Theorem isrnghmd 20353

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 20060	. . . 4 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel)
11		ablgrp 19705	. . . 4 ⊢ (𝑅 ∈ Abel → 𝑅 ∈ Grp)
12	4, 10, 11	3syl 18	. . 3 ⊢ (𝜑 → 𝑅 ∈ Grp)
13		rngabl 20060	. . . 4 ⊢ (𝑆 ∈ Rng → 𝑆 ∈ Abel)
14		ablgrp 19705	. . . 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 19150	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑅 GrpHom 𝑆))
19	1, 2, 3, 4, 5, 6, 18	isrnghm2d 20352	1 ⊢ (𝜑 → 𝐹 ∈ (𝑅 RngHom 𝑆))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  ⟶wf 6533  ‘cfv 6537  (class class class)co 7405  Basecbs 17153  +_gcplusg 17206  ._rcmulr 17207  Grpcgrp 18863  Abelcabl 19701  Rngcrng 20057   RngHom crnghm 20336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-map 8824  df-ghm 19139  df-abl 19703  df-rng 20058  df-rnghm 20338

This theorem is referenced by: (None)