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Theorem rngokerinj 32944
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 RngHom 𝑆)) → (𝐹:𝑋1-1𝑌 ↔ (𝐹 “ {𝑍}) = {𝑊}))

Proof of Theorem rngokerinj
StepHypRef Expression
1 eqid 2610 . . . 4 (1st𝑅) = (1st𝑅)
21rngogrpo 32879 . . 3 (𝑅 ∈ RingOps → (1st𝑅) ∈ GrpOp)
323ad2ant1 1075 . 2 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (1st𝑅) ∈ GrpOp)
4 eqid 2610 . . . 4 (1st𝑆) = (1st𝑆)
54rngogrpo 32879 . . 3 (𝑆 ∈ RingOps → (1st𝑆) ∈ GrpOp)
653ad2ant2 1076 . 2 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (1st𝑆) ∈ GrpOp)
71, 4rngogrphom 32940 . 2 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐹 ∈ ((1st𝑅) GrpOpHom (1st𝑆)))
8 rngkerinj.2 . . . 4 𝑋 = ran 𝐺
9 rngkerinj.1 . . . . 5 𝐺 = (1st𝑅)
109rneqi 5273 . . . 4 ran 𝐺 = ran (1st𝑅)
118, 10eqtri 2632 . . 3 𝑋 = ran (1st𝑅)
12 rngkerinj.3 . . . 4 𝑊 = (GId‘𝐺)
139fveq2i 6106 . . . 4 (GId‘𝐺) = (GId‘(1st𝑅))
1412, 13eqtri 2632 . . 3 𝑊 = (GId‘(1st𝑅))
15 rngkerinj.5 . . . 4 𝑌 = ran 𝐽
16 rngkerinj.4 . . . . 5 𝐽 = (1st𝑆)
1716rneqi 5273 . . . 4 ran 𝐽 = ran (1st𝑆)
1815, 17eqtri 2632 . . 3 𝑌 = ran (1st𝑆)
19 rngkerinj.6 . . . 4 𝑍 = (GId‘𝐽)
2016fveq2i 6106 . . . 4 (GId‘𝐽) = (GId‘(1st𝑆))
2119, 20eqtri 2632 . . 3 𝑍 = (GId‘(1st𝑆))
2211, 14, 18, 21grpokerinj 32862 . 2 (((1st𝑅) ∈ GrpOp ∧ (1st𝑆) ∈ GrpOp ∧ 𝐹 ∈ ((1st𝑅) GrpOpHom (1st𝑆))) → (𝐹:𝑋1-1𝑌 ↔ (𝐹 “ {𝑍}) = {𝑊}))
233, 6, 7, 22syl3anc 1318 1 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹:𝑋1-1𝑌 ↔ (𝐹 “ {𝑍}) = {𝑊}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  w3a 1031   = wceq 1475  wcel 1977  {csn 4125  ccnv 5037  ran crn 5039  cima 5041  1-1wf1 5801  cfv 5804  (class class class)co 6549  1st c1st 7057  GrpOpcgr 26727  GIdcgi 26728   GrpOpHom cghomOLD 32852  RingOpscrngo 32863   RngHom crnghom 32929
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746  df-grpo 26731  df-gid 26732  df-ginv 26733  df-gdiv 26734  df-ablo 26783  df-ghomOLD 32853  df-rngo 32864  df-rngohom 32932
This theorem is referenced by: (None)
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