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Theorem rngohomcl 37975
Description: Closure law for a ring homomorphism. (Contributed by Jeff Madsen, 3-Jan-2011.)
Hypotheses
Ref Expression
rnghomf.1 𝐺 = (1st𝑅)
rnghomf.2 𝑋 = ran 𝐺
rnghomf.3 𝐽 = (1st𝑆)
rnghomf.4 𝑌 = ran 𝐽
Assertion
Ref Expression
rngohomcl (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ 𝐴𝑋) → (𝐹𝐴) ∈ 𝑌)

Proof of Theorem rngohomcl
StepHypRef Expression
1 rnghomf.1 . . 3 𝐺 = (1st𝑅)
2 rnghomf.2 . . 3 𝑋 = ran 𝐺
3 rnghomf.3 . . 3 𝐽 = (1st𝑆)
4 rnghomf.4 . . 3 𝑌 = ran 𝐽
51, 2, 3, 4rngohomf 37974 . 2 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → 𝐹:𝑋𝑌)
65ffvelcdmda 7103 1 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ 𝐴𝑋) → (𝐹𝐴) ∈ 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1539  wcel 2107  ran crn 5685  cfv 6560  (class class class)co 7432  1st c1st 8013  RingOpscrngo 37902   RingOpsHom crngohom 37968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-map 8869  df-rngohom 37971
This theorem is referenced by:  rngohomco  37982  keridl  38040
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