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Theorem isrnghmd 20468
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 20173 . . . 4 (𝑅 ∈ Rng → 𝑅 ∈ Abel)
11 ablgrp 19818 . . . 4 (𝑅 ∈ Abel → 𝑅 ∈ Grp)
124, 10, 113syl 18 . . 3 (𝜑𝑅 ∈ Grp)
13 rngabl 20173 . . . 4 (𝑆 ∈ Rng → 𝑆 ∈ Abel)
14 ablgrp 19818 . . . 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 19256 . 2 (𝜑𝐹 ∈ (𝑅 GrpHom 𝑆))
191, 2, 3, 4, 5, 6, 18isrnghm2d 20467 1 (𝜑𝐹 ∈ (𝑅 RngHom 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wf 6559  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  .rcmulr 17299  Grpcgrp 18964  Abelcabl 19814  Rngcrng 20170   RngHom crnghm 20451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-ghm 19244  df-abl 19816  df-rng 20171  df-rnghm 20453
This theorem is referenced by: (None)
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