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 45435 | . . . 4 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel) | |
11 | ablgrp 19391 | . . . 4 ⊢ (𝑅 ∈ Abel → 𝑅 ∈ Grp) | |
12 | 4, 10, 11 | 3syl 18 | . . 3 ⊢ (𝜑 → 𝑅 ∈ Grp) |
13 | rngabl 45435 | . . . 4 ⊢ (𝑆 ∈ Rng → 𝑆 ∈ Abel) | |
14 | ablgrp 19391 | . . . 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 18843 | . 2 ⊢ (𝜑 → 𝐹 ∈ (𝑅 GrpHom 𝑆)) |
19 | 1, 2, 3, 4, 5, 6, 18 | isrnghm2d 45459 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝑅 RngHomo 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 Basecbs 16912 +gcplusg 16962 .rcmulr 16963 Grpcgrp 18577 Abelcabl 19387 Rngcrng 45432 RngHomo crngh 45443 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-map 8617 df-ghm 18832 df-abl 19389 df-rng0 45433 df-rnghomo 45445 |
This theorem is referenced by: (None) |
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