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Theorem rngohomadd 35128
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 RngHom 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 2818 . . . . . . 7 (2nd𝑅) = (2nd𝑅)
3 rnghomadd.2 . . . . . . 7 𝑋 = ran 𝐺
4 eqid 2818 . . . . . . 7 (GId‘(2nd𝑅)) = (GId‘(2nd𝑅))
5 rnghomadd.3 . . . . . . 7 𝐽 = (1st𝑆)
6 eqid 2818 . . . . . . 7 (2nd𝑆) = (2nd𝑆)
7 eqid 2818 . . . . . . 7 ran 𝐽 = ran 𝐽
8 eqid 2818 . . . . . . 7 (GId‘(2nd𝑆)) = (GId‘(2nd𝑆))
91, 2, 3, 4, 5, 6, 7, 8isrngohom 35124 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RngHom 𝑆) ↔ (𝐹:𝑋⟶ran 𝐽 ∧ (𝐹‘(GId‘(2nd𝑅))) = (GId‘(2nd𝑆)) ∧ ∀𝑥𝑋𝑦𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)) ∧ (𝐹‘(𝑥(2nd𝑅)𝑦)) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑦))))))
109biimpa 477 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹:𝑋⟶ran 𝐽 ∧ (𝐹‘(GId‘(2nd𝑅))) = (GId‘(2nd𝑆)) ∧ ∀𝑥𝑋𝑦𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)) ∧ (𝐹‘(𝑥(2nd𝑅)𝑦)) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑦)))))
1110simp3d 1136 . . . 4 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ∀𝑥𝑋𝑦𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)) ∧ (𝐹‘(𝑥(2nd𝑅)𝑦)) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑦))))
12113impa 1102 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ∀𝑥𝑋𝑦𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)) ∧ (𝐹‘(𝑥(2nd𝑅)𝑦)) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑦))))
13 simpl 483 . . . 4 (((𝐹‘(𝑥𝐺𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)) ∧ (𝐹‘(𝑥(2nd𝑅)𝑦)) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑦))) → (𝐹‘(𝑥𝐺𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)))
14132ralimi 3158 . . 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 RngHom 𝑆)) → ∀𝑥𝑋𝑦𝑋 (𝐹‘(𝑥𝐺𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)))
16 fvoveq1 7168 . . . 4 (𝑥 = 𝐴 → (𝐹‘(𝑥𝐺𝑦)) = (𝐹‘(𝐴𝐺𝑦)))
17 fveq2 6663 . . . . 5 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
1817oveq1d 7160 . . . 4 (𝑥 = 𝐴 → ((𝐹𝑥)𝐽(𝐹𝑦)) = ((𝐹𝐴)𝐽(𝐹𝑦)))
1916, 18eqeq12d 2834 . . 3 (𝑥 = 𝐴 → ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)) ↔ (𝐹‘(𝐴𝐺𝑦)) = ((𝐹𝐴)𝐽(𝐹𝑦))))
20 oveq2 7153 . . . . 5 (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵))
2120fveq2d 6667 . . . 4 (𝑦 = 𝐵 → (𝐹‘(𝐴𝐺𝑦)) = (𝐹‘(𝐴𝐺𝐵)))
22 fveq2 6663 . . . . 5 (𝑦 = 𝐵 → (𝐹𝑦) = (𝐹𝐵))
2322oveq2d 7161 . . . 4 (𝑦 = 𝐵 → ((𝐹𝐴)𝐽(𝐹𝑦)) = ((𝐹𝐴)𝐽(𝐹𝐵)))
2421, 23eqeq12d 2834 . . 3 (𝑦 = 𝐵 → ((𝐹‘(𝐴𝐺𝑦)) = ((𝐹𝐴)𝐽(𝐹𝑦)) ↔ (𝐹‘(𝐴𝐺𝐵)) = ((𝐹𝐴)𝐽(𝐹𝐵))))
2519, 24rspc2v 3630 . 2 ((𝐴𝑋𝐵𝑋) → (∀𝑥𝑋𝑦𝑋 (𝐹‘(𝑥𝐺𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)) → (𝐹‘(𝐴𝐺𝐵)) = ((𝐹𝐴)𝐽(𝐹𝐵))))
2615, 25mpan9 507 1 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝐴𝑋𝐵𝑋)) → (𝐹‘(𝐴𝐺𝐵)) = ((𝐹𝐴)𝐽(𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  ran crn 5549  wf 6344  cfv 6348  (class class class)co 7145  1st c1st 7676  2nd c2nd 7677  GIdcgi 28194  RingOpscrngo 35053   RngHom crnghom 35119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-map 8397  df-rngohom 35122
This theorem is referenced by:  rngogrphom  35130  rngohomco  35133  rngoisocnv  35140  keridl  35191
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