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Theorem isrnghm2d 44468
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 515 . 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 3181 . . 3 (𝜑 → ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) × (𝐹𝑦)))
74, 6jca 515 . 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 44459 . 2 (𝐹 ∈ (𝑅 RngHomo 𝑆) ↔ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) × (𝐹𝑦)))))
123, 7, 11sylanbrc 586 1 (𝜑𝐹 ∈ (𝑅 RngHomo 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2114  wral 3130  cfv 6334  (class class class)co 7140  Basecbs 16474  .rcmulr 16557   GrpHom cghm 18346  Rngcrng 44441   RngHomo crngh 44452
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-map 8395  df-ghm 18347  df-abl 18900  df-rng0 44442  df-rnghomo 44454
This theorem is referenced by:  isrnghmd  44469
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