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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 2733 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | eqid 2733 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
3 | eqid 2733 | . . 3 ⊢ (.r‘𝑆) = (.r‘𝑆) | |
4 | 1, 2, 3 | isrnghm 46276 | . 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 46262 | . . . . . . . . . 10 ⊢ (𝑅 ∈ Rng → 𝑀 ∈ Smgrp) |
7 | sgrpmgm 18556 | . . . . . . . . . 10 ⊢ (𝑀 ∈ Smgrp → 𝑀 ∈ Mgm) | |
8 | 6, 7 | syl 17 | . . . . . . . . 9 ⊢ (𝑅 ∈ Rng → 𝑀 ∈ Mgm) |
9 | isrnghmmul.n | . . . . . . . . . . 11 ⊢ 𝑁 = (mulGrp‘𝑆) | |
10 | 9 | rngmgp 46262 | . . . . . . . . . 10 ⊢ (𝑆 ∈ Rng → 𝑁 ∈ Smgrp) |
11 | sgrpmgm 18556 | . . . . . . . . . 10 ⊢ (𝑁 ∈ Smgrp → 𝑁 ∈ Mgm) | |
12 | 10, 11 | syl 17 | . . . . . . . . 9 ⊢ (𝑆 ∈ Rng → 𝑁 ∈ Mgm) |
13 | 8, 12 | anim12i 614 | . . . . . . . 8 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → (𝑀 ∈ Mgm ∧ 𝑁 ∈ Mgm)) |
14 | eqid 2733 | . . . . . . . . 9 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
15 | 1, 14 | ghmf 19017 | . . . . . . . 8 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆)) |
16 | 13, 15 | anim12i 614 | . . . . . . 7 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ 𝐹 ∈ (𝑅 GrpHom 𝑆)) → ((𝑀 ∈ Mgm ∧ 𝑁 ∈ Mgm) ∧ 𝐹:(Base‘𝑅)⟶(Base‘𝑆))) |
17 | 16 | biantrurd 534 | . . . . . 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 470 | . . . . . 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 | bitrdi 287 | . . . . 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 19907 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑀) |
21 | 9, 14 | mgpbas 19907 | . . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑁) |
22 | 5, 2 | mgpplusg 19905 | . . . . . 6 ⊢ (.r‘𝑅) = (+g‘𝑀) |
23 | 9, 3 | mgpplusg 19905 | . . . . . 6 ⊢ (.r‘𝑆) = (+g‘𝑁) |
24 | 20, 21, 22, 23 | ismgmhm 46163 | . . . . 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 | bitr4di 289 | . . . 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 580 | . . 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 576 | . 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 275 | 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 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3061 ⟶wf 6493 ‘cfv 6497 (class class class)co 7358 Basecbs 17088 .rcmulr 17139 Mgmcmgm 18500 Smgrpcsgrp 18550 GrpHom cghm 19010 mulGrpcmgp 19901 MgmHom cmgmhm 46157 Rngcrng 46258 RngHomo crngh 46269 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-map 8770 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-plusg 17151 df-sgrp 18551 df-ghm 19011 df-abl 19570 df-mgp 19902 df-mgmhm 46159 df-rng 46259 df-rnghomo 46271 |
This theorem is referenced by: rnghmmgmhm 46278 rnghmval2 46279 rnghmf1o 46287 rnghmco 46291 idrnghm 46292 c0rnghm 46297 rhmisrnghm 46304 |
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