Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rngohomadd Structured version   Visualization version   GIF version

Theorem rngohomadd 37293
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 2724 . . . . . . 7 (2nd𝑅) = (2nd𝑅)
3 rnghomadd.2 . . . . . . 7 𝑋 = ran 𝐺
4 eqid 2724 . . . . . . 7 (GId‘(2nd𝑅)) = (GId‘(2nd𝑅))
5 rnghomadd.3 . . . . . . 7 𝐽 = (1st𝑆)
6 eqid 2724 . . . . . . 7 (2nd𝑆) = (2nd𝑆)
7 eqid 2724 . . . . . . 7 ran 𝐽 = ran 𝐽
8 eqid 2724 . . . . . . 7 (GId‘(2nd𝑆)) = (GId‘(2nd𝑆))
91, 2, 3, 4, 5, 6, 7, 8isrngohom 37289 . . . . . 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 1141 . . . 4 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → ∀𝑥𝑋𝑦𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)) ∧ (𝐹‘(𝑥(2nd𝑅)𝑦)) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑦))))
12113impa 1107 . . 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 3115 . . 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 7424 . . . 4 (𝑥 = 𝐴 → (𝐹‘(𝑥𝐺𝑦)) = (𝐹‘(𝐴𝐺𝑦)))
17 fveq2 6881 . . . . 5 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
1817oveq1d 7416 . . . 4 (𝑥 = 𝐴 → ((𝐹𝑥)𝐽(𝐹𝑦)) = ((𝐹𝐴)𝐽(𝐹𝑦)))
1916, 18eqeq12d 2740 . . 3 (𝑥 = 𝐴 → ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)) ↔ (𝐹‘(𝐴𝐺𝑦)) = ((𝐹𝐴)𝐽(𝐹𝑦))))
20 oveq2 7409 . . . . 5 (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵))
2120fveq2d 6885 . . . 4 (𝑦 = 𝐵 → (𝐹‘(𝐴𝐺𝑦)) = (𝐹‘(𝐴𝐺𝐵)))
22 fveq2 6881 . . . . 5 (𝑦 = 𝐵 → (𝐹𝑦) = (𝐹𝐵))
2322oveq2d 7417 . . . 4 (𝑦 = 𝐵 → ((𝐹𝐴)𝐽(𝐹𝑦)) = ((𝐹𝐴)𝐽(𝐹𝐵)))
2421, 23eqeq12d 2740 . . 3 (𝑦 = 𝐵 → ((𝐹‘(𝐴𝐺𝑦)) = ((𝐹𝐴)𝐽(𝐹𝑦)) ↔ (𝐹‘(𝐴𝐺𝐵)) = ((𝐹𝐴)𝐽(𝐹𝐵))))
2519, 24rspc2v 3614 . 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 1084   = wceq 1533  wcel 2098  wral 3053  ran crn 5667  wf 6529  cfv 6533  (class class class)co 7401  1st c1st 7966  2nd c2nd 7967  GIdcgi 30167  RingOpscrngo 37218   RingOpsHom crngohom 37284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3770  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-map 8817  df-rngohom 37287
This theorem is referenced by:  rngogrphom  37295  rngohomco  37298  rngoisocnv  37305  keridl  37356
  Copyright terms: Public domain W3C validator