Theorem isrnghm2d 20366

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)))
isrnghm2d.f	⊢ (𝜑 → 𝐹 ∈ (𝑅 GrpHom 𝑆))

Assertion

Ref	Expression
isrnghm2d	⊢ (𝜑 → 𝐹 ∈ (𝑅 RngHom 𝑆))

Distinct variable groups:   𝜑,𝑥,𝑦   𝑥,𝐵,𝑦   𝑥,𝐹,𝑦   𝑥,𝑅,𝑦   𝑥,𝑆,𝑦

Allowed substitution hints:   · (𝑥,𝑦)   × (𝑥,𝑦)

Proof of Theorem isrnghm2d

Step	Hyp	Ref	Expression
1		isrnghmd.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ Rng)
2		isrnghmd.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ Rng)
3	1, 2	jca 511	. 2 ⊢ (𝜑 → (𝑅 ∈ Rng ∧ 𝑆 ∈ Rng))
4		isrnghm2d.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑅 GrpHom 𝑆))
5		isrnghmd.ht	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) × (𝐹‘𝑦)))
6	5	ralrimivva 3181	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) × (𝐹‘𝑦)))
7	4, 6	jca 511	. 2 ⊢ (𝜑 → (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) × (𝐹‘𝑦))))
8		isrnghmd.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
9		isrnghmd.t	. . 3 ⊢ · = (.r‘𝑅)
10		isrnghmd.u	. . 3 ⊢ × = (.r‘𝑆)
11	8, 9, 10	isrnghm 20357	. 2 ⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) ↔ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) × (𝐹‘𝑦)))))
12	3, 7, 11	sylanbrc 583	1 ⊢ (𝜑 → 𝐹 ∈ (𝑅 RngHom 𝑆))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3045  ‘cfv 6514  (class class class)co 7390  Basecbs 17186  ._rcmulr 17228   GrpHom cghm 19151  Rngcrng 20068   RngHom crnghm 20350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-map 8804  df-ghm 19152  df-abl 19720  df-rng 20069  df-rnghm 20352

This theorem is referenced by:  isrnghmd  20367