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Theorem ghmeqker 19261
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 4637 . . . . . 6 { 0 } = {(0g𝑇)}
43imaeq2i 6076 . . . . 5 (𝐹 “ { 0 }) = (𝐹 “ {(0g𝑇)})
51, 4eqtri 2765 . . . 4 𝐾 = (𝐹 “ {(0g𝑇)})
65eleq2i 2833 . . 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 19238 . . . . . 6 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:𝐵⟶(Base‘𝑇))
109ffnd 6737 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹 Fn 𝐵)
11103ad2ant1 1134 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → 𝐹 Fn 𝐵)
12 fniniseg 7080 . . . 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, 13bitrid 283 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → ((𝑈 𝑉) ∈ 𝐾 ↔ ((𝑈 𝑉) ∈ 𝐵 ∧ (𝐹‘(𝑈 𝑉)) = (0g𝑇))))
15 ghmgrp1 19236 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
16 ghmeqker.m . . . . . 6 = (-g𝑆)
177, 16grpsubcl 19038 . . . . 5 ((𝑆 ∈ Grp ∧ 𝑈𝐵𝑉𝐵) → (𝑈 𝑉) ∈ 𝐵)
1815, 17syl3an1 1164 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → (𝑈 𝑉) ∈ 𝐵)
1918biantrurd 532 . . 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 19242 . . . 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 281 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → (((𝑈 𝑉) ∈ 𝐵 ∧ (𝐹‘(𝑈 𝑉)) = (0g𝑇)) ↔ ((𝐹𝑈)(-g𝑇)(𝐹𝑉)) = (0g𝑇)))
24 ghmgrp2 19237 . . . 4 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
25243ad2ant1 1134 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → 𝑇 ∈ Grp)
2693ad2ant1 1134 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → 𝐹:𝐵⟶(Base‘𝑇))
27 simp2 1138 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → 𝑈𝐵)
2826, 27ffvelcdmd 7105 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → (𝐹𝑈) ∈ (Base‘𝑇))
29 simp3 1139 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → 𝑉𝐵)
3026, 29ffvelcdmd 7105 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → (𝐹𝑉) ∈ (Base‘𝑇))
31 eqid 2737 . . . 4 (0g𝑇) = (0g𝑇)
328, 31, 20grpsubeq0 19044 . . 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 306 1 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → ((𝐹𝑈) = (𝐹𝑉) ↔ (𝑈 𝑉) ∈ 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  {csn 4626  ccnv 5684  cima 5688   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  Basecbs 17247  0gc0g 17484  Grpcgrp 18951  -gcsg 18953   GrpHom cghm 19230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955  df-sbg 18956  df-ghm 19231
This theorem is referenced by:  kerf1ghm  19265  kercvrlsm  43095
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