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Mirrors > Home > MPE Home > Th. List > rhmisrnghm | Structured version Visualization version GIF version |
Description: Each unital ring homomorphism is a non-unital ring homomorphism. (Contributed by AV, 29-Feb-2020.) |
Ref | Expression |
---|---|
rhmisrnghm | ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 RngHom 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringrng 20174 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Rng) | |
2 | ringrng 20174 | . . . 4 ⊢ (𝑆 ∈ Ring → 𝑆 ∈ Rng) | |
3 | 1, 2 | anim12i 612 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Ring) → (𝑅 ∈ Rng ∧ 𝑆 ∈ Rng)) |
4 | mhmismgmhm 18714 | . . . 4 ⊢ (𝐹 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑆)) → 𝐹 ∈ ((mulGrp‘𝑅) MgmHom (mulGrp‘𝑆))) | |
5 | 4 | anim2i 616 | . . 3 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑆))) → (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ ((mulGrp‘𝑅) MgmHom (mulGrp‘𝑆)))) |
6 | 3, 5 | anim12i 612 | . 2 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Ring) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑆)))) → ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ ((mulGrp‘𝑅) MgmHom (mulGrp‘𝑆))))) |
7 | eqid 2731 | . . 3 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
8 | eqid 2731 | . . 3 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆) | |
9 | 7, 8 | isrhm 20370 | . 2 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) ↔ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Ring) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑆))))) |
10 | 7, 8 | isrnghmmul 20334 | . 2 ⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) ↔ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ ((mulGrp‘𝑅) MgmHom (mulGrp‘𝑆))))) |
11 | 6, 9, 10 | 3imtr4i 291 | 1 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 RngHom 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2105 ‘cfv 6544 (class class class)co 7412 MgmHom cmgmhm 18616 MndHom cmhm 18704 GrpHom cghm 19128 mulGrpcmgp 20029 Rngcrng 20047 Ringcrg 20128 RngHom crnghm 20326 RingHom crh 20361 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7859 df-2nd 7979 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-er 8706 df-map 8825 df-en 8943 df-dom 8944 df-sdom 8945 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-2 12280 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-plusg 17215 df-0g 17392 df-mgm 18566 df-mgmhm 18618 df-sgrp 18645 df-mnd 18661 df-mhm 18706 df-grp 18859 df-minusg 18860 df-ghm 19129 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-rnghm 20328 df-rhm 20364 |
This theorem is referenced by: rimisrngim 20390 rhmsscrnghm 47014 rhmsubcALTVlem4 47095 |
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