Theorem isrnghmd 41902

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 RngHomo 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 41877	. . . 4 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel)
11		ablgrp 18198	. . . 4 ⊢ (𝑅 ∈ Abel → 𝑅 ∈ Grp)
12	4, 10, 11	3syl 18	. . 3 ⊢ (𝜑 → 𝑅 ∈ Grp)
13		rngabl 41877	. . . 4 ⊢ (𝑆 ∈ Rng → 𝑆 ∈ Abel)
14		ablgrp 18198	. . . 4 ⊢ (𝑆 ∈ Abel → 𝑆 ∈ Grp)
15	5, 13, 14	3syl 18	. . 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 17669	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑅 GrpHom 𝑆))
19	1, 2, 3, 4, 5, 6, 18	isrnghm2d 41901	1 ⊢ (𝜑 → 𝐹 ∈ (𝑅 RngHomo 𝑆))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  Basecbs 15857  +_gcplusg 15941  ._rcmulr 15942  Grpcgrp 17422  Abelcabl 18194  Rngcrng 41874   RngHomo crngh 41885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-ghm 17658  df-abl 18196  df-rng0 41875  df-rnghomo 41887

This theorem is referenced by: (None)