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Theorem ghmeqker 18380
 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 4536 . . . . . 6 { 0 } = {(0g𝑇)}
43imaeq2i 5894 . . . . 5 (𝐹 “ { 0 }) = (𝐹 “ {(0g𝑇)})
51, 4eqtri 2821 . . . 4 𝐾 = (𝐹 “ {(0g𝑇)})
65eleq2i 2881 . . 3 ((𝑈 𝑉) ∈ 𝐾 ↔ (𝑈 𝑉) ∈ (𝐹 “ {(0g𝑇)}))
7 ghmeqker.b . . . . . . 7 𝐵 = (Base‘𝑆)
8 eqid 2798 . . . . . . 7 (Base‘𝑇) = (Base‘𝑇)
97, 8ghmf 18357 . . . . . 6 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:𝐵⟶(Base‘𝑇))
109ffnd 6488 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹 Fn 𝐵)
11103ad2ant1 1130 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → 𝐹 Fn 𝐵)
12 fniniseg 6807 . . . 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 18355 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
16 ghmeqker.m . . . . . 6 = (-g𝑆)
177, 16grpsubcl 18174 . . . . 5 ((𝑆 ∈ Grp ∧ 𝑈𝐵𝑉𝐵) → (𝑈 𝑉) ∈ 𝐵)
1815, 17syl3an1 1160 . . . 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 2798 . . . . 5 (-g𝑇) = (-g𝑇)
217, 16, 20ghmsub 18361 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → (𝐹‘(𝑈 𝑉)) = ((𝐹𝑈)(-g𝑇)(𝐹𝑉)))
2221eqeq1d 2800 . . 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 18356 . . . 4 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
25243ad2ant1 1130 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → 𝑇 ∈ Grp)
2693ad2ant1 1130 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → 𝐹:𝐵⟶(Base‘𝑇))
27 simp2 1134 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → 𝑈𝐵)
2826, 27ffvelrnd 6829 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → (𝐹𝑈) ∈ (Base‘𝑇))
29 simp3 1135 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → 𝑉𝐵)
3026, 29ffvelrnd 6829 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → (𝐹𝑉) ∈ (Base‘𝑇))
31 eqid 2798 . . . 4 (0g𝑇) = (0g𝑇)
328, 31, 20grpsubeq0 18180 . . 3 ((𝑇 ∈ Grp ∧ (𝐹𝑈) ∈ (Base‘𝑇) ∧ (𝐹𝑉) ∈ (Base‘𝑇)) → (((𝐹𝑈)(-g𝑇)(𝐹𝑉)) = (0g𝑇) ↔ (𝐹𝑈) = (𝐹𝑉)))
3325, 28, 30, 32syl3anc 1368 . 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 1084   = wceq 1538   ∈ wcel 2111  {csn 4525  ◡ccnv 5518   “ cima 5522   Fn wfn 6319  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135  Basecbs 16477  0gc0g 16707  Grpcgrp 18097  -gcsg 18099   GrpHom cghm 18350 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7673  df-2nd 7674  df-0g 16709  df-mgm 17846  df-sgrp 17895  df-mnd 17906  df-grp 18100  df-minusg 18101  df-sbg 18102  df-ghm 18351 This theorem is referenced by:  kerf1ghm  19494  kercvrlsm  40042
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