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Theorem rngohomcl 32936
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 RngHom 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 32935 . 2 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐹:𝑋𝑌)
65ffvelrnda 6267 1 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ 𝐴𝑋) → (𝐹𝐴) ∈ 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1031   = wceq 1475  wcel 1977  ran crn 5039  cfv 5804  (class class class)co 6549  1st c1st 7057  RingOpscrngo 32863   RngHom crnghom 32929
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-rngohom 32932
This theorem is referenced by:  rngohomco  32943  keridl  33001
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