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Theorem ghmeqker 19274
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 4642 . . . . . 6 { 0 } = {(0g𝑇)}
43imaeq2i 6078 . . . . 5 (𝐹 “ { 0 }) = (𝐹 “ {(0g𝑇)})
51, 4eqtri 2763 . . . 4 𝐾 = (𝐹 “ {(0g𝑇)})
65eleq2i 2831 . . 3 ((𝑈 𝑉) ∈ 𝐾 ↔ (𝑈 𝑉) ∈ (𝐹 “ {(0g𝑇)}))
7 ghmeqker.b . . . . . . 7 𝐵 = (Base‘𝑆)
8 eqid 2735 . . . . . . 7 (Base‘𝑇) = (Base‘𝑇)
97, 8ghmf 19251 . . . . . 6 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:𝐵⟶(Base‘𝑇))
109ffnd 6738 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹 Fn 𝐵)
11103ad2ant1 1132 . . . 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 19249 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
16 ghmeqker.m . . . . . 6 = (-g𝑆)
177, 16grpsubcl 19051 . . . . 5 ((𝑆 ∈ Grp ∧ 𝑈𝐵𝑉𝐵) → (𝑈 𝑉) ∈ 𝐵)
1815, 17syl3an1 1162 . . . 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 2735 . . . . 5 (-g𝑇) = (-g𝑇)
217, 16, 20ghmsub 19255 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → (𝐹‘(𝑈 𝑉)) = ((𝐹𝑈)(-g𝑇)(𝐹𝑉)))
2221eqeq1d 2737 . . 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 19250 . . . 4 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
25243ad2ant1 1132 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → 𝑇 ∈ Grp)
2693ad2ant1 1132 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → 𝐹:𝐵⟶(Base‘𝑇))
27 simp2 1136 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → 𝑈𝐵)
2826, 27ffvelcdmd 7105 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → (𝐹𝑈) ∈ (Base‘𝑇))
29 simp3 1137 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → 𝑉𝐵)
3026, 29ffvelcdmd 7105 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → (𝐹𝑉) ∈ (Base‘𝑇))
31 eqid 2735 . . . 4 (0g𝑇) = (0g𝑇)
328, 31, 20grpsubeq0 19057 . . 3 ((𝑇 ∈ Grp ∧ (𝐹𝑈) ∈ (Base‘𝑇) ∧ (𝐹𝑉) ∈ (Base‘𝑇)) → (((𝐹𝑈)(-g𝑇)(𝐹𝑉)) = (0g𝑇) ↔ (𝐹𝑈) = (𝐹𝑉)))
3325, 28, 30, 32syl3anc 1370 . 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 1086   = wceq 1537  wcel 2106  {csn 4631  ccnv 5688  cima 5692   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  Basecbs 17245  0gc0g 17486  Grpcgrp 18964  -gcsg 18966   GrpHom cghm 19243
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-sbg 18969  df-ghm 19244
This theorem is referenced by:  kerf1ghm  19278  kercvrlsm  43072
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