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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 2823 | . . . . . 6 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
5 | 4 | ringmgp 19305 | . . . . 5 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑅) ∈ Mnd) |
6 | 1, 5 | syl 17 | . . . 4 ⊢ (𝜑 → (mulGrp‘𝑅) ∈ Mnd) |
7 | eqid 2823 | . . . . . 6 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆) | |
8 | 7 | ringmgp 19305 | . . . . 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 2823 | . . . . . . 7 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
12 | 10, 11 | ghmf 18364 | . . . . . 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 3193 | . . . . 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 19255 | . . . . . . 7 ⊢ 1 = (0g‘(mulGrp‘𝑅)) |
19 | 18 | fveq2i 6675 | . . . . . 6 ⊢ (𝐹‘ 1 ) = (𝐹‘(0g‘(mulGrp‘𝑅))) |
20 | isrhmd.n | . . . . . . 7 ⊢ 𝑁 = (1r‘𝑆) | |
21 | 7, 20 | ringidval 19255 | . . . . . 6 ⊢ 𝑁 = (0g‘(mulGrp‘𝑆)) |
22 | 16, 19, 21 | 3eqtr3g 2881 | . . . . 5 ⊢ (𝜑 → (𝐹‘(0g‘(mulGrp‘𝑅))) = (0g‘(mulGrp‘𝑆))) |
23 | 13, 15, 22 | 3jca 1124 | . . . 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 19247 | . . . . 5 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅)) |
25 | 7, 11 | mgpbas 19247 | . . . . 5 ⊢ (Base‘𝑆) = (Base‘(mulGrp‘𝑆)) |
26 | isrhmd.t | . . . . . 6 ⊢ · = (.r‘𝑅) | |
27 | 4, 26 | mgpplusg 19245 | . . . . 5 ⊢ · = (+g‘(mulGrp‘𝑅)) |
28 | isrhmd.u | . . . . . 6 ⊢ × = (.r‘𝑆) | |
29 | 7, 28 | mgpplusg 19245 | . . . . 5 ⊢ × = (+g‘(mulGrp‘𝑆)) |
30 | eqid 2823 | . . . . 5 ⊢ (0g‘(mulGrp‘𝑅)) = (0g‘(mulGrp‘𝑅)) | |
31 | eqid 2823 | . . . . 5 ⊢ (0g‘(mulGrp‘𝑆)) = (0g‘(mulGrp‘𝑆)) | |
32 | 24, 25, 27, 29, 30, 31 | ismhm 17960 | . . . 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 1340 | . . 3 ⊢ (𝜑 → 𝐹 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑆))) |
34 | 3, 33 | jca 514 | . 2 ⊢ (𝜑 → (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑆)))) |
35 | 4, 7 | isrhm 19475 | . 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 1340 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 .rcmulr 16568 0gc0g 16715 Mndcmnd 17913 MndHom cmhm 17956 GrpHom cghm 18357 mulGrpcmgp 19241 1rcur 19253 Ringcrg 19299 RingHom crh 19466 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-plusg 16580 df-0g 16717 df-mhm 17958 df-ghm 18358 df-mgp 19242 df-ur 19254 df-ring 19301 df-rnghom 19469 |
This theorem is referenced by: isrhmd 19483 qusrhm 20012 asclrhm 20121 mulgrhm 20647 rhmopp 30894 |
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