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Mirrors > Home > MPE Home > Th. List > isrnghm2d | Structured version Visualization version GIF version |
Description: Demonstration of non-unital ring homomorphism. (Contributed by AV, 23-Feb-2020.) |
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)) |
Ref | Expression |
---|---|
isrnghm2d | ⊢ (𝜑 → 𝐹 ∈ (𝑅 RngHom 𝑆)) |
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 3200 | . . 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 20458 | . 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)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 .rcmulr 17299 GrpHom cghm 19243 Rngcrng 20170 RngHom crnghm 20451 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-map 8867 df-ghm 19244 df-abl 19816 df-rng 20171 df-rnghm 20453 |
This theorem is referenced by: isrnghmd 20468 |
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