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Mirrors > Home > MPE Home > Th. List > isrhmd | Structured version Visualization version GIF version |
Description: Demonstration of ring homomorphism. (Contributed by Stefan O'Rear, 8-Mar-2015.) |
Ref | Expression |
---|---|
isrhmd.b | ⊢ 𝐵 = (Base‘𝑅) |
isrhmd.o | ⊢ 1 = (1r‘𝑅) |
isrhmd.n | ⊢ 𝑁 = (1r‘𝑆) |
isrhmd.t | ⊢ · = (.r‘𝑅) |
isrhmd.u | ⊢ × = (.r‘𝑆) |
isrhmd.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
isrhmd.s | ⊢ (𝜑 → 𝑆 ∈ Ring) |
isrhmd.ho | ⊢ (𝜑 → (𝐹‘ 1 ) = 𝑁) |
isrhmd.ht | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) × (𝐹‘𝑦))) |
isrhmd.c | ⊢ 𝐶 = (Base‘𝑆) |
isrhmd.p | ⊢ + = (+g‘𝑅) |
isrhmd.q | ⊢ ⨣ = (+g‘𝑆) |
isrhmd.f | ⊢ (𝜑 → 𝐹:𝐵⟶𝐶) |
isrhmd.hp | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) |
Ref | Expression |
---|---|
isrhmd | ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isrhmd.b | . 2 ⊢ 𝐵 = (Base‘𝑅) | |
2 | isrhmd.o | . 2 ⊢ 1 = (1r‘𝑅) | |
3 | isrhmd.n | . 2 ⊢ 𝑁 = (1r‘𝑆) | |
4 | isrhmd.t | . 2 ⊢ · = (.r‘𝑅) | |
5 | isrhmd.u | . 2 ⊢ × = (.r‘𝑆) | |
6 | isrhmd.r | . 2 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
7 | isrhmd.s | . 2 ⊢ (𝜑 → 𝑆 ∈ Ring) | |
8 | isrhmd.ho | . 2 ⊢ (𝜑 → (𝐹‘ 1 ) = 𝑁) | |
9 | isrhmd.ht | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) × (𝐹‘𝑦))) | |
10 | isrhmd.c | . . 3 ⊢ 𝐶 = (Base‘𝑆) | |
11 | isrhmd.p | . . 3 ⊢ + = (+g‘𝑅) | |
12 | isrhmd.q | . . 3 ⊢ ⨣ = (+g‘𝑆) | |
13 | ringgrp 18772 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
14 | 6, 13 | syl 17 | . . 3 ⊢ (𝜑 → 𝑅 ∈ Grp) |
15 | ringgrp 18772 | . . . 4 ⊢ (𝑆 ∈ Ring → 𝑆 ∈ Grp) | |
16 | 7, 15 | syl 17 | . . 3 ⊢ (𝜑 → 𝑆 ∈ Grp) |
17 | isrhmd.f | . . 3 ⊢ (𝜑 → 𝐹:𝐵⟶𝐶) | |
18 | isrhmd.hp | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) | |
19 | 1, 10, 11, 12, 14, 16, 17, 18 | isghmd 17890 | . 2 ⊢ (𝜑 → 𝐹 ∈ (𝑅 GrpHom 𝑆)) |
20 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 19 | isrhm2d 18950 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2139 ⟶wf 6045 ‘cfv 6049 (class class class)co 6814 Basecbs 16079 +gcplusg 16163 .rcmulr 16164 Grpcgrp 17643 1rcur 18721 Ringcrg 18767 RingHom crh 18934 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-cnex 10204 ax-resscn 10205 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-addrcl 10209 ax-mulcl 10210 ax-mulrcl 10211 ax-mulcom 10212 ax-addass 10213 ax-mulass 10214 ax-distr 10215 ax-i2m1 10216 ax-1ne0 10217 ax-1rid 10218 ax-rnegex 10219 ax-rrecex 10220 ax-cnre 10221 ax-pre-lttri 10222 ax-pre-lttrn 10223 ax-pre-ltadd 10224 ax-pre-mulgt0 10225 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6775 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-om 7232 df-wrecs 7577 df-recs 7638 df-rdg 7676 df-er 7913 df-map 8027 df-en 8124 df-dom 8125 df-sdom 8126 df-pnf 10288 df-mnf 10289 df-xr 10290 df-ltxr 10291 df-le 10292 df-sub 10480 df-neg 10481 df-nn 11233 df-2 11291 df-ndx 16082 df-slot 16083 df-base 16085 df-sets 16086 df-plusg 16176 df-0g 16324 df-mhm 17556 df-ghm 17879 df-mgp 18710 df-ur 18722 df-ring 18769 df-rnghom 18937 |
This theorem is referenced by: issrngd 19083 evlslem1 19737 qqhrhm 30363 |
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