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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 553 | . 2 ⊢ (𝜑 → (𝑅 ∈ Ring ∧ 𝑆 ∈ Ring)) |
4 | isrhm2d.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑅 GrpHom 𝑆)) | |
5 | eqid 2651 | . . . . . . 7 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
6 | 5 | ringmgp 18599 | . . . . . 6 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑅) ∈ Mnd) |
7 | 1, 6 | syl 17 | . . . . 5 ⊢ (𝜑 → (mulGrp‘𝑅) ∈ Mnd) |
8 | eqid 2651 | . . . . . . 7 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆) | |
9 | 8 | ringmgp 18599 | . . . . . 6 ⊢ (𝑆 ∈ Ring → (mulGrp‘𝑆) ∈ Mnd) |
10 | 2, 9 | syl 17 | . . . . 5 ⊢ (𝜑 → (mulGrp‘𝑆) ∈ Mnd) |
11 | 7, 10 | jca 553 | . . . 4 ⊢ (𝜑 → ((mulGrp‘𝑅) ∈ Mnd ∧ (mulGrp‘𝑆) ∈ Mnd)) |
12 | isrhmd.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑅) | |
13 | eqid 2651 | . . . . . . 7 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
14 | 12, 13 | ghmf 17711 | . . . . . 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 3000 | . . . . 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 18549 | . . . . . . 7 ⊢ 1 = (0g‘(mulGrp‘𝑅)) |
21 | 20 | fveq2i 6232 | . . . . . 6 ⊢ (𝐹‘ 1 ) = (𝐹‘(0g‘(mulGrp‘𝑅))) |
22 | isrhmd.n | . . . . . . 7 ⊢ 𝑁 = (1r‘𝑆) | |
23 | 8, 22 | ringidval 18549 | . . . . . 6 ⊢ 𝑁 = (0g‘(mulGrp‘𝑆)) |
24 | 18, 21, 23 | 3eqtr3g 2708 | . . . . 5 ⊢ (𝜑 → (𝐹‘(0g‘(mulGrp‘𝑅))) = (0g‘(mulGrp‘𝑆))) |
25 | 15, 17, 24 | 3jca 1261 | . . . 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 18541 | . . . . 5 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅)) |
27 | 8, 13 | mgpbas 18541 | . . . . 5 ⊢ (Base‘𝑆) = (Base‘(mulGrp‘𝑆)) |
28 | isrhmd.t | . . . . . 6 ⊢ · = (.r‘𝑅) | |
29 | 5, 28 | mgpplusg 18539 | . . . . 5 ⊢ · = (+g‘(mulGrp‘𝑅)) |
30 | isrhmd.u | . . . . . 6 ⊢ × = (.r‘𝑆) | |
31 | 8, 30 | mgpplusg 18539 | . . . . 5 ⊢ × = (+g‘(mulGrp‘𝑆)) |
32 | eqid 2651 | . . . . 5 ⊢ (0g‘(mulGrp‘𝑅)) = (0g‘(mulGrp‘𝑅)) | |
33 | eqid 2651 | . . . . 5 ⊢ (0g‘(mulGrp‘𝑆)) = (0g‘(mulGrp‘𝑆)) | |
34 | 26, 27, 29, 31, 32, 33 | ismhm 17384 | . . . 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 699 | . . 3 ⊢ (𝜑 → 𝐹 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑆))) |
36 | 4, 35 | jca 553 | . 2 ⊢ (𝜑 → (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑆)))) |
37 | 5, 8 | isrhm 18769 | . 2 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) ↔ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Ring) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑆))))) |
38 | 3, 36, 37 | sylanbrc 699 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 ∀wral 2941 ⟶wf 5922 ‘cfv 5926 (class class class)co 6690 Basecbs 15904 .rcmulr 15989 0gc0g 16147 Mndcmnd 17341 MndHom cmhm 17380 GrpHom cghm 17704 mulGrpcmgp 18535 1rcur 18547 Ringcrg 18593 RingHom crh 18760 |
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-0g 16149 df-mhm 17382 df-ghm 17705 df-mgp 18536 df-ur 18548 df-ring 18595 df-rnghom 18763 |
This theorem is referenced by: isrhmd 18777 qusrhm 19285 asclrhm 19390 mulgrhm 19894 rhmopp 29947 |
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