇))))
14	6, 13	bitrid 286	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵) → ((𝑈 − 𝑉) ∈ 𝐾 ↔ ((𝑈 − 𝑉) ∈ 𝐵 ∧ (𝐹‘(𝑈 − 𝑉)) = (0g‘𝑇))))
15		ghmgrp1 19336	. . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
16		ghmeqker.m	. . . . . 6 ⊢ − = (-g‘𝑆)
17	7, 16	grpsubcl 19134	. . . . 5 ⊢ ((𝑆 ∈ Grp ∧ 𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵) → (𝑈 − 𝑉) ∈ 𝐵)
18	15, 17	syl3an1 1181	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵) → (𝑈 − 𝑉) ∈ 𝐵)
19	18	biantrurd 542	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵) → ((𝐹‘(𝑈 − 𝑉)) = (0g‘𝑇) ↔ ((𝑈 − 𝑉) ∈ 𝐵 ∧ (𝐹‘(𝑈 − 𝑉)) = (0g‘𝑇))))
20		eqid 2765	. . . . 5 ⊢ (-g‘𝑇) = (-g‘𝑇)
21	7, 16, 20	ghmsub 19342	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵) → (𝐹‘(𝑈 − 𝑉)) = ((𝐹‘𝑈)(-g‘𝑇)(𝐹‘𝑉)))
22	21	eqeq1d 2767	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵) → ((𝐹‘(𝑈 − 𝑉)) = (0g‘𝑇) ↔ ((𝐹‘𝑈)(-g‘𝑇)(𝐹‘𝑉)) = (0g‘𝑇)))
23	19, 22	bitr3d 284	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵) → (((𝑈 − 𝑉) ∈ 𝐵 ∧ (𝐹‘(𝑈 − 𝑉)) = (0g‘𝑇)) ↔ ((𝐹‘𝑈)(-g‘𝑇)(𝐹‘𝑉)) = (0g‘𝑇)))
24		ghmgrp2 19337	. . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
25	24	3ad2ant1 1151	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵) → 𝑇 ∈ Grp)
26	9	3ad2ant1 1151	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵) → 𝐹:𝐵⟶(Base‘𝑇))
27		simp2 1155	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵) → 𝑈 ∈ 𝐵)
28	26, 27	ffvelcdmd 7084	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵) → (𝐹‘𝑈) ∈ (Base‘𝑇))
29		simp3 1156	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵) → 𝑉 ∈ 𝐵)
30	26, 29	ffvelcdmd 7084	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵) → (𝐹‘𝑉) ∈ (Base‘𝑇))
31		eqid 2765	. . . 4 ⊢ (0g‘𝑇) = (0g‘𝑇)
32	8, 31, 20	grpsubeq0 19140	. . 3 ⊢ ((𝑇 ∈ Grp ∧ (𝐹‘𝑈) ∈ (Base‘𝑇) ∧ (𝐹‘𝑉) ∈ (Base‘𝑇)) → (((𝐹‘𝑈)(-g‘𝑇)(𝐹‘𝑉)) = (0g‘𝑇) ↔ (𝐹‘𝑈) = (𝐹‘𝑉)))
33	25, 28, 30, 32	syl3anc 1398	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵) → (((𝐹‘𝑈)(-g‘𝑇)(𝐹‘𝑉)) = (0g‘𝑇) ↔ (𝐹‘𝑈) = (𝐹‘𝑉)))
34	14, 23, 33	3bitrrd 309	1 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵) → ((𝐹‘𝑈) = (𝐹‘𝑉) ↔ (𝑈 − 𝑉) ∈ 𝐾))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  {csn 4591  ^◡ccnv 5662   “ cima 5666   Fn wfn 6535  ⟶wf 6536  ‘cfv 6540  (class class class)co 7419  Basecbs 17295  0_gc0g 17518  Grpcgrp 19048  -_gcsg 19050   GrpHom cghm 19331

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7992  df-2nd 7993  df-map 8832  df-0g 17520  df-mgm 18724  df-sgrp 18813  df-mnd 18829  df-grp 19051  df-minusg 19052  df-sbg 19053  df-ghm 19332

This theorem is used by:  kerf1ghm  19365  kercvrlsm  43887