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Theorem rngohomadd 37956
Description: Ring homomorphisms preserve addition. (Contributed by Jeff Madsen, 3-Jan-2011.)
Hypotheses
Ref Expression
rnghomadd.1 𝐺 = (1st𝑅)
rnghomadd.2 𝑋 = ran 𝐺
rnghomadd.3 𝐽 = (1st𝑆)
Assertion
Ref Expression
rngohomadd (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝐴𝑋𝐵𝑋)) → (𝐹‘(𝐴𝐺𝐵)) = ((𝐹𝐴)𝐽(𝐹𝐵)))

Proof of Theorem rngohomadd
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rnghomadd.1 . . . . . . 7 𝐺 = (1st𝑅)
2 eqid 2735 . . . . . . 7 (2nd𝑅) = (2nd𝑅)
3 rnghomadd.2 . . . . . . 7 𝑋 = ran 𝐺
4 eqid 2735 . . . . . . 7 (GId‘(2nd𝑅)) = (GId‘(2nd𝑅))
5 rnghomadd.3 . . . . . . 7 𝐽 = (1st𝑆)
6 eqid 2735 . . . . . . 7 (2nd𝑆) = (2nd𝑆)
7 eqid 2735 . . . . . . 7 ran 𝐽 = ran 𝐽
8 eqid 2735 . . . . . . 7 (GId‘(2nd𝑆)) = (GId‘(2nd𝑆))
91, 2, 3, 4, 5, 6, 7, 8isrngohom 37952 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RingOpsHom 𝑆) ↔ (𝐹:𝑋⟶ran 𝐽 ∧ (𝐹‘(GId‘(2nd𝑅))) = (GId‘(2nd𝑆)) ∧ ∀𝑥𝑋𝑦𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)) ∧ (𝐹‘(𝑥(2nd𝑅)𝑦)) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑦))))))
109biimpa 476 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (𝐹:𝑋⟶ran 𝐽 ∧ (𝐹‘(GId‘(2nd𝑅))) = (GId‘(2nd𝑆)) ∧ ∀𝑥𝑋𝑦𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)) ∧ (𝐹‘(𝑥(2nd𝑅)𝑦)) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑦)))))
1110simp3d 1143 . . . 4 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → ∀𝑥𝑋𝑦𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)) ∧ (𝐹‘(𝑥(2nd𝑅)𝑦)) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑦))))
12113impa 1109 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → ∀𝑥𝑋𝑦𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)) ∧ (𝐹‘(𝑥(2nd𝑅)𝑦)) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑦))))
13 simpl 482 . . . 4 (((𝐹‘(𝑥𝐺𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)) ∧ (𝐹‘(𝑥(2nd𝑅)𝑦)) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑦))) → (𝐹‘(𝑥𝐺𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)))
14132ralimi 3121 . . 3 (∀𝑥𝑋𝑦𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)) ∧ (𝐹‘(𝑥(2nd𝑅)𝑦)) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑦))) → ∀𝑥𝑋𝑦𝑋 (𝐹‘(𝑥𝐺𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)))
1512, 14syl 17 . 2 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → ∀𝑥𝑋𝑦𝑋 (𝐹‘(𝑥𝐺𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)))
16 fvoveq1 7454 . . . 4 (𝑥 = 𝐴 → (𝐹‘(𝑥𝐺𝑦)) = (𝐹‘(𝐴𝐺𝑦)))
17 fveq2 6907 . . . . 5 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
1817oveq1d 7446 . . . 4 (𝑥 = 𝐴 → ((𝐹𝑥)𝐽(𝐹𝑦)) = ((𝐹𝐴)𝐽(𝐹𝑦)))
1916, 18eqeq12d 2751 . . 3 (𝑥 = 𝐴 → ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)) ↔ (𝐹‘(𝐴𝐺𝑦)) = ((𝐹𝐴)𝐽(𝐹𝑦))))
20 oveq2 7439 . . . . 5 (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵))
2120fveq2d 6911 . . . 4 (𝑦 = 𝐵 → (𝐹‘(𝐴𝐺𝑦)) = (𝐹‘(𝐴𝐺𝐵)))
22 fveq2 6907 . . . . 5 (𝑦 = 𝐵 → (𝐹𝑦) = (𝐹𝐵))
2322oveq2d 7447 . . . 4 (𝑦 = 𝐵 → ((𝐹𝐴)𝐽(𝐹𝑦)) = ((𝐹𝐴)𝐽(𝐹𝐵)))
2421, 23eqeq12d 2751 . . 3 (𝑦 = 𝐵 → ((𝐹‘(𝐴𝐺𝑦)) = ((𝐹𝐴)𝐽(𝐹𝑦)) ↔ (𝐹‘(𝐴𝐺𝐵)) = ((𝐹𝐴)𝐽(𝐹𝐵))))
2519, 24rspc2v 3633 . 2 ((𝐴𝑋𝐵𝑋) → (∀𝑥𝑋𝑦𝑋 (𝐹‘(𝑥𝐺𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)) → (𝐹‘(𝐴𝐺𝐵)) = ((𝐹𝐴)𝐽(𝐹𝐵))))
2615, 25mpan9 506 1 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝐴𝑋𝐵𝑋)) → (𝐹‘(𝐴𝐺𝐵)) = ((𝐹𝐴)𝐽(𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  ran crn 5690  wf 6559  cfv 6563  (class class class)co 7431  1st c1st 8011  2nd c2nd 8012  GIdcgi 30519  RingOpscrngo 37881   RingOpsHom crngohom 37947
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-rngohom 37950
This theorem is referenced by:  rngogrphom  37958  rngohomco  37961  rngoisocnv  37968  keridl  38019
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