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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 2736 | . . 3 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
2 | eqid 2736 | . . 3 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆) | |
3 | 1, 2 | rhmmhm 19696 | . 2 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑆))) |
4 | rhmmul.x | . . . 4 ⊢ 𝑋 = (Base‘𝑅) | |
5 | 1, 4 | mgpbas 19464 | . . 3 ⊢ 𝑋 = (Base‘(mulGrp‘𝑅)) |
6 | rhmmul.m | . . . 4 ⊢ · = (.r‘𝑅) | |
7 | 1, 6 | mgpplusg 19462 | . . 3 ⊢ · = (+g‘(mulGrp‘𝑅)) |
8 | rhmmul.n | . . . 4 ⊢ × = (.r‘𝑆) | |
9 | 2, 8 | mgpplusg 19462 | . . 3 ⊢ × = (+g‘(mulGrp‘𝑆)) |
10 | 5, 7, 9 | mhmlin 18179 | . 2 ⊢ ((𝐹 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑆)) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐹‘(𝐴 · 𝐵)) = ((𝐹‘𝐴) × (𝐹‘𝐵))) |
11 | 3, 10 | syl3an1 1165 | 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 1089 = wceq 1543 ∈ wcel 2112 ‘cfv 6358 (class class class)co 7191 Basecbs 16666 .rcmulr 16750 MndHom cmhm 18170 mulGrpcmgp 19458 RingHom crh 19686 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-er 8369 df-map 8488 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-plusg 16762 df-0g 16900 df-mhm 18172 df-ghm 18574 df-mgp 19459 df-ur 19471 df-ring 19518 df-rnghom 19689 |
This theorem is referenced by: srngmul 19848 domnchr 20451 znfld 20479 znidomb 20480 znunit 20482 znrrg 20484 evl1muld 21213 evl1scvarpw 21233 mat2pmatmul 21582 mat2pmatlin 21586 cayhamlem4 21739 ply1rem 25015 fta1glem2 25018 fta1blem 25020 dchrzrhmul 26081 lgsdchr 26190 lgseisenlem3 26212 lgseisenlem4 26213 rhmdvdsr 31190 rhmopp 31191 rhmdvd 31193 rhmunitinv 31194 kerunit 31195 rhmpreimaidl 31271 rhmimaidl 31277 rhmpreimaprmidl 31295 mdetpmtr1 31441 mdetpmtr12 31443 qqhghm 31604 qqhrhm 31605 evlsmulval 39929 |
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