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Theorem ghabloOLD 26095
 Description: Obsolete version of ghmabl 17472 as of 14-Mar-2020. The image of an Abelian group under a group homomorphism is an Abelian group (Contributed by Paul Chapman, 25-Apr-2008.) (Revised by Mario Carneiro, 12-May-2014.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
ghgrpOLD.1
ghgrpOLD.2
ghgrpOLD.3
ghgrpOLD.4
ghgrpOLD.5
ghgrpOLD.6
ghabloOLD.7
Assertion
Ref Expression
ghabloOLD
Distinct variable groups:   ,,   ,,   ,,   ,,   ,,   ,,   ,,
Allowed substitution hints:   (,)

Proof of Theorem ghabloOLD
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ghgrpOLD.1 . . 3
2 ghgrpOLD.2 . . 3
3 ghgrpOLD.3 . . 3
4 ghgrpOLD.4 . . 3
5 ghgrpOLD.5 . . 3
6 ghgrpOLD.6 . . 3
7 ghabloOLD.7 . . . 4
8 ablogrpo 26010 . . . 4
97, 8syl 17 . . 3
101, 2, 3, 4, 5, 6, 9ghgrpOLD 26094 . 2
11 fndm 5693 . . . . . . . . 9
126, 11syl 17 . . . . . . . 8
133resgrprn 26006 . . . . . . . 8
1412, 10, 5, 13syl3anc 1264 . . . . . . 7
1514eleq2d 2492 . . . . . 6
1614eleq2d 2492 . . . . . 6
1715, 16anbi12d 715 . . . . 5
1817biimpar 487 . . . 4
197adantr 466 . . . . . . . . . . . . 13
20 simprl 762 . . . . . . . . . . . . 13
21 simprr 764 . . . . . . . . . . . . 13
224ablocom 26011 . . . . . . . . . . . . 13
2319, 20, 21, 22syl3anc 1264 . . . . . . . . . . . 12
2423fveq2d 5885 . . . . . . . . . . 11
251, 2, 3ghgrplem2OLD 26093 . . . . . . . . . . 11
261, 2, 3ghgrplem2OLD 26093 . . . . . . . . . . . 12
2726ancom2s 809 . . . . . . . . . . 11
2824, 25, 273eqtr3d 2471 . . . . . . . . . 10
2928ancom2s 809 . . . . . . . . 9
3029expr 618 . . . . . . . 8
31 oveq2 6313 . . . . . . . . . 10
32 oveq1 6312 . . . . . . . . . 10
3331, 32eqeq12d 2444 . . . . . . . . 9
3433imbi2d 317 . . . . . . . 8
351, 30, 34ghgrplem1OLD 26092 . . . . . . 7
3635impancom 441 . . . . . 6
37 oveq1 6312 . . . . . . . 8
38 oveq2 6313 . . . . . . . 8
3937, 38eqeq12d 2444 . . . . . . 7
4039imbi2d 317 . . . . . 6
411, 36, 40ghgrplem1OLD 26092 . . . . 5
4241impr 623 . . . 4
4318, 42syldan 472 . . 3
4443ralrimivva 2843 . 2
45 eqid 2422 . . 3
4645isablo 26009 . 2
4710, 44, 46sylanbrc 668 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 370   wceq 1437   wcel 1872  wral 2771   wss 3436   cxp 4851   cdm 4853   crn 4854   cres 4855   wfn 5596  wfo 5599  cfv 5601  (class class class)co 6305  cgr 25912  cablo 26007 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-grpo 25917  df-gid 25918  df-ginv 25919  df-gdiv 25920  df-ablo 26008 This theorem is referenced by:  ghsubgolemOLD  26096
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