Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > rnghmco | Structured version Visualization version GIF version |
Description: The composition of non-unital ring homomorphisms is a homomorphism. (Contributed by AV, 27-Feb-2020.) |
Ref | Expression |
---|---|
rnghmco | ⊢ ((𝐹 ∈ (𝑇 RngHomo 𝑈) ∧ 𝐺 ∈ (𝑆 RngHomo 𝑇)) → (𝐹 ∘ 𝐺) ∈ (𝑆 RngHomo 𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rnghmrcl 44513 | . . . 4 ⊢ (𝐹 ∈ (𝑇 RngHomo 𝑈) → (𝑇 ∈ Rng ∧ 𝑈 ∈ Rng)) | |
2 | 1 | simprd 499 | . . 3 ⊢ (𝐹 ∈ (𝑇 RngHomo 𝑈) → 𝑈 ∈ Rng) |
3 | rnghmrcl 44513 | . . . 4 ⊢ (𝐺 ∈ (𝑆 RngHomo 𝑇) → (𝑆 ∈ Rng ∧ 𝑇 ∈ Rng)) | |
4 | 3 | simpld 498 | . . 3 ⊢ (𝐺 ∈ (𝑆 RngHomo 𝑇) → 𝑆 ∈ Rng) |
5 | 2, 4 | anim12ci 616 | . 2 ⊢ ((𝐹 ∈ (𝑇 RngHomo 𝑈) ∧ 𝐺 ∈ (𝑆 RngHomo 𝑇)) → (𝑆 ∈ Rng ∧ 𝑈 ∈ Rng)) |
6 | rnghmghm 44522 | . . . 4 ⊢ (𝐹 ∈ (𝑇 RngHomo 𝑈) → 𝐹 ∈ (𝑇 GrpHom 𝑈)) | |
7 | rnghmghm 44522 | . . . 4 ⊢ (𝐺 ∈ (𝑆 RngHomo 𝑇) → 𝐺 ∈ (𝑆 GrpHom 𝑇)) | |
8 | ghmco 18370 | . . . 4 ⊢ ((𝐹 ∈ (𝑇 GrpHom 𝑈) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → (𝐹 ∘ 𝐺) ∈ (𝑆 GrpHom 𝑈)) | |
9 | 6, 7, 8 | syl2an 598 | . . 3 ⊢ ((𝐹 ∈ (𝑇 RngHomo 𝑈) ∧ 𝐺 ∈ (𝑆 RngHomo 𝑇)) → (𝐹 ∘ 𝐺) ∈ (𝑆 GrpHom 𝑈)) |
10 | eqid 2798 | . . . . 5 ⊢ (mulGrp‘𝑇) = (mulGrp‘𝑇) | |
11 | eqid 2798 | . . . . 5 ⊢ (mulGrp‘𝑈) = (mulGrp‘𝑈) | |
12 | 10, 11 | rnghmmgmhm 44518 | . . . 4 ⊢ (𝐹 ∈ (𝑇 RngHomo 𝑈) → 𝐹 ∈ ((mulGrp‘𝑇) MgmHom (mulGrp‘𝑈))) |
13 | eqid 2798 | . . . . 5 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆) | |
14 | 13, 10 | rnghmmgmhm 44518 | . . . 4 ⊢ (𝐺 ∈ (𝑆 RngHomo 𝑇) → 𝐺 ∈ ((mulGrp‘𝑆) MgmHom (mulGrp‘𝑇))) |
15 | mgmhmco 44421 | . . . 4 ⊢ ((𝐹 ∈ ((mulGrp‘𝑇) MgmHom (mulGrp‘𝑈)) ∧ 𝐺 ∈ ((mulGrp‘𝑆) MgmHom (mulGrp‘𝑇))) → (𝐹 ∘ 𝐺) ∈ ((mulGrp‘𝑆) MgmHom (mulGrp‘𝑈))) | |
16 | 12, 14, 15 | syl2an 598 | . . 3 ⊢ ((𝐹 ∈ (𝑇 RngHomo 𝑈) ∧ 𝐺 ∈ (𝑆 RngHomo 𝑇)) → (𝐹 ∘ 𝐺) ∈ ((mulGrp‘𝑆) MgmHom (mulGrp‘𝑈))) |
17 | 9, 16 | jca 515 | . 2 ⊢ ((𝐹 ∈ (𝑇 RngHomo 𝑈) ∧ 𝐺 ∈ (𝑆 RngHomo 𝑇)) → ((𝐹 ∘ 𝐺) ∈ (𝑆 GrpHom 𝑈) ∧ (𝐹 ∘ 𝐺) ∈ ((mulGrp‘𝑆) MgmHom (mulGrp‘𝑈)))) |
18 | 13, 11 | isrnghmmul 44517 | . 2 ⊢ ((𝐹 ∘ 𝐺) ∈ (𝑆 RngHomo 𝑈) ↔ ((𝑆 ∈ Rng ∧ 𝑈 ∈ Rng) ∧ ((𝐹 ∘ 𝐺) ∈ (𝑆 GrpHom 𝑈) ∧ (𝐹 ∘ 𝐺) ∈ ((mulGrp‘𝑆) MgmHom (mulGrp‘𝑈))))) |
19 | 5, 17, 18 | sylanbrc 586 | 1 ⊢ ((𝐹 ∈ (𝑇 RngHomo 𝑈) ∧ 𝐺 ∈ (𝑆 RngHomo 𝑇)) → (𝐹 ∘ 𝐺) ∈ (𝑆 RngHomo 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2111 ∘ ccom 5523 ‘cfv 6324 (class class class)co 7135 GrpHom cghm 18347 mulGrpcmgp 19232 MgmHom cmgmhm 44397 Rngcrng 44498 RngHomo crngh 44509 |
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-rmo 3114 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-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-grp 18098 df-ghm 18348 df-abl 18901 df-mgp 19233 df-mgmhm 44399 df-rng0 44499 df-rnghomo 44511 |
This theorem is referenced by: rnghmsubcsetclem2 44600 rngccatidALTV 44613 |
Copyright terms: Public domain | W3C validator |