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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 | . 2 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
2 | isrhmd.s | . 2 ⊢ (𝜑 → 𝑆 ∈ Ring) | |
3 | isrhm2d.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑅 GrpHom 𝑆)) | |
4 | eqid 2725 | . . . . . 6 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
5 | 4 | ringmgp 20217 | . . . . 5 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑅) ∈ Mnd) |
6 | 1, 5 | syl 17 | . . . 4 ⊢ (𝜑 → (mulGrp‘𝑅) ∈ Mnd) |
7 | eqid 2725 | . . . . . 6 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆) | |
8 | 7 | ringmgp 20217 | . . . . 5 ⊢ (𝑆 ∈ Ring → (mulGrp‘𝑆) ∈ Mnd) |
9 | 2, 8 | syl 17 | . . . 4 ⊢ (𝜑 → (mulGrp‘𝑆) ∈ Mnd) |
10 | isrhmd.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑅) | |
11 | eqid 2725 | . . . . . . 7 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
12 | 10, 11 | ghmf 19209 | . . . . . 6 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝐹:𝐵⟶(Base‘𝑆)) |
13 | 3, 12 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐹:𝐵⟶(Base‘𝑆)) |
14 | isrhmd.ht | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) × (𝐹‘𝑦))) | |
15 | 14 | ralrimivva 3190 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) × (𝐹‘𝑦))) |
16 | isrhmd.ho | . . . . . 6 ⊢ (𝜑 → (𝐹‘ 1 ) = 𝑁) | |
17 | isrhmd.o | . . . . . . . 8 ⊢ 1 = (1r‘𝑅) | |
18 | 4, 17 | ringidval 20161 | . . . . . . 7 ⊢ 1 = (0g‘(mulGrp‘𝑅)) |
19 | 18 | fveq2i 6903 | . . . . . 6 ⊢ (𝐹‘ 1 ) = (𝐹‘(0g‘(mulGrp‘𝑅))) |
20 | isrhmd.n | . . . . . . 7 ⊢ 𝑁 = (1r‘𝑆) | |
21 | 7, 20 | ringidval 20161 | . . . . . 6 ⊢ 𝑁 = (0g‘(mulGrp‘𝑆)) |
22 | 16, 19, 21 | 3eqtr3g 2788 | . . . . 5 ⊢ (𝜑 → (𝐹‘(0g‘(mulGrp‘𝑅))) = (0g‘(mulGrp‘𝑆))) |
23 | 13, 15, 22 | 3jca 1125 | . . . 4 ⊢ (𝜑 → (𝐹:𝐵⟶(Base‘𝑆) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) × (𝐹‘𝑦)) ∧ (𝐹‘(0g‘(mulGrp‘𝑅))) = (0g‘(mulGrp‘𝑆)))) |
24 | 4, 10 | mgpbas 20118 | . . . . 5 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅)) |
25 | 7, 11 | mgpbas 20118 | . . . . 5 ⊢ (Base‘𝑆) = (Base‘(mulGrp‘𝑆)) |
26 | isrhmd.t | . . . . . 6 ⊢ · = (.r‘𝑅) | |
27 | 4, 26 | mgpplusg 20116 | . . . . 5 ⊢ · = (+g‘(mulGrp‘𝑅)) |
28 | isrhmd.u | . . . . . 6 ⊢ × = (.r‘𝑆) | |
29 | 7, 28 | mgpplusg 20116 | . . . . 5 ⊢ × = (+g‘(mulGrp‘𝑆)) |
30 | eqid 2725 | . . . . 5 ⊢ (0g‘(mulGrp‘𝑅)) = (0g‘(mulGrp‘𝑅)) | |
31 | eqid 2725 | . . . . 5 ⊢ (0g‘(mulGrp‘𝑆)) = (0g‘(mulGrp‘𝑆)) | |
32 | 24, 25, 27, 29, 30, 31 | ismhm 18770 | . . . 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))))) |
33 | 6, 9, 23, 32 | syl21anbrc 1341 | . . 3 ⊢ (𝜑 → 𝐹 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑆))) |
34 | 3, 33 | jca 510 | . 2 ⊢ (𝜑 → (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑆)))) |
35 | 4, 7 | isrhm 20455 | . 2 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) ↔ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Ring) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑆))))) |
36 | 1, 2, 34, 35 | syl21anbrc 1341 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3050 ⟶wf 6549 ‘cfv 6553 (class class class)co 7423 Basecbs 17208 .rcmulr 17262 0gc0g 17449 Mndcmnd 18722 MndHom cmhm 18766 GrpHom cghm 19201 mulGrpcmgp 20112 1rcur 20159 Ringcrg 20211 RingHom crh 20446 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 ax-cnex 11210 ax-resscn 11211 ax-1cn 11212 ax-icn 11213 ax-addcl 11214 ax-addrcl 11215 ax-mulcl 11216 ax-mulrcl 11217 ax-mulcom 11218 ax-addass 11219 ax-mulass 11220 ax-distr 11221 ax-i2m1 11222 ax-1ne0 11223 ax-1rid 11224 ax-rnegex 11225 ax-rrecex 11226 ax-cnre 11227 ax-pre-lttri 11228 ax-pre-lttrn 11229 ax-pre-ltadd 11230 ax-pre-mulgt0 11231 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5579 df-eprel 5585 df-po 5593 df-so 5594 df-fr 5636 df-we 5638 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-pred 6311 df-ord 6378 df-on 6379 df-lim 6380 df-suc 6381 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7379 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7876 df-1st 8002 df-2nd 8003 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-er 8733 df-map 8856 df-en 8974 df-dom 8975 df-sdom 8976 df-pnf 11296 df-mnf 11297 df-xr 11298 df-ltxr 11299 df-le 11300 df-sub 11492 df-neg 11493 df-nn 12260 df-2 12322 df-sets 17161 df-slot 17179 df-ndx 17191 df-base 17209 df-plusg 17274 df-0g 17451 df-mhm 18768 df-ghm 19202 df-mgp 20113 df-ur 20160 df-ring 20213 df-rhm 20449 |
This theorem is referenced by: isrhmd 20465 rhmopp 20486 qusrhm 21212 rhmqusnsg 21221 mulgrhm 21459 asclrhm 21879 rhmquskerlem 33277 |
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