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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 3202 | . . 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 20441 | . 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 1540 ∈ wcel 2108 ∀wral 3061 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 .rcmulr 17298 GrpHom cghm 19230 Rngcrng 20149 RngHom crnghm 20434 | 
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-map 8868 df-ghm 19231 df-abl 19801 df-rng 20150 df-rnghm 20436 | 
| This theorem is referenced by: isrnghmd 20451 | 
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