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Theorem isrnghm2d 42226
Description: Demonstration of non-unital ring homomorphism. (Contributed by AV, 23-Feb-2020.)
Hypotheses
Ref Expression
isrnghmd.b 𝐵 = (Base‘𝑅)
isrnghmd.t · = (.r𝑅)
isrnghmd.u × = (.r𝑆)
isrnghmd.r (𝜑𝑅 ∈ Rng)
isrnghmd.s (𝜑𝑆 ∈ Rng)
isrnghmd.ht ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) × (𝐹𝑦)))
isrnghm2d.f (𝜑𝐹 ∈ (𝑅 GrpHom 𝑆))
Assertion
Ref Expression
isrnghm2d (𝜑𝐹 ∈ (𝑅 RngHomo 𝑆))
Distinct variable groups:   𝜑,𝑥,𝑦   𝑥,𝐵,𝑦   𝑥,𝐹,𝑦   𝑥,𝑅,𝑦   𝑥,𝑆,𝑦
Allowed substitution hints:   · (𝑥,𝑦)   × (𝑥,𝑦)

Proof of Theorem isrnghm2d
StepHypRef Expression
1 isrnghmd.r . . 3 (𝜑𝑅 ∈ Rng)
2 isrnghmd.s . . 3 (𝜑𝑆 ∈ Rng)
31, 2jca 553 . 2 (𝜑 → (𝑅 ∈ Rng ∧ 𝑆 ∈ Rng))
4 isrnghm2d.f . . 3 (𝜑𝐹 ∈ (𝑅 GrpHom 𝑆))
5 isrnghmd.ht . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) × (𝐹𝑦)))
65ralrimivva 3000 . . 3 (𝜑 → ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) × (𝐹𝑦)))
74, 6jca 553 . 2 (𝜑 → (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) × (𝐹𝑦))))
8 isrnghmd.b . . 3 𝐵 = (Base‘𝑅)
9 isrnghmd.t . . 3 · = (.r𝑅)
10 isrnghmd.u . . 3 × = (.r𝑆)
118, 9, 10isrnghm 42217 . 2 (𝐹 ∈ (𝑅 RngHomo 𝑆) ↔ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) × (𝐹𝑦)))))
123, 7, 11sylanbrc 699 1 (𝜑𝐹 ∈ (𝑅 RngHomo 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wral 2941  cfv 5926  (class class class)co 6690  Basecbs 15904  .rcmulr 15989   GrpHom cghm 17704  Rngcrng 42199   RngHomo crngh 42210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-map 7901  df-ghm 17705  df-abl 18242  df-rng0 42200  df-rnghomo 42212
This theorem is referenced by:  isrnghmd  42227
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