Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 RngHomo 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 44155 | . . . 4 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel) | |
11 | ablgrp 18913 | . . . 4 ⊢ (𝑅 ∈ Abel → 𝑅 ∈ Grp) | |
12 | 4, 10, 11 | 3syl 18 | . . 3 ⊢ (𝜑 → 𝑅 ∈ Grp) |
13 | rngabl 44155 | . . . 4 ⊢ (𝑆 ∈ Rng → 𝑆 ∈ Abel) | |
14 | ablgrp 18913 | . . . 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 18369 | . 2 ⊢ (𝜑 → 𝐹 ∈ (𝑅 GrpHom 𝑆)) |
19 | 1, 2, 3, 4, 5, 6, 18 | isrnghm2d 44179 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝑅 RngHomo 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 +gcplusg 16567 .rcmulr 16568 Grpcgrp 18105 Abelcabl 18909 Rngcrng 44152 RngHomo crngh 44163 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-map 8410 df-ghm 18358 df-abl 18911 df-rng0 44153 df-rnghomo 44165 |
This theorem is referenced by: (None) |
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