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Mirrors > Home > MPE Home > Th. List > rhmmul | Structured version Visualization version GIF version |
Description: A homomorphism of rings preserves multiplication. (Contributed by Mario Carneiro, 12-Jun-2015.) |
Ref | Expression |
---|---|
rhmmul.x | ⊢ 𝑋 = (Base‘𝑅) |
rhmmul.m | ⊢ · = (.r‘𝑅) |
rhmmul.n | ⊢ × = (.r‘𝑆) |
Ref | Expression |
---|---|
rhmmul | ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐹‘(𝐴 · 𝐵)) = ((𝐹‘𝐴) × (𝐹‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2724 | . . 3 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
2 | eqid 2724 | . . 3 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆) | |
3 | 1, 2 | rhmmhm 20371 | . 2 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑆))) |
4 | rhmmul.x | . . . 4 ⊢ 𝑋 = (Base‘𝑅) | |
5 | 1, 4 | mgpbas 20035 | . . 3 ⊢ 𝑋 = (Base‘(mulGrp‘𝑅)) |
6 | rhmmul.m | . . . 4 ⊢ · = (.r‘𝑅) | |
7 | 1, 6 | mgpplusg 20033 | . . 3 ⊢ · = (+g‘(mulGrp‘𝑅)) |
8 | rhmmul.n | . . . 4 ⊢ × = (.r‘𝑆) | |
9 | 2, 8 | mgpplusg 20033 | . . 3 ⊢ × = (+g‘(mulGrp‘𝑆)) |
10 | 5, 7, 9 | mhmlin 18713 | . 2 ⊢ ((𝐹 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑆)) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐹‘(𝐴 · 𝐵)) = ((𝐹‘𝐴) × (𝐹‘𝐵))) |
11 | 3, 10 | syl3an1 1160 | 1 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐹‘(𝐴 · 𝐵)) = ((𝐹‘𝐴) × (𝐹‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ‘cfv 6533 (class class class)co 7401 Basecbs 17143 .rcmulr 17197 MndHom cmhm 18701 mulGrpcmgp 20029 RingHom crh 20361 |
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 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-plusg 17209 df-0g 17386 df-mhm 18703 df-ghm 19129 df-mgp 20030 df-ur 20077 df-ring 20130 df-rhm 20364 |
This theorem is referenced by: rhmdvdsr 20400 rhmopp 20401 rhmunitinv 20403 srngmul 20691 domnchr 21391 znfld 21423 znidomb 21424 znunit 21426 znrrg 21428 evl1muld 22184 evl1scvarpw 22204 mat2pmatmul 22555 mat2pmatlin 22559 cayhamlem4 22712 ply1rem 26021 fta1glem2 26024 fta1blem 26026 dchrzrhmul 27095 lgsdchr 27204 lgseisenlem3 27226 lgseisenlem4 27227 rhmdvd 32902 kerunit 32903 rhmpreimaidl 33006 rhmquskerlem 33012 rhmimaidl 33019 rhmpreimaprmidl 33039 evls1fpws 33113 mdetpmtr1 33292 mdetpmtr12 33294 qqhghm 33457 qqhrhm 33458 fldhmf1 41448 imacrhmcl 41580 rhmcomulmpl 41613 evlsmulval 41630 evlmulval 41637 |
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