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Hypotheses
Ref Expression
Assertion
Ref Expression
rngohomadd (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝐴𝑋𝐵𝑋)) → (𝐹‘(𝐴𝐺𝐵)) = ((𝐹𝐴)𝐽(𝐹𝐵)))

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rnghomadd.1 . . . . . . 7 𝐺 = (1st𝑅)
2 eqid 2610 . . . . . . 7 (2nd𝑅) = (2nd𝑅)
3 rnghomadd.2 . . . . . . 7 𝑋 = ran 𝐺
4 eqid 2610 . . . . . . 7 (GId‘(2nd𝑅)) = (GId‘(2nd𝑅))
5 rnghomadd.3 . . . . . . 7 𝐽 = (1st𝑆)
6 eqid 2610 . . . . . . 7 (2nd𝑆) = (2nd𝑆)
7 eqid 2610 . . . . . . 7 ran 𝐽 = ran 𝐽
8 eqid 2610 . . . . . . 7 (GId‘(2nd𝑆)) = (GId‘(2nd𝑆))
91, 2, 3, 4, 5, 6, 7, 8isrngohom 32934 . . . . . 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 500 . . . . 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 1068 . . . 4 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ∀𝑥𝑋𝑦𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)) ∧ (𝐹‘(𝑥(2nd𝑅)𝑦)) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑦))))
12113impa 1251 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ∀𝑥𝑋𝑦𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)) ∧ (𝐹‘(𝑥(2nd𝑅)𝑦)) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑦))))
13 simpl 472 . . . . 5 (((𝐹‘(𝑥𝐺𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)) ∧ (𝐹‘(𝑥(2nd𝑅)𝑦)) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑦))) → (𝐹‘(𝑥𝐺𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)))
1413ralimi 2936 . . . 4 (∀𝑦𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)) ∧ (𝐹‘(𝑥(2nd𝑅)𝑦)) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑦))) → ∀𝑦𝑋 (𝐹‘(𝑥𝐺𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)))
1514ralimi 2936 . . 3 (∀𝑥𝑋𝑦𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)) ∧ (𝐹‘(𝑥(2nd𝑅)𝑦)) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑦))) → ∀𝑥𝑋𝑦𝑋 (𝐹‘(𝑥𝐺𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)))
1612, 15syl 17 . 2 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ∀𝑥𝑋𝑦𝑋 (𝐹‘(𝑥𝐺𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)))
17 oveq1 6556 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐺𝑦) = (𝐴𝐺𝑦))
1817fveq2d 6107 . . . 4 (𝑥 = 𝐴 → (𝐹‘(𝑥𝐺𝑦)) = (𝐹‘(𝐴𝐺𝑦)))
19 fveq2 6103 . . . . 5 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
2019oveq1d 6564 . . . 4 (𝑥 = 𝐴 → ((𝐹𝑥)𝐽(𝐹𝑦)) = ((𝐹𝐴)𝐽(𝐹𝑦)))
2118, 20eqeq12d 2625 . . 3 (𝑥 = 𝐴 → ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)) ↔ (𝐹‘(𝐴𝐺𝑦)) = ((𝐹𝐴)𝐽(𝐹𝑦))))
22 oveq2 6557 . . . . 5 (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵))
2322fveq2d 6107 . . . 4 (𝑦 = 𝐵 → (𝐹‘(𝐴𝐺𝑦)) = (𝐹‘(𝐴𝐺𝐵)))
24 fveq2 6103 . . . . 5 (𝑦 = 𝐵 → (𝐹𝑦) = (𝐹𝐵))
2524oveq2d 6565 . . . 4 (𝑦 = 𝐵 → ((𝐹𝐴)𝐽(𝐹𝑦)) = ((𝐹𝐴)𝐽(𝐹𝐵)))
2623, 25eqeq12d 2625 . . 3 (𝑦 = 𝐵 → ((𝐹‘(𝐴𝐺𝑦)) = ((𝐹𝐴)𝐽(𝐹𝑦)) ↔ (𝐹‘(𝐴𝐺𝐵)) = ((𝐹𝐴)𝐽(𝐹𝐵))))
2721, 26rspc2v 3293 . 2 ((𝐴𝑋𝐵𝑋) → (∀𝑥𝑋𝑦𝑋 (𝐹‘(𝑥𝐺𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)) → (𝐹‘(𝐴𝐺𝐵)) = ((𝐹𝐴)𝐽(𝐹𝐵))))
2816, 27mpan9 485 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 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ran crn 5039  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  GIdcgi 26728  RingOpscrngo 32863   RngHom crnghom 32929 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-rngohom 32932 This theorem is referenced by:  rngogrphom  32940  rngohomco  32943  rngoisocnv  32950  keridl  33001
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