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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 2798 | . . . . . 6 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
5 | 4 | ringmgp 19296 | . . . . 5 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑅) ∈ Mnd) |
6 | 1, 5 | syl 17 | . . . 4 ⊢ (𝜑 → (mulGrp‘𝑅) ∈ Mnd) |
7 | eqid 2798 | . . . . . 6 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆) | |
8 | 7 | ringmgp 19296 | . . . . 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 2798 | . . . . . . 7 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
12 | 10, 11 | ghmf 18354 | . . . . . 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 3156 | . . . . 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 19246 | . . . . . . 7 ⊢ 1 = (0g‘(mulGrp‘𝑅)) |
19 | 18 | fveq2i 6648 | . . . . . 6 ⊢ (𝐹‘ 1 ) = (𝐹‘(0g‘(mulGrp‘𝑅))) |
20 | isrhmd.n | . . . . . . 7 ⊢ 𝑁 = (1r‘𝑆) | |
21 | 7, 20 | ringidval 19246 | . . . . . 6 ⊢ 𝑁 = (0g‘(mulGrp‘𝑆)) |
22 | 16, 19, 21 | 3eqtr3g 2856 | . . . . 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 19238 | . . . . 5 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅)) |
25 | 7, 11 | mgpbas 19238 | . . . . 5 ⊢ (Base‘𝑆) = (Base‘(mulGrp‘𝑆)) |
26 | isrhmd.t | . . . . . 6 ⊢ · = (.r‘𝑅) | |
27 | 4, 26 | mgpplusg 19236 | . . . . 5 ⊢ · = (+g‘(mulGrp‘𝑅)) |
28 | isrhmd.u | . . . . . 6 ⊢ × = (.r‘𝑆) | |
29 | 7, 28 | mgpplusg 19236 | . . . . 5 ⊢ × = (+g‘(mulGrp‘𝑆)) |
30 | eqid 2798 | . . . . 5 ⊢ (0g‘(mulGrp‘𝑅)) = (0g‘(mulGrp‘𝑅)) | |
31 | eqid 2798 | . . . . 5 ⊢ (0g‘(mulGrp‘𝑆)) = (0g‘(mulGrp‘𝑆)) | |
32 | 24, 25, 27, 29, 30, 31 | ismhm 17950 | . . . 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 515 | . 2 ⊢ (𝜑 → (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑆)))) |
35 | 4, 7 | isrhm 19469 | . 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 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 .rcmulr 16558 0gc0g 16705 Mndcmnd 17903 MndHom cmhm 17946 GrpHom cghm 18347 mulGrpcmgp 19232 1rcur 19244 Ringcrg 19290 RingHom crh 19460 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-plusg 16570 df-0g 16707 df-mhm 17948 df-ghm 18348 df-mgp 19233 df-ur 19245 df-ring 19292 df-rnghom 19463 |
This theorem is referenced by: isrhmd 19477 qusrhm 20003 mulgrhm 20191 asclrhm 20576 rhmopp 30943 |
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