Theorem isrnghmd 46700

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 46651	. . . 4 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel)
11		ablgrp 19653	. . . 4 ⊢ (𝑅 ∈ Abel → 𝑅 ∈ Grp)
12	4, 10, 11	3syl 18	. . 3 ⊢ (𝜑 → 𝑅 ∈ Grp)
13		rngabl 46651	. . . 4 ⊢ (𝑆 ∈ Rng → 𝑆 ∈ Abel)
14		ablgrp 19653	. . . 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 19101	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑅 GrpHom 𝑆))
19	1, 2, 3, 4, 5, 6, 18	isrnghm2d 46699	1 ⊢ (𝜑 → 𝐹 ∈ (𝑅 RngHomo 𝑆))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ⟶wf 6540  ‘cfv 6544  (class class class)co 7409  Basecbs 17144  +_gcplusg 17197  ._rcmulr 17198  Grpcgrp 18819  Abelcabl 19649  Rngcrng 46648   RngHomo crngh 46683

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-map 8822  df-ghm 19090  df-abl 19651  df-rng 46649  df-rnghomo 46685

This theorem is referenced by: (None)