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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 3201 | . . 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 20442 | . 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 1539 ∈ wcel 2107 ∀wral 3060 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 .rcmulr 17299 GrpHom cghm 19231 Rngcrng 20150 RngHom crnghm 20435 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-1st 8015 df-2nd 8016 df-map 8869 df-ghm 19232 df-abl 19802 df-rng 20151 df-rnghm 20437 | 
| This theorem is referenced by: isrnghmd 20452 | 
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