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Theorem ghmeqker 18649
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 4552 . . . . . 6 { 0 } = {(0g𝑇)}
43imaeq2i 5927 . . . . 5 (𝐹 “ { 0 }) = (𝐹 “ {(0g𝑇)})
51, 4eqtri 2765 . . . 4 𝐾 = (𝐹 “ {(0g𝑇)})
65eleq2i 2829 . . 3 ((𝑈 𝑉) ∈ 𝐾 ↔ (𝑈 𝑉) ∈ (𝐹 “ {(0g𝑇)}))
7 ghmeqker.b . . . . . . 7 𝐵 = (Base‘𝑆)
8 eqid 2737 . . . . . . 7 (Base‘𝑇) = (Base‘𝑇)
97, 8ghmf 18626 . . . . . 6 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:𝐵⟶(Base‘𝑇))
109ffnd 6546 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹 Fn 𝐵)
11103ad2ant1 1135 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → 𝐹 Fn 𝐵)
12 fniniseg 6880 . . . 4 (𝐹 Fn 𝐵 → ((𝑈 𝑉) ∈ (𝐹 “ {(0g𝑇)}) ↔ ((𝑈 𝑉) ∈ 𝐵 ∧ (𝐹‘(𝑈 𝑉)) = (0g𝑇))))
1311, 12syl 17 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → ((𝑈 𝑉) ∈ (𝐹 “ {(0g𝑇)}) ↔ ((𝑈 𝑉) ∈ 𝐵 ∧ (𝐹‘(𝑈 𝑉)) = (0g𝑇))))
146, 13syl5bb 286 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → ((𝑈 𝑉) ∈ 𝐾 ↔ ((𝑈 𝑉) ∈ 𝐵 ∧ (𝐹‘(𝑈 𝑉)) = (0g𝑇))))
15 ghmgrp1 18624 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
16 ghmeqker.m . . . . . 6 = (-g𝑆)
177, 16grpsubcl 18443 . . . . 5 ((𝑆 ∈ Grp ∧ 𝑈𝐵𝑉𝐵) → (𝑈 𝑉) ∈ 𝐵)
1815, 17syl3an1 1165 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → (𝑈 𝑉) ∈ 𝐵)
1918biantrurd 536 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → ((𝐹‘(𝑈 𝑉)) = (0g𝑇) ↔ ((𝑈 𝑉) ∈ 𝐵 ∧ (𝐹‘(𝑈 𝑉)) = (0g𝑇))))
20 eqid 2737 . . . . 5 (-g𝑇) = (-g𝑇)
217, 16, 20ghmsub 18630 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → (𝐹‘(𝑈 𝑉)) = ((𝐹𝑈)(-g𝑇)(𝐹𝑉)))
2221eqeq1d 2739 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → ((𝐹‘(𝑈 𝑉)) = (0g𝑇) ↔ ((𝐹𝑈)(-g𝑇)(𝐹𝑉)) = (0g𝑇)))
2319, 22bitr3d 284 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → (((𝑈 𝑉) ∈ 𝐵 ∧ (𝐹‘(𝑈 𝑉)) = (0g𝑇)) ↔ ((𝐹𝑈)(-g𝑇)(𝐹𝑉)) = (0g𝑇)))
24 ghmgrp2 18625 . . . 4 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
25243ad2ant1 1135 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → 𝑇 ∈ Grp)
2693ad2ant1 1135 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → 𝐹:𝐵⟶(Base‘𝑇))
27 simp2 1139 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → 𝑈𝐵)
2826, 27ffvelrnd 6905 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → (𝐹𝑈) ∈ (Base‘𝑇))
29 simp3 1140 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → 𝑉𝐵)
3026, 29ffvelrnd 6905 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → (𝐹𝑉) ∈ (Base‘𝑇))
31 eqid 2737 . . . 4 (0g𝑇) = (0g𝑇)
328, 31, 20grpsubeq0 18449 . . 3 ((𝑇 ∈ Grp ∧ (𝐹𝑈) ∈ (Base‘𝑇) ∧ (𝐹𝑉) ∈ (Base‘𝑇)) → (((𝐹𝑈)(-g𝑇)(𝐹𝑉)) = (0g𝑇) ↔ (𝐹𝑈) = (𝐹𝑉)))
3325, 28, 30, 32syl3anc 1373 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → (((𝐹𝑈)(-g𝑇)(𝐹𝑉)) = (0g𝑇) ↔ (𝐹𝑈) = (𝐹𝑉)))
3414, 23, 333bitrrd 309 1 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → ((𝐹𝑈) = (𝐹𝑉) ↔ (𝑈 𝑉) ∈ 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1089   = wceq 1543  wcel 2110  {csn 4541  ccnv 5550  cima 5554   Fn wfn 6375  wf 6376  cfv 6380  (class class class)co 7213  Basecbs 16760  0gc0g 16944  Grpcgrp 18365  -gcsg 18367   GrpHom cghm 18619
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-1st 7761  df-2nd 7762  df-0g 16946  df-mgm 18114  df-sgrp 18163  df-mnd 18174  df-grp 18368  df-minusg 18369  df-sbg 18370  df-ghm 18620
This theorem is referenced by:  kerf1ghm  19763  kercvrlsm  40611
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