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Theorem rngohom1 32937
Description: A ring homomorphism preserves 1. (Contributed by Jeff Madsen, 24-Jun-2011.)
Hypotheses
Ref Expression
rnghom1.1 𝐻 = (2nd𝑅)
rnghom1.2 𝑈 = (GId‘𝐻)
rnghom1.3 𝐾 = (2nd𝑆)
rnghom1.4 𝑉 = (GId‘𝐾)
Assertion
Ref Expression
rngohom1 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹𝑈) = 𝑉)

Proof of Theorem rngohom1
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2610 . . . . 5 (1st𝑅) = (1st𝑅)
2 rnghom1.1 . . . . 5 𝐻 = (2nd𝑅)
3 eqid 2610 . . . . 5 ran (1st𝑅) = ran (1st𝑅)
4 rnghom1.2 . . . . 5 𝑈 = (GId‘𝐻)
5 eqid 2610 . . . . 5 (1st𝑆) = (1st𝑆)
6 rnghom1.3 . . . . 5 𝐾 = (2nd𝑆)
7 eqid 2610 . . . . 5 ran (1st𝑆) = ran (1st𝑆)
8 rnghom1.4 . . . . 5 𝑉 = (GId‘𝐾)
91, 2, 3, 4, 5, 6, 7, 8isrngohom 32934 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RngHom 𝑆) ↔ (𝐹:ran (1st𝑅)⟶ran (1st𝑆) ∧ (𝐹𝑈) = 𝑉 ∧ ∀𝑥 ∈ ran (1st𝑅)∀𝑦 ∈ ran (1st𝑅)((𝐹‘(𝑥(1st𝑅)𝑦)) = ((𝐹𝑥)(1st𝑆)(𝐹𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹𝑥)𝐾(𝐹𝑦))))))
109biimpa 500 . . 3 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹:ran (1st𝑅)⟶ran (1st𝑆) ∧ (𝐹𝑈) = 𝑉 ∧ ∀𝑥 ∈ ran (1st𝑅)∀𝑦 ∈ ran (1st𝑅)((𝐹‘(𝑥(1st𝑅)𝑦)) = ((𝐹𝑥)(1st𝑆)(𝐹𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹𝑥)𝐾(𝐹𝑦)))))
1110simp2d 1067 . 2 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹𝑈) = 𝑉)
12113impa 1251 1 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹𝑈) = 𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1031   = wceq 1475  wcel 1977  wral 2896  ran crn 5039  wf 5800  cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  GIdcgi 26728  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  rngoisocnv  32950
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