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Theorem rngohom1 34391
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 2778 . . . . 5 (1st𝑅) = (1st𝑅)
2 rnghom1.1 . . . . 5 𝐻 = (2nd𝑅)
3 eqid 2778 . . . . 5 ran (1st𝑅) = ran (1st𝑅)
4 rnghom1.2 . . . . 5 𝑈 = (GId‘𝐻)
5 eqid 2778 . . . . 5 (1st𝑆) = (1st𝑆)
6 rnghom1.3 . . . . 5 𝐾 = (2nd𝑆)
7 eqid 2778 . . . . 5 ran (1st𝑆) = ran (1st𝑆)
8 rnghom1.4 . . . . 5 𝑉 = (GId‘𝐾)
91, 2, 3, 4, 5, 6, 7, 8isrngohom 34388 . . . 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 470 . . 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 1134 . 2 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹𝑈) = 𝑉)
12113impa 1097 1 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹𝑈) = 𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1071   = wceq 1601  wcel 2107  wral 3090  ran crn 5356  wf 6131  cfv 6135  (class class class)co 6922  1st c1st 7443  2nd c2nd 7444  GIdcgi 27917  RingOpscrngo 34317   RngHom crnghom 34383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-map 8142  df-rngohom 34386
This theorem is referenced by:  rngohomco  34397  rngoisocnv  34404
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