Mathbox for Jeff Madsen < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rngohommul Structured version   Unicode version

Theorem rngohommul 26578
 Description: Ring homomorphisms preserve multiplication. (Contributed by Jeff Madsen, 3-Jan-2011.)
Hypotheses
Ref Expression
rnghommul.1
rnghommul.2
rnghommul.3
rnghommul.4
Assertion
Ref Expression
rngohommul

Proof of Theorem rngohommul
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rnghommul.1 . . . . . . 7
2 rnghommul.3 . . . . . . 7
3 rnghommul.2 . . . . . . 7
4 eqid 2436 . . . . . . 7 GId GId
5 eqid 2436 . . . . . . 7
6 rnghommul.4 . . . . . . 7
7 eqid 2436 . . . . . . 7
8 eqid 2436 . . . . . . 7 GId GId
91, 2, 3, 4, 5, 6, 7, 8isrngohom 26573 . . . . . 6 GId GId
109biimpa 471 . . . . 5 GId GId
1110simp3d 971 . . . 4
12113impa 1148 . . 3
13 simpr 448 . . . . 5
1413ralimi 2774 . . . 4
1514ralimi 2774 . . 3
1612, 15syl 16 . 2
17 oveq1 6081 . . . . 5
1817fveq2d 5725 . . . 4
19 fveq2 5721 . . . . 5
2019oveq1d 6089 . . . 4
2118, 20eqeq12d 2450 . . 3
22 oveq2 6082 . . . . 5
2322fveq2d 5725 . . . 4
24 fveq2 5721 . . . . 5
2524oveq2d 6090 . . . 4
2623, 25eqeq12d 2450 . . 3
2721, 26rspc2v 3051 . 2
2816, 27mpan9 456 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   w3a 936   wceq 1652   wcel 1725  wral 2698   crn 4872  wf 5443  cfv 5447  (class class class)co 6074  c1st 6340  c2nd 6341  GIdcgi 21768  crngo 21956   crnghom 26568 This theorem is referenced by:  rngohomco  26582  rngoisocnv  26589  crngohomfo  26608  keridl  26634 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 2417  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694 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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-sbc 3155  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-br 4206  df-opab 4260  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-map 7013  df-rngohom 26571
 Copyright terms: Public domain W3C validator