Theorem isrnghmd 20533

Description: Demonstration of non-unital ring homomorphism. (Contributed by AV, 23-Feb-2020.)

Hypotheses

Ref	Expression
isrnghmd.b	⊢ 𝐵 = (Base‘𝑅)
isrnghmd.t	⊢ · = (.r‘𝑅)
isrnghmd.u	⊢ × = (.r‘𝑆)
isrnghmd.r	⊢ (𝜑 → 𝑅 ∈ Rng)
isrnghmd.s	⊢ (𝜑 → 𝑆 ∈ Rng)
isrnghmd.ht	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) × (𝐹‘𝑦)))
isrnghmd.c	⊢ 𝐶 = (Base‘𝑆)
isrnghmd.p	⊢ + = (+g‘𝑅)
isrnghmd.q	⊢ ⨣ = (+g‘𝑆)
isrnghmd.f	⊢ (𝜑 → 𝐹:𝐵⟶𝐶)
isrnghmd.hp	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)))

Assertion

Ref	Expression
isrnghmd	⊢ (𝜑 → 𝐹 ∈ (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

Step	Hyp	Ref	Expression
1		isrnghmd.b	. 2 ⊢ 𝐵 = (Base‘𝑅)
2		isrnghmd.t	. 2 ⊢ · = (.r‘𝑅)
3		isrnghmd.u	. 2 ⊢ × = (.r‘𝑆)
4		isrnghmd.r	. 2 ⊢ (𝜑 → 𝑅 ∈ Rng)
5		isrnghmd.s	. 2 ⊢ (𝜑 → 𝑆 ∈ Rng)
6		isrnghmd.ht	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) × (𝐹‘𝑦)))
7		isrnghmd.c	. . 3 ⊢ 𝐶 = (Base‘𝑆)
8		isrnghmd.p	. . 3 ⊢ + = (+g‘𝑅)
9		isrnghmd.q	. . 3 ⊢ ⨣ = (+g‘𝑆)
10		rngabl 20233	. . . 4 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel)
11		ablgrp 19855	. . . 4 ⊢ (𝑅 ∈ Abel → 𝑅 ∈ Grp)
12	4, 10, 11	3syl 19	. . 3 ⊢ (𝜑 → 𝑅 ∈ Grp)
13		rngabl 20233	. . . 4 ⊢ (𝑆 ∈ Rng → 𝑆 ∈ Abel)
14		ablgrp 19855	. . . 4 ⊢ (𝑆 ∈ Abel → 𝑆 ∈ Grp)
15	5, 13, 14	3syl 19	. . 3 ⊢ (𝜑 → 𝑆 ∈ Grp)
16		isrnghmd.f	. . 3 ⊢ (𝜑 → 𝐹:𝐵⟶𝐶)
17		isrnghmd.hp	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)))
18	1, 7, 8, 9, 12, 15, 16, 17	isghmd 19295	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑅 GrpHom 𝑆))
19	1, 2, 3, 4, 5, 6, 18	isrnghm2d 20532	1 ⊢ (𝜑 → 𝐹 ∈ (𝑅 RngHom 𝑆))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ⟶wf 6533  ‘cfv 6537  (class class class)co 7411  Basecbs 17269  +_gcplusg 17310  ._rcmulr 17311  Grpcgrp 19000  Abelcabl 19851  Rngcrng 20230   RngHom crnghm 20516

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7986  df-2nd 7987  df-map 8826  df-ghm 19284  df-abl 19853  df-rng 20231  df-rnghm 20518

This theorem is referenced by: (None)