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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > isrnghmmul | Structured version Visualization version GIF version |
Description: A function is a non-unital ring homomorphism iff it preserves both addition and multiplication. (Contributed by AV, 27-Feb-2020.) |
Ref | Expression |
---|---|
isrnghmmul.m | ⊢ 𝑀 = (mulGrp‘𝑅) |
isrnghmmul.n | ⊢ 𝑁 = (mulGrp‘𝑆) |
Ref | Expression |
---|---|
isrnghmmul | ⊢ (𝐹 ∈ (𝑅 RngHomo 𝑆) ↔ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ (𝑀 MgmHom 𝑁)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2651 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | eqid 2651 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
3 | eqid 2651 | . . 3 ⊢ (.r‘𝑆) = (.r‘𝑆) | |
4 | 1, 2, 3 | isrnghm 42217 | . 2 ⊢ (𝐹 ∈ (𝑅 RngHomo 𝑆) ↔ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(.r‘𝑅)𝑦)) = ((𝐹‘𝑥)(.r‘𝑆)(𝐹‘𝑦))))) |
5 | isrnghmmul.m | . . . . . . . . . . 11 ⊢ 𝑀 = (mulGrp‘𝑅) | |
6 | 5 | rngmgp 42203 | . . . . . . . . . 10 ⊢ (𝑅 ∈ Rng → 𝑀 ∈ SGrp) |
7 | sgrpmgm 17336 | . . . . . . . . . 10 ⊢ (𝑀 ∈ SGrp → 𝑀 ∈ Mgm) | |
8 | 6, 7 | syl 17 | . . . . . . . . 9 ⊢ (𝑅 ∈ Rng → 𝑀 ∈ Mgm) |
9 | isrnghmmul.n | . . . . . . . . . . 11 ⊢ 𝑁 = (mulGrp‘𝑆) | |
10 | 9 | rngmgp 42203 | . . . . . . . . . 10 ⊢ (𝑆 ∈ Rng → 𝑁 ∈ SGrp) |
11 | sgrpmgm 17336 | . . . . . . . . . 10 ⊢ (𝑁 ∈ SGrp → 𝑁 ∈ Mgm) | |
12 | 10, 11 | syl 17 | . . . . . . . . 9 ⊢ (𝑆 ∈ Rng → 𝑁 ∈ Mgm) |
13 | 8, 12 | anim12i 589 | . . . . . . . 8 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → (𝑀 ∈ Mgm ∧ 𝑁 ∈ Mgm)) |
14 | eqid 2651 | . . . . . . . . 9 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
15 | 1, 14 | ghmf 17711 | . . . . . . . 8 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆)) |
16 | 13, 15 | anim12i 589 | . . . . . . 7 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ 𝐹 ∈ (𝑅 GrpHom 𝑆)) → ((𝑀 ∈ Mgm ∧ 𝑁 ∈ Mgm) ∧ 𝐹:(Base‘𝑅)⟶(Base‘𝑆))) |
17 | 16 | biantrurd 528 | . . . . . 6 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ 𝐹 ∈ (𝑅 GrpHom 𝑆)) → (∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(.r‘𝑅)𝑦)) = ((𝐹‘𝑥)(.r‘𝑆)(𝐹‘𝑦)) ↔ (((𝑀 ∈ Mgm ∧ 𝑁 ∈ Mgm) ∧ 𝐹:(Base‘𝑅)⟶(Base‘𝑆)) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(.r‘𝑅)𝑦)) = ((𝐹‘𝑥)(.r‘𝑆)(𝐹‘𝑦))))) |
18 | anass 682 | . . . . . 6 ⊢ ((((𝑀 ∈ Mgm ∧ 𝑁 ∈ Mgm) ∧ 𝐹:(Base‘𝑅)⟶(Base‘𝑆)) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(.r‘𝑅)𝑦)) = ((𝐹‘𝑥)(.r‘𝑆)(𝐹‘𝑦))) ↔ ((𝑀 ∈ Mgm ∧ 𝑁 ∈ Mgm) ∧ (𝐹:(Base‘𝑅)⟶(Base‘𝑆) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(.r‘𝑅)𝑦)) = ((𝐹‘𝑥)(.r‘𝑆)(𝐹‘𝑦))))) | |
19 | 17, 18 | syl6bb 276 | . . . . 5 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ 𝐹 ∈ (𝑅 GrpHom 𝑆)) → (∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(.r‘𝑅)𝑦)) = ((𝐹‘𝑥)(.r‘𝑆)(𝐹‘𝑦)) ↔ ((𝑀 ∈ Mgm ∧ 𝑁 ∈ Mgm) ∧ (𝐹:(Base‘𝑅)⟶(Base‘𝑆) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(.r‘𝑅)𝑦)) = ((𝐹‘𝑥)(.r‘𝑆)(𝐹‘𝑦)))))) |
20 | 5, 1 | mgpbas 18541 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑀) |
21 | 9, 14 | mgpbas 18541 | . . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑁) |
22 | 5, 2 | mgpplusg 18539 | . . . . . 6 ⊢ (.r‘𝑅) = (+g‘𝑀) |
23 | 9, 3 | mgpplusg 18539 | . . . . . 6 ⊢ (.r‘𝑆) = (+g‘𝑁) |
24 | 20, 21, 22, 23 | ismgmhm 42108 | . . . . 5 ⊢ (𝐹 ∈ (𝑀 MgmHom 𝑁) ↔ ((𝑀 ∈ Mgm ∧ 𝑁 ∈ Mgm) ∧ (𝐹:(Base‘𝑅)⟶(Base‘𝑆) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(.r‘𝑅)𝑦)) = ((𝐹‘𝑥)(.r‘𝑆)(𝐹‘𝑦))))) |
25 | 19, 24 | syl6bbr 278 | . . . 4 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ 𝐹 ∈ (𝑅 GrpHom 𝑆)) → (∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(.r‘𝑅)𝑦)) = ((𝐹‘𝑥)(.r‘𝑆)(𝐹‘𝑦)) ↔ 𝐹 ∈ (𝑀 MgmHom 𝑁))) |
26 | 25 | pm5.32da 674 | . . 3 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(.r‘𝑅)𝑦)) = ((𝐹‘𝑥)(.r‘𝑆)(𝐹‘𝑦))) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ (𝑀 MgmHom 𝑁)))) |
27 | 26 | pm5.32i 670 | . 2 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(.r‘𝑅)𝑦)) = ((𝐹‘𝑥)(.r‘𝑆)(𝐹‘𝑦)))) ↔ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ (𝑀 MgmHom 𝑁)))) |
28 | 4, 27 | bitri 264 | 1 ⊢ (𝐹 ∈ (𝑅 RngHomo 𝑆) ↔ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ (𝑀 MgmHom 𝑁)))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ∀wral 2941 ⟶wf 5922 ‘cfv 5926 (class class class)co 6690 Basecbs 15904 .rcmulr 15989 Mgmcmgm 17287 SGrpcsgrp 17330 GrpHom cghm 17704 mulGrpcmgp 18535 MgmHom cmgmhm 42102 Rngcrng 42199 RngHomo crngh 42210 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-map 7901 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-plusg 16001 df-sgrp 17331 df-ghm 17705 df-abl 18242 df-mgp 18536 df-mgmhm 42104 df-rng0 42200 df-rnghomo 42212 |
This theorem is referenced by: rnghmmgmhm 42219 rnghmval2 42220 rnghmf1o 42228 rnghmco 42232 idrnghm 42233 c0rnghm 42238 rhmisrnghm 42245 |
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