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

Theorem crngohomfo 26607
 Description: The image of a homomorphism from a commutative ring is commutative. (Contributed by Jeff Madsen, 4-Jan-2011.)
Hypotheses
Ref Expression
crnghomfo.1
crnghomfo.2
crnghomfo.3
crnghomfo.4
Assertion
Ref Expression
crngohomfo CRingOps CRingOps

Proof of Theorem crngohomfo
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simplr 732 . 2 CRingOps
2 foelrn 5880 . . . . . . . 8
32ex 424 . . . . . . 7
4 foelrn 5880 . . . . . . . 8
54ex 424 . . . . . . 7
63, 5anim12d 547 . . . . . 6
7 reeanv 2867 . . . . . 6
86, 7syl6ibr 219 . . . . 5
98ad2antll 710 . . . 4 CRingOps
10 crnghomfo.1 . . . . . . . . . . . . . 14
11 eqid 2435 . . . . . . . . . . . . . 14
12 crnghomfo.2 . . . . . . . . . . . . . 14
1310, 11, 12crngocom 26602 . . . . . . . . . . . . 13 CRingOps
14133expb 1154 . . . . . . . . . . . 12 CRingOps
15143ad2antl1 1119 . . . . . . . . . . 11 CRingOps
1615fveq2d 5724 . . . . . . . . . 10 CRingOps
17 crngorngo 26601 . . . . . . . . . . 11 CRingOps
18 eqid 2435 . . . . . . . . . . . 12
1910, 12, 11, 18rngohommul 26577 . . . . . . . . . . 11
2017, 19syl3anl1 1232 . . . . . . . . . 10 CRingOps
2110, 12, 11, 18rngohommul 26577 . . . . . . . . . . . 12
2221ancom2s 778 . . . . . . . . . . 11
2317, 22syl3anl1 1232 . . . . . . . . . 10 CRingOps
2416, 20, 233eqtr3d 2475 . . . . . . . . 9 CRingOps
25 oveq12 6082 . . . . . . . . . 10
26 oveq12 6082 . . . . . . . . . . 11
2726ancoms 440 . . . . . . . . . 10
2825, 27eqeq12d 2449 . . . . . . . . 9
2924, 28syl5ibrcom 214 . . . . . . . 8 CRingOps
3029ex 424 . . . . . . 7 CRingOps
31303expa 1153 . . . . . 6 CRingOps
3231adantrr 698 . . . . 5 CRingOps
3332rexlimdvv 2828 . . . 4 CRingOps
349, 33syld 42 . . 3 CRingOps
3534ralrimivv 2789 . 2 CRingOps
36 crnghomfo.3 . . 3
37 crnghomfo.4 . . 3
3836, 18, 37iscrngo2 26599 . 2 CRingOps
391, 35, 38sylanbrc 646 1 CRingOps CRingOps
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   w3a 936   wceq 1652   wcel 1725  wral 2697  wrex 2698   crn 4871  wfo 5444  cfv 5446  (class class class)co 6073  c1st 6339  c2nd 6340  crngo 21955   crnghom 26567  CRingOpsccring 26596 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-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-rab 2706  df-v 2950  df-sbc 3154  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-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-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fo 5452  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-map 7012  df-rngo 21956  df-com2 21991  df-rngohom 26570  df-crngo 26597
 Copyright terms: Public domain W3C validator