Theorem ghmeqker 19170

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 4589	. . . . . 6 ⊢ { 0 } = {(0g‘𝑇)}
4	3	imaeq2i 6015	. . . . 5 ⊢ (◡𝐹 “ { 0 }) = (◡𝐹 “ {(0g‘𝑇)})
5	1, 4	eqtri 2757	. . . 4 ⊢ 𝐾 = (◡𝐹 “ {(0g‘𝑇)})
6	5	eleq2i 2826	. . 3 ⊢ ((𝑈 − 𝑉) ∈ 𝐾 ↔ (𝑈 − 𝑉) ∈ (◡𝐹 “ {(0g‘𝑇)}))
7		ghmeqker.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑆)
8		eqid 2734	. . . . . . 7 ⊢ (Base‘𝑇) = (Base‘𝑇)
9	7, 8	ghmf 19147	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:𝐵⟶(Base‘𝑇))
10	9	ffnd 6661	. . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹 Fn 𝐵)
11	10	3ad2ant1 1133	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵) → 𝐹 Fn 𝐵)
12		fniniseg 7003	. . . 4 ⊢ (𝐹 Fn 𝐵 → ((𝑈 − 𝑉) ∈ (◡𝐹 “ {(0g‘𝑇)}) ↔ ((𝑈 − 𝑉) ∈ 𝐵 ∧ (𝐹‘(𝑈 − 𝑉)) = (0g‘𝑇))))
13	11, 12	syl 17	. . 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 283	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵) → ((𝑈 − 𝑉) ∈ 𝐾 ↔ ((𝑈 − 𝑉) ∈ 𝐵 ∧ (𝐹‘(𝑈 − 𝑉)) = (0g‘𝑇))))
15		ghmgrp1 19145	. . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
16		ghmeqker.m	. . . . . 6 ⊢ − = (-g‘𝑆)
17	7, 16	grpsubcl 18948	. . . . 5 ⊢ ((𝑆 ∈ Grp ∧ 𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵) → (𝑈 − 𝑉) ∈ 𝐵)
18	15, 17	syl3an1 1163	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵) → (𝑈 − 𝑉) ∈ 𝐵)
19	18	biantrurd 532	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵) → ((𝐹‘(𝑈 − 𝑉)) = (0g‘𝑇) ↔ ((𝑈 − 𝑉) ∈ 𝐵 ∧ (𝐹‘(𝑈 − 𝑉)) = (0g‘𝑇))))
20		eqid 2734	. . . . 5 ⊢ (-g‘𝑇) = (-g‘𝑇)
21	7, 16, 20	ghmsub 19151	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵) → (𝐹‘(𝑈 − 𝑉)) = ((𝐹‘𝑈)(-g‘𝑇)(𝐹‘𝑉)))
22	21	eqeq1d 2736	. . 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 281	. 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 19146	. . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
25	24	3ad2ant1 1133	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵) → 𝑇 ∈ Grp)
26	9	3ad2ant1 1133	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵) → 𝐹:𝐵⟶(Base‘𝑇))
27		simp2 1137	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵) → 𝑈 ∈ 𝐵)
28	26, 27	ffvelcdmd 7028	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵) → (𝐹‘𝑈) ∈ (Base‘𝑇))
29		simp3 1138	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵) → 𝑉 ∈ 𝐵)
30	26, 29	ffvelcdmd 7028	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵) → (𝐹‘𝑉) ∈ (Base‘𝑇))
31		eqid 2734	. . . 4 ⊢ (0g‘𝑇) = (0g‘𝑇)
32	8, 31, 20	grpsubeq0 18954	. . 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 1373	. 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 306	1 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵) → ((𝐹‘𝑈) = (𝐹‘𝑉) ↔ (𝑈 − 𝑉) ∈ 𝐾))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  {csn 4578  ^◡ccnv 5621   “ cima 5625   Fn wfn 6485  ⟶wf 6486  ‘cfv 6490  (class class class)co 7356  Basecbs 17134  0_gc0g 17357  Grpcgrp 18861  -_gcsg 18863   GrpHom cghm 19139

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-fv 6498  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-map 8763  df-0g 17359  df-mgm 18563  df-sgrp 18642  df-mnd 18658  df-grp 18864  df-minusg 18865  df-sbg 18866  df-ghm 19140

This theorem is referenced by:  kerf1ghm  19174  kercvrlsm  43267