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Theorem rngokerinj 26582
 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
rngkerinj.2
rngkerinj.3 GId
rngkerinj.4
rngkerinj.5
rngkerinj.6 GId
Assertion
Ref Expression
rngokerinj

Proof of Theorem rngokerinj
StepHypRef Expression
1 eqid 2435 . . . 4
21rngogrpo 21970 . . 3
323ad2ant1 978 . 2
4 eqid 2435 . . . 4
54rngogrpo 21970 . . 3
653ad2ant2 979 . 2
71, 4rngogrphom 26578 . 2 GrpOpHom
8 rngkerinj.2 . . . 4
9 rngkerinj.1 . . . . 5
109rneqi 5088 . . . 4
118, 10eqtri 2455 . . 3
12 rngkerinj.3 . . . 4 GId
139fveq2i 5723 . . . 4 GId GId
1412, 13eqtri 2455 . . 3 GId
15 rngkerinj.5 . . . 4
16 rngkerinj.4 . . . . 5
1716rneqi 5088 . . . 4
1815, 17eqtri 2455 . . 3
19 rngkerinj.6 . . . 4 GId
2016fveq2i 5723 . . . 4 GId GId
2119, 20eqtri 2455 . . 3 GId
2211, 14, 18, 21grpokerinj 26551 . 2 GrpOpHom
233, 6, 7, 22syl3anc 1184 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   w3a 936   wceq 1652   wcel 1725  csn 3806  ccnv 4869   crn 4871  cima 4873  wf1 5443  cfv 5446  (class class class)co 6073  c1st 6339  cgr 21766  GIdcgi 21767   GrpOpHom cghom 21937  crngo 21955   crnghom 26567 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-map 7012  df-grpo 21771  df-gid 21772  df-ginv 21773  df-gdiv 21774  df-ablo 21862  df-ghom 21938  df-rngo 21956  df-rngohom 26570
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