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Mirrors > Home > MPE Home > Th. List > c0rnghm | Structured version Visualization version GIF version |
Description: The constant mapping to zero is a non-unital ring homomorphism from any non-unital ring to the zero ring. (Contributed by AV, 17-Apr-2020.) |
Ref | Expression |
---|---|
c0rhm.b | ⊢ 𝐵 = (Base‘𝑆) |
c0rhm.0 | ⊢ 0 = (0g‘𝑇) |
c0rhm.h | ⊢ 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 ) |
Ref | Expression |
---|---|
c0rnghm | ⊢ ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝐻 ∈ (𝑆 RngHom 𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringssrng 20182 | . . . . . 6 ⊢ Ring ⊆ Rng | |
2 | 1 | a1i 11 | . . . . 5 ⊢ (𝑆 ∈ Rng → Ring ⊆ Rng) |
3 | 2 | ssdifssd 4137 | . . . 4 ⊢ (𝑆 ∈ Rng → (Ring ∖ NzRing) ⊆ Rng) |
4 | 3 | sseld 3976 | . . 3 ⊢ (𝑆 ∈ Rng → (𝑇 ∈ (Ring ∖ NzRing) → 𝑇 ∈ Rng)) |
5 | 4 | imdistani 568 | . 2 ⊢ ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → (𝑆 ∈ Rng ∧ 𝑇 ∈ Rng)) |
6 | rngabl 20057 | . . . . 5 ⊢ (𝑆 ∈ Rng → 𝑆 ∈ Abel) | |
7 | ablgrp 19702 | . . . . 5 ⊢ (𝑆 ∈ Abel → 𝑆 ∈ Grp) | |
8 | 6, 7 | syl 17 | . . . 4 ⊢ (𝑆 ∈ Rng → 𝑆 ∈ Grp) |
9 | eldifi 4121 | . . . . 5 ⊢ (𝑇 ∈ (Ring ∖ NzRing) → 𝑇 ∈ Ring) | |
10 | ringgrp 20140 | . . . . 5 ⊢ (𝑇 ∈ Ring → 𝑇 ∈ Grp) | |
11 | 9, 10 | syl 17 | . . . 4 ⊢ (𝑇 ∈ (Ring ∖ NzRing) → 𝑇 ∈ Grp) |
12 | c0rhm.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑆) | |
13 | c0rhm.0 | . . . . 5 ⊢ 0 = (0g‘𝑇) | |
14 | c0rhm.h | . . . . 5 ⊢ 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 ) | |
15 | 12, 13, 14 | c0ghm 20360 | . . . 4 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → 𝐻 ∈ (𝑆 GrpHom 𝑇)) |
16 | 8, 11, 15 | syl2an 595 | . . 3 ⊢ ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝐻 ∈ (𝑆 GrpHom 𝑇)) |
17 | eqid 2726 | . . . . . . . . 9 ⊢ (Base‘𝑇) = (Base‘𝑇) | |
18 | eqid 2726 | . . . . . . . . 9 ⊢ (1r‘𝑇) = (1r‘𝑇) | |
19 | 17, 13, 18 | 0ring1eq0 20430 | . . . . . . . 8 ⊢ (𝑇 ∈ (Ring ∖ NzRing) → (1r‘𝑇) = 0 ) |
20 | 19 | eqcomd 2732 | . . . . . . 7 ⊢ (𝑇 ∈ (Ring ∖ NzRing) → 0 = (1r‘𝑇)) |
21 | 20 | mpteq2dv 5243 | . . . . . 6 ⊢ (𝑇 ∈ (Ring ∖ NzRing) → (𝑥 ∈ 𝐵 ↦ 0 ) = (𝑥 ∈ 𝐵 ↦ (1r‘𝑇))) |
22 | 21 | adantl 481 | . . . . 5 ⊢ ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → (𝑥 ∈ 𝐵 ↦ 0 ) = (𝑥 ∈ 𝐵 ↦ (1r‘𝑇))) |
23 | 14, 22 | eqtrid 2778 | . . . 4 ⊢ ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝐻 = (𝑥 ∈ 𝐵 ↦ (1r‘𝑇))) |
24 | eqid 2726 | . . . . . . 7 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆) | |
25 | 24 | rngmgp 20058 | . . . . . 6 ⊢ (𝑆 ∈ Rng → (mulGrp‘𝑆) ∈ Smgrp) |
26 | sgrpmgm 18654 | . . . . . 6 ⊢ ((mulGrp‘𝑆) ∈ Smgrp → (mulGrp‘𝑆) ∈ Mgm) | |
27 | 25, 26 | syl 17 | . . . . 5 ⊢ (𝑆 ∈ Rng → (mulGrp‘𝑆) ∈ Mgm) |
28 | eqid 2726 | . . . . . . 7 ⊢ (mulGrp‘𝑇) = (mulGrp‘𝑇) | |
29 | 28 | ringmgp 20141 | . . . . . 6 ⊢ (𝑇 ∈ Ring → (mulGrp‘𝑇) ∈ Mnd) |
30 | 9, 29 | syl 17 | . . . . 5 ⊢ (𝑇 ∈ (Ring ∖ NzRing) → (mulGrp‘𝑇) ∈ Mnd) |
31 | 24, 12 | mgpbas 20042 | . . . . . 6 ⊢ 𝐵 = (Base‘(mulGrp‘𝑆)) |
32 | 28, 18 | ringidval 20085 | . . . . . 6 ⊢ (1r‘𝑇) = (0g‘(mulGrp‘𝑇)) |
33 | eqid 2726 | . . . . . 6 ⊢ (𝑥 ∈ 𝐵 ↦ (1r‘𝑇)) = (𝑥 ∈ 𝐵 ↦ (1r‘𝑇)) | |
34 | 31, 32, 33 | c0mgm 20358 | . . . . 5 ⊢ (((mulGrp‘𝑆) ∈ Mgm ∧ (mulGrp‘𝑇) ∈ Mnd) → (𝑥 ∈ 𝐵 ↦ (1r‘𝑇)) ∈ ((mulGrp‘𝑆) MgmHom (mulGrp‘𝑇))) |
35 | 27, 30, 34 | syl2an 595 | . . . 4 ⊢ ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → (𝑥 ∈ 𝐵 ↦ (1r‘𝑇)) ∈ ((mulGrp‘𝑆) MgmHom (mulGrp‘𝑇))) |
36 | 23, 35 | eqeltrd 2827 | . . 3 ⊢ ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝐻 ∈ ((mulGrp‘𝑆) MgmHom (mulGrp‘𝑇))) |
37 | 16, 36 | jca 511 | . 2 ⊢ ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → (𝐻 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐻 ∈ ((mulGrp‘𝑆) MgmHom (mulGrp‘𝑇)))) |
38 | 24, 28 | isrnghmmul 20341 | . 2 ⊢ (𝐻 ∈ (𝑆 RngHom 𝑇) ↔ ((𝑆 ∈ Rng ∧ 𝑇 ∈ Rng) ∧ (𝐻 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐻 ∈ ((mulGrp‘𝑆) MgmHom (mulGrp‘𝑇))))) |
39 | 5, 37, 38 | sylanbrc 582 | 1 ⊢ ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝐻 ∈ (𝑆 RngHom 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∖ cdif 3940 ⊆ wss 3943 ↦ cmpt 5224 ‘cfv 6536 (class class class)co 7404 Basecbs 17150 0gc0g 17391 Mgmcmgm 18568 MgmHom cmgmhm 18620 Smgrpcsgrp 18648 Mndcmnd 18664 Grpcgrp 18860 GrpHom cghm 19135 Abelcabl 19698 mulGrpcmgp 20036 Rngcrng 20054 1rcur 20083 Ringcrg 20135 RngHom crnghm 20333 NzRingcnzr 20411 |
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 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-oadd 8468 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-dju 9895 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-n0 12474 df-xnn0 12546 df-z 12560 df-uz 12824 df-fz 13488 df-hash 14293 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17151 df-plusg 17216 df-0g 17393 df-mgm 18570 df-mgmhm 18622 df-sgrp 18649 df-mnd 18665 df-mhm 18710 df-grp 18863 df-minusg 18864 df-ghm 19136 df-cmn 19699 df-abl 19700 df-mgp 20037 df-rng 20055 df-ur 20084 df-ring 20137 df-rnghm 20335 df-nzr 20412 |
This theorem is referenced by: zrtermorngc 20536 |
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