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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 RngHomo 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isrnghmd.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ Rng) | |
2 | isrnghmd.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ Rng) | |
3 | 1, 2 | jca 515 | . 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 3156 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) × (𝐹‘𝑦))) |
7 | 4, 6 | jca 515 | . 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 44516 | . 2 ⊢ (𝐹 ∈ (𝑅 RngHomo 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 586 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝑅 RngHomo 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 .rcmulr 16558 GrpHom cghm 18347 Rngcrng 44498 RngHomo crngh 44509 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-map 8391 df-ghm 18348 df-abl 18901 df-rng0 44499 df-rnghomo 44511 |
This theorem is referenced by: isrnghmd 44526 |
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