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Theorem rngokerinj 37505
Description: A ring homomorphism is injective if and only if its kernel is zero. (Contributed by Jeff Madsen, 16-Jun-2011.)
Hypotheses
Ref Expression
rngkerinj.1 𝐺 = (1st𝑅)
rngkerinj.2 𝑋 = ran 𝐺
rngkerinj.3 𝑊 = (GId‘𝐺)
rngkerinj.4 𝐽 = (1st𝑆)
rngkerinj.5 𝑌 = ran 𝐽
rngkerinj.6 𝑍 = (GId‘𝐽)
Assertion
Ref Expression
rngokerinj ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (𝐹:𝑋1-1𝑌 ↔ (𝐹 “ {𝑍}) = {𝑊}))

Proof of Theorem rngokerinj
StepHypRef Expression
1 eqid 2725 . . . 4 (1st𝑅) = (1st𝑅)
21rngogrpo 37440 . . 3 (𝑅 ∈ RingOps → (1st𝑅) ∈ GrpOp)
323ad2ant1 1130 . 2 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (1st𝑅) ∈ GrpOp)
4 eqid 2725 . . . 4 (1st𝑆) = (1st𝑆)
54rngogrpo 37440 . . 3 (𝑆 ∈ RingOps → (1st𝑆) ∈ GrpOp)
653ad2ant2 1131 . 2 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (1st𝑆) ∈ GrpOp)
71, 4rngogrphom 37501 . 2 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → 𝐹 ∈ ((1st𝑅) GrpOpHom (1st𝑆)))
8 rngkerinj.2 . . . 4 𝑋 = ran 𝐺
9 rngkerinj.1 . . . . 5 𝐺 = (1st𝑅)
109rneqi 5933 . . . 4 ran 𝐺 = ran (1st𝑅)
118, 10eqtri 2753 . . 3 𝑋 = ran (1st𝑅)
12 rngkerinj.3 . . . 4 𝑊 = (GId‘𝐺)
139fveq2i 6895 . . . 4 (GId‘𝐺) = (GId‘(1st𝑅))
1412, 13eqtri 2753 . . 3 𝑊 = (GId‘(1st𝑅))
15 rngkerinj.5 . . . 4 𝑌 = ran 𝐽
16 rngkerinj.4 . . . . 5 𝐽 = (1st𝑆)
1716rneqi 5933 . . . 4 ran 𝐽 = ran (1st𝑆)
1815, 17eqtri 2753 . . 3 𝑌 = ran (1st𝑆)
19 rngkerinj.6 . . . 4 𝑍 = (GId‘𝐽)
2016fveq2i 6895 . . . 4 (GId‘𝐽) = (GId‘(1st𝑆))
2119, 20eqtri 2753 . . 3 𝑍 = (GId‘(1st𝑆))
2211, 14, 18, 21grpokerinj 37423 . 2 (((1st𝑅) ∈ GrpOp ∧ (1st𝑆) ∈ GrpOp ∧ 𝐹 ∈ ((1st𝑅) GrpOpHom (1st𝑆))) → (𝐹:𝑋1-1𝑌 ↔ (𝐹 “ {𝑍}) = {𝑊}))
233, 6, 7, 22syl3anc 1368 1 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (𝐹:𝑋1-1𝑌 ↔ (𝐹 “ {𝑍}) = {𝑊}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3a 1084   = wceq 1533  wcel 2098  {csn 4624  ccnv 5671  ran crn 5673  cima 5675  1-1wf1 6540  cfv 6543  (class class class)co 7416  1st c1st 7989  GrpOpcgr 30343  GIdcgi 30344   GrpOpHom cghomOLD 37413  RingOpscrngo 37424   RingOpsHom crngohom 37490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7991  df-2nd 7992  df-map 8845  df-grpo 30347  df-gid 30348  df-ginv 30349  df-gdiv 30350  df-ablo 30399  df-ghomOLD 37414  df-rngo 37425  df-rngohom 37493
This theorem is referenced by: (None)
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