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| Mirrors > Home > MPE Home > Th. List > isrnghmd | 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))) | 
| isrnghmd.c | ⊢ 𝐶 = (Base‘𝑆) | 
| isrnghmd.p | ⊢ + = (+g‘𝑅) | 
| isrnghmd.q | ⊢ ⨣ = (+g‘𝑆) | 
| isrnghmd.f | ⊢ (𝜑 → 𝐹:𝐵⟶𝐶) | 
| isrnghmd.hp | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) | 
| Ref | Expression | 
|---|---|
| isrnghmd | ⊢ (𝜑 → 𝐹 ∈ (𝑅 RngHom 𝑆)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | isrnghmd.b | . 2 ⊢ 𝐵 = (Base‘𝑅) | |
| 2 | isrnghmd.t | . 2 ⊢ · = (.r‘𝑅) | |
| 3 | isrnghmd.u | . 2 ⊢ × = (.r‘𝑆) | |
| 4 | isrnghmd.r | . 2 ⊢ (𝜑 → 𝑅 ∈ Rng) | |
| 5 | isrnghmd.s | . 2 ⊢ (𝜑 → 𝑆 ∈ Rng) | |
| 6 | isrnghmd.ht | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) × (𝐹‘𝑦))) | |
| 7 | isrnghmd.c | . . 3 ⊢ 𝐶 = (Base‘𝑆) | |
| 8 | isrnghmd.p | . . 3 ⊢ + = (+g‘𝑅) | |
| 9 | isrnghmd.q | . . 3 ⊢ ⨣ = (+g‘𝑆) | |
| 10 | rngabl 20152 | . . . 4 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel) | |
| 11 | ablgrp 19803 | . . . 4 ⊢ (𝑅 ∈ Abel → 𝑅 ∈ Grp) | |
| 12 | 4, 10, 11 | 3syl 18 | . . 3 ⊢ (𝜑 → 𝑅 ∈ Grp) | 
| 13 | rngabl 20152 | . . . 4 ⊢ (𝑆 ∈ Rng → 𝑆 ∈ Abel) | |
| 14 | ablgrp 19803 | . . . 4 ⊢ (𝑆 ∈ Abel → 𝑆 ∈ Grp) | |
| 15 | 5, 13, 14 | 3syl 18 | . . 3 ⊢ (𝜑 → 𝑆 ∈ Grp) | 
| 16 | isrnghmd.f | . . 3 ⊢ (𝜑 → 𝐹:𝐵⟶𝐶) | |
| 17 | isrnghmd.hp | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) | |
| 18 | 1, 7, 8, 9, 12, 15, 16, 17 | isghmd 19243 | . 2 ⊢ (𝜑 → 𝐹 ∈ (𝑅 GrpHom 𝑆)) | 
| 19 | 1, 2, 3, 4, 5, 6, 18 | isrnghm2d 20450 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝑅 RngHom 𝑆)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 .rcmulr 17298 Grpcgrp 18951 Abelcabl 19799 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: (None) | 
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