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Theorem rngohom1 35351
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 2824 . . . . 5 (1st𝑅) = (1st𝑅)
2 rnghom1.1 . . . . 5 𝐻 = (2nd𝑅)
3 eqid 2824 . . . . 5 ran (1st𝑅) = ran (1st𝑅)
4 rnghom1.2 . . . . 5 𝑈 = (GId‘𝐻)
5 eqid 2824 . . . . 5 (1st𝑆) = (1st𝑆)
6 rnghom1.3 . . . . 5 𝐾 = (2nd𝑆)
7 eqid 2824 . . . . 5 ran (1st𝑆) = ran (1st𝑆)
8 rnghom1.4 . . . . 5 𝑉 = (GId‘𝐾)
91, 2, 3, 4, 5, 6, 7, 8isrngohom 35348 . . . 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 480 . . 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 1140 . 2 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹𝑈) = 𝑉)
12113impa 1107 1 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹𝑈) = 𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2115  wral 3133  ran crn 5543  wf 6339  cfv 6343  (class class class)co 7149  1st c1st 7682  2nd c2nd 7683  GIdcgi 28276  RingOpscrngo 35277   RngHom crnghom 35343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-map 8404  df-rngohom 35346
This theorem is referenced by:  rngohomco  35357  rngoisocnv  35364
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