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Mirrors > Home > MPE Home > Th. List > isrhm2d | Structured version Visualization version GIF version |
Description: Demonstration of ring homomorphism. (Contributed by Mario Carneiro, 13-Jun-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))) |
isrhm2d.f | ⊢ (𝜑 → 𝐹 ∈ (𝑅 GrpHom 𝑆)) |
Ref | Expression |
---|---|
isrhm2d | ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isrhmd.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
2 | isrhmd.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ Ring) | |
3 | 1, 2 | jca 507 | . 2 ⊢ (𝜑 → (𝑅 ∈ Ring ∧ 𝑆 ∈ Ring)) |
4 | isrhm2d.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑅 GrpHom 𝑆)) | |
5 | eqid 2777 | . . . . . . 7 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
6 | 5 | ringmgp 18940 | . . . . . 6 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑅) ∈ Mnd) |
7 | 1, 6 | syl 17 | . . . . 5 ⊢ (𝜑 → (mulGrp‘𝑅) ∈ Mnd) |
8 | eqid 2777 | . . . . . . 7 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆) | |
9 | 8 | ringmgp 18940 | . . . . . 6 ⊢ (𝑆 ∈ Ring → (mulGrp‘𝑆) ∈ Mnd) |
10 | 2, 9 | syl 17 | . . . . 5 ⊢ (𝜑 → (mulGrp‘𝑆) ∈ Mnd) |
11 | 7, 10 | jca 507 | . . . 4 ⊢ (𝜑 → ((mulGrp‘𝑅) ∈ Mnd ∧ (mulGrp‘𝑆) ∈ Mnd)) |
12 | isrhmd.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑅) | |
13 | eqid 2777 | . . . . . . 7 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
14 | 12, 13 | ghmf 18048 | . . . . . 6 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝐹:𝐵⟶(Base‘𝑆)) |
15 | 4, 14 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐹:𝐵⟶(Base‘𝑆)) |
16 | isrhmd.ht | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) × (𝐹‘𝑦))) | |
17 | 16 | ralrimivva 3152 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) × (𝐹‘𝑦))) |
18 | isrhmd.ho | . . . . . 6 ⊢ (𝜑 → (𝐹‘ 1 ) = 𝑁) | |
19 | isrhmd.o | . . . . . . . 8 ⊢ 1 = (1r‘𝑅) | |
20 | 5, 19 | ringidval 18890 | . . . . . . 7 ⊢ 1 = (0g‘(mulGrp‘𝑅)) |
21 | 20 | fveq2i 6449 | . . . . . 6 ⊢ (𝐹‘ 1 ) = (𝐹‘(0g‘(mulGrp‘𝑅))) |
22 | isrhmd.n | . . . . . . 7 ⊢ 𝑁 = (1r‘𝑆) | |
23 | 8, 22 | ringidval 18890 | . . . . . 6 ⊢ 𝑁 = (0g‘(mulGrp‘𝑆)) |
24 | 18, 21, 23 | 3eqtr3g 2836 | . . . . 5 ⊢ (𝜑 → (𝐹‘(0g‘(mulGrp‘𝑅))) = (0g‘(mulGrp‘𝑆))) |
25 | 15, 17, 24 | 3jca 1119 | . . . 4 ⊢ (𝜑 → (𝐹:𝐵⟶(Base‘𝑆) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) × (𝐹‘𝑦)) ∧ (𝐹‘(0g‘(mulGrp‘𝑅))) = (0g‘(mulGrp‘𝑆)))) |
26 | 5, 12 | mgpbas 18882 | . . . . 5 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅)) |
27 | 8, 13 | mgpbas 18882 | . . . . 5 ⊢ (Base‘𝑆) = (Base‘(mulGrp‘𝑆)) |
28 | isrhmd.t | . . . . . 6 ⊢ · = (.r‘𝑅) | |
29 | 5, 28 | mgpplusg 18880 | . . . . 5 ⊢ · = (+g‘(mulGrp‘𝑅)) |
30 | isrhmd.u | . . . . . 6 ⊢ × = (.r‘𝑆) | |
31 | 8, 30 | mgpplusg 18880 | . . . . 5 ⊢ × = (+g‘(mulGrp‘𝑆)) |
32 | eqid 2777 | . . . . 5 ⊢ (0g‘(mulGrp‘𝑅)) = (0g‘(mulGrp‘𝑅)) | |
33 | eqid 2777 | . . . . 5 ⊢ (0g‘(mulGrp‘𝑆)) = (0g‘(mulGrp‘𝑆)) | |
34 | 26, 27, 29, 31, 32, 33 | ismhm 17723 | . . . 4 ⊢ (𝐹 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑆)) ↔ (((mulGrp‘𝑅) ∈ Mnd ∧ (mulGrp‘𝑆) ∈ Mnd) ∧ (𝐹:𝐵⟶(Base‘𝑆) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) × (𝐹‘𝑦)) ∧ (𝐹‘(0g‘(mulGrp‘𝑅))) = (0g‘(mulGrp‘𝑆))))) |
35 | 11, 25, 34 | sylanbrc 578 | . . 3 ⊢ (𝜑 → 𝐹 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑆))) |
36 | 4, 35 | jca 507 | . 2 ⊢ (𝜑 → (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑆)))) |
37 | 5, 8 | isrhm 19110 | . 2 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) ↔ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Ring) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑆))))) |
38 | 3, 36, 37 | sylanbrc 578 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2106 ∀wral 3089 ⟶wf 6131 ‘cfv 6135 (class class class)co 6922 Basecbs 16255 .rcmulr 16339 0gc0g 16486 Mndcmnd 17680 MndHom cmhm 17719 GrpHom cghm 18041 mulGrpcmgp 18876 1rcur 18888 Ringcrg 18934 RingHom crh 19101 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-plusg 16351 df-0g 16488 df-mhm 17721 df-ghm 18042 df-mgp 18877 df-ur 18889 df-ring 18936 df-rnghom 19104 |
This theorem is referenced by: isrhmd 19118 qusrhm 19634 asclrhm 19739 mulgrhm 20242 rhmopp 30381 |
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