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Theorem isrnghmd 20451
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)))
isrnghmd.c 𝐶 = (Base‘𝑆)
isrnghmd.p + = (+g𝑅)
isrnghmd.q = (+g𝑆)
isrnghmd.f (𝜑𝐹:𝐵𝐶)
isrnghmd.hp ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
Assertion
Ref Expression
isrnghmd (𝜑𝐹 ∈ (𝑅 RngHom 𝑆))
Distinct variable groups:   𝜑,𝑥,𝑦   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝑥,𝐹,𝑦   𝑥, + ,𝑦   𝑥, ,𝑦   𝑥,𝑅,𝑦   𝑥,𝑆,𝑦
Allowed substitution hints:   · (𝑥,𝑦)   × (𝑥,𝑦)

Proof of Theorem isrnghmd
StepHypRef Expression
1 isrnghmd.b . 2 𝐵 = (Base‘𝑅)
2 isrnghmd.t . 2 · = (.r𝑅)
3 isrnghmd.u . 2 × = (.r𝑆)
4 isrnghmd.r . 2 (𝜑𝑅 ∈ Rng)
5 isrnghmd.s . 2 (𝜑𝑆 ∈ Rng)
6 isrnghmd.ht . 2 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) × (𝐹𝑦)))
7 isrnghmd.c . . 3 𝐶 = (Base‘𝑆)
8 isrnghmd.p . . 3 + = (+g𝑅)
9 isrnghmd.q . . 3 = (+g𝑆)
10 rngabl 20152 . . . 4 (𝑅 ∈ Rng → 𝑅 ∈ Abel)
11 ablgrp 19803 . . . 4 (𝑅 ∈ Abel → 𝑅 ∈ Grp)
124, 10, 113syl 18 . . 3 (𝜑𝑅 ∈ Grp)
13 rngabl 20152 . . . 4 (𝑆 ∈ Rng → 𝑆 ∈ Abel)
14 ablgrp 19803 . . . 4 (𝑆 ∈ Abel → 𝑆 ∈ Grp)
155, 13, 143syl 18 . . 3 (𝜑𝑆 ∈ Grp)
16 isrnghmd.f . . 3 (𝜑𝐹:𝐵𝐶)
17 isrnghmd.hp . . 3 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
181, 7, 8, 9, 12, 15, 16, 17isghmd 19243 . 2 (𝜑𝐹 ∈ (𝑅 GrpHom 𝑆))
191, 2, 3, 4, 5, 6, 18isrnghm2d 20450 1 (𝜑𝐹 ∈ (𝑅 RngHom 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wf 6557  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  .rcmulr 17298  Grpcgrp 18951  Abelcabl 19799  Rngcrng 20149   RngHom crnghm 20434
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868  df-ghm 19231  df-abl 19801  df-rng 20150  df-rnghm 20436
This theorem is referenced by: (None)
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