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Theorem ghmeqker 17681
 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 = (0g𝑇)
ghmeqker.k 𝐾 = (𝐹 “ { 0 })
ghmeqker.m = (-g𝑆)
Assertion
Ref Expression
ghmeqker ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → ((𝐹𝑈) = (𝐹𝑉) ↔ (𝑈 𝑉) ∈ 𝐾))

Proof of Theorem ghmeqker
StepHypRef Expression
1 ghmeqker.k . . . . 5 𝐾 = (𝐹 “ { 0 })
2 ghmeqker.z . . . . . . 7 0 = (0g𝑇)
32sneqi 4186 . . . . . 6 { 0 } = {(0g𝑇)}
43imaeq2i 5462 . . . . 5 (𝐹 “ { 0 }) = (𝐹 “ {(0g𝑇)})
51, 4eqtri 2643 . . . 4 𝐾 = (𝐹 “ {(0g𝑇)})
65eleq2i 2692 . . 3 ((𝑈 𝑉) ∈ 𝐾 ↔ (𝑈 𝑉) ∈ (𝐹 “ {(0g𝑇)}))
7 ghmeqker.b . . . . . . 7 𝐵 = (Base‘𝑆)
8 eqid 2621 . . . . . . 7 (Base‘𝑇) = (Base‘𝑇)
97, 8ghmf 17658 . . . . . 6 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:𝐵⟶(Base‘𝑇))
10 ffn 6043 . . . . . 6 (𝐹:𝐵⟶(Base‘𝑇) → 𝐹 Fn 𝐵)
119, 10syl 17 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹 Fn 𝐵)
12113ad2ant1 1081 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → 𝐹 Fn 𝐵)
13 fniniseg 6336 . . . 4 (𝐹 Fn 𝐵 → ((𝑈 𝑉) ∈ (𝐹 “ {(0g𝑇)}) ↔ ((𝑈 𝑉) ∈ 𝐵 ∧ (𝐹‘(𝑈 𝑉)) = (0g𝑇))))
1412, 13syl 17 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → ((𝑈 𝑉) ∈ (𝐹 “ {(0g𝑇)}) ↔ ((𝑈 𝑉) ∈ 𝐵 ∧ (𝐹‘(𝑈 𝑉)) = (0g𝑇))))
156, 14syl5bb 272 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → ((𝑈 𝑉) ∈ 𝐾 ↔ ((𝑈 𝑉) ∈ 𝐵 ∧ (𝐹‘(𝑈 𝑉)) = (0g𝑇))))
16 ghmgrp1 17656 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
17 ghmeqker.m . . . . . 6 = (-g𝑆)
187, 17grpsubcl 17489 . . . . 5 ((𝑆 ∈ Grp ∧ 𝑈𝐵𝑉𝐵) → (𝑈 𝑉) ∈ 𝐵)
1916, 18syl3an1 1358 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → (𝑈 𝑉) ∈ 𝐵)
2019biantrurd 529 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → ((𝐹‘(𝑈 𝑉)) = (0g𝑇) ↔ ((𝑈 𝑉) ∈ 𝐵 ∧ (𝐹‘(𝑈 𝑉)) = (0g𝑇))))
21 eqid 2621 . . . . 5 (-g𝑇) = (-g𝑇)
227, 17, 21ghmsub 17662 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → (𝐹‘(𝑈 𝑉)) = ((𝐹𝑈)(-g𝑇)(𝐹𝑉)))
2322eqeq1d 2623 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → ((𝐹‘(𝑈 𝑉)) = (0g𝑇) ↔ ((𝐹𝑈)(-g𝑇)(𝐹𝑉)) = (0g𝑇)))
2420, 23bitr3d 270 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → (((𝑈 𝑉) ∈ 𝐵 ∧ (𝐹‘(𝑈 𝑉)) = (0g𝑇)) ↔ ((𝐹𝑈)(-g𝑇)(𝐹𝑉)) = (0g𝑇)))
25 ghmgrp2 17657 . . . 4 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
26253ad2ant1 1081 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → 𝑇 ∈ Grp)
2793ad2ant1 1081 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → 𝐹:𝐵⟶(Base‘𝑇))
28 simp2 1061 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → 𝑈𝐵)
2927, 28ffvelrnd 6358 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → (𝐹𝑈) ∈ (Base‘𝑇))
30 simp3 1062 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → 𝑉𝐵)
3127, 30ffvelrnd 6358 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → (𝐹𝑉) ∈ (Base‘𝑇))
32 eqid 2621 . . . 4 (0g𝑇) = (0g𝑇)
338, 32, 21grpsubeq0 17495 . . 3 ((𝑇 ∈ Grp ∧ (𝐹𝑈) ∈ (Base‘𝑇) ∧ (𝐹𝑉) ∈ (Base‘𝑇)) → (((𝐹𝑈)(-g𝑇)(𝐹𝑉)) = (0g𝑇) ↔ (𝐹𝑈) = (𝐹𝑉)))
3426, 29, 31, 33syl3anc 1325 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → (((𝐹𝑈)(-g𝑇)(𝐹𝑉)) = (0g𝑇) ↔ (𝐹𝑈) = (𝐹𝑉)))
3515, 24, 343bitrrd 295 1 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → ((𝐹𝑈) = (𝐹𝑉) ↔ (𝑈 𝑉) ∈ 𝐾))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1482   ∈ wcel 1989  {csn 4175  ◡ccnv 5111   “ cima 5115   Fn wfn 5881  ⟶wf 5882  ‘cfv 5886  (class class class)co 6647  Basecbs 15851  0gc0g 16094  Grpcgrp 17416  -gcsg 17418   GrpHom cghm 17651 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rmo 2919  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-riota 6608  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-1st 7165  df-2nd 7166  df-0g 16096  df-mgm 17236  df-sgrp 17278  df-mnd 17289  df-grp 17419  df-minusg 17420  df-sbg 17421  df-ghm 17652 This theorem is referenced by:  kerf1hrm  18737  kercvrlsm  37479
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