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Theorem grpokerinj 32247
 Description: A group homomorphism is injective if and only if its kernel is zero. (Contributed by Jeff Madsen, 16-Jun-2011.)
Hypotheses
Ref Expression
grpkerinj.1
grpkerinj.2 GId
grpkerinj.3
grpkerinj.4 GId
Assertion
Ref Expression
grpokerinj GrpOpHom

Proof of Theorem grpokerinj
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 grpkerinj.2 . . . . . . . . 9 GId
2 grpkerinj.4 . . . . . . . . 9 GId
31, 2ghomidOLD 26174 . . . . . . . 8 GrpOpHom
43sneqd 3971 . . . . . . 7 GrpOpHom
5 grpkerinj.1 . . . . . . . . . 10
6 grpkerinj.3 . . . . . . . . . 10
75, 6ghomf 32244 . . . . . . . . 9 GrpOpHom
8 ffn 5739 . . . . . . . . 9
97, 8syl 17 . . . . . . . 8 GrpOpHom
105, 1grpoidcl 26026 . . . . . . . . 9
11103ad2ant1 1051 . . . . . . . 8 GrpOpHom
12 fnsnfv 5940 . . . . . . . 8
139, 11, 12syl2anc 673 . . . . . . 7 GrpOpHom
144, 13eqtr3d 2507 . . . . . 6 GrpOpHom
1514imaeq2d 5174 . . . . 5 GrpOpHom
1615adantl 473 . . . 4 GrpOpHom
1710snssd 4108 . . . . . 6
18173ad2ant1 1051 . . . . 5 GrpOpHom
19 f1imacnv 5844 . . . . 5
2018, 19sylan2 482 . . . 4 GrpOpHom
2116, 20eqtrd 2505 . . 3 GrpOpHom
2221expcom 442 . 2 GrpOpHom
237adantr 472 . . . 4 GrpOpHom
24 simpl2 1034 . . . . . . . 8 GrpOpHom
257ffvelrnda 6037 . . . . . . . . 9 GrpOpHom
2625adantrr 731 . . . . . . . 8 GrpOpHom
277ffvelrnda 6037 . . . . . . . . 9 GrpOpHom
2827adantrl 730 . . . . . . . 8 GrpOpHom
29 eqid 2471 . . . . . . . . 9
306, 2, 29grpoeqdivid 32243 . . . . . . . 8
3124, 26, 28, 30syl3anc 1292 . . . . . . 7 GrpOpHom
3231adantlr 729 . . . . . 6 GrpOpHom
33 eqid 2471 . . . . . . . . . 10
345, 33, 29ghomdiv 32246 . . . . . . . . 9 GrpOpHom
3534adantlr 729 . . . . . . . 8 GrpOpHom
3635eqeq1d 2473 . . . . . . 7 GrpOpHom
37 fvex 5889 . . . . . . . . . . 11 GId
382, 37eqeltri 2545 . . . . . . . . . 10
3938snid 3988 . . . . . . . . 9
40 eleq1 2537 . . . . . . . . 9
4139, 40mpbiri 241 . . . . . . . 8
42 ffun 5742 . . . . . . . . . . . . . 14
437, 42syl 17 . . . . . . . . . . . . 13 GrpOpHom
4443adantr 472 . . . . . . . . . . . 12 GrpOpHom
455, 33grpodivcl 26056 . . . . . . . . . . . . . . 15
46453expb 1232 . . . . . . . . . . . . . 14
47463ad2antl1 1192 . . . . . . . . . . . . 13 GrpOpHom
48 fdm 5745 . . . . . . . . . . . . . . 15
497, 48syl 17 . . . . . . . . . . . . . 14 GrpOpHom
5049adantr 472 . . . . . . . . . . . . 13 GrpOpHom
5147, 50eleqtrrd 2552 . . . . . . . . . . . 12 GrpOpHom
52 fvimacnv 6012 . . . . . . . . . . . 12
5344, 51, 52syl2anc 673 . . . . . . . . . . 11 GrpOpHom
54 eleq2 2538 . . . . . . . . . . 11
5553, 54sylan9bb 714 . . . . . . . . . 10 GrpOpHom
5655an32s 821 . . . . . . . . 9 GrpOpHom
57 elsni 3985 . . . . . . . . . . 11
585, 1, 33grpoeqdivid 32243 . . . . . . . . . . . . . 14
5958biimprd 231 . . . . . . . . . . . . 13
60593expb 1232 . . . . . . . . . . . 12
61603ad2antl1 1192 . . . . . . . . . . 11 GrpOpHom
6257, 61syl5 32 . . . . . . . . . 10 GrpOpHom
6362adantlr 729 . . . . . . . . 9 GrpOpHom
6456, 63sylbid 223 . . . . . . . 8 GrpOpHom
6541, 64syl5 32 . . . . . . 7 GrpOpHom
6636, 65sylbird 243 . . . . . 6 GrpOpHom
6732, 66sylbid 223 . . . . 5 GrpOpHom
6867ralrimivva 2814 . . . 4 GrpOpHom
69 dff13 6177 . . . 4
7023, 68, 69sylanbrc 677 . . 3 GrpOpHom
7170ex 441 . 2 GrpOpHom
7222, 71impbid 195 1 GrpOpHom
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 189   wa 376   w3a 1007   wceq 1452   wcel 1904  wral 2756  cvv 3031   wss 3390  csn 3959  ccnv 4838   cdm 4839   crn 4840  cima 4842   wfun 5583   wfn 5584  wf 5585  wf1 5586  cfv 5589  (class class class)co 6308  cgr 25995  GIdcgi 25996   cgs 25998   GrpOpHom cghomOLD 26166 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813  df-grpo 26000  df-gid 26001  df-ginv 26002  df-gdiv 26003  df-ghomOLD 26167 This theorem is referenced by:  rngokerinj  32278
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