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Mirrors > Home > MPE Home > Th. List > pwsdiagrhm | Structured version Visualization version GIF version |
Description: Diagonal homomorphism into a structure power (Rings). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.) |
Ref | Expression |
---|---|
pwsdiagrhm.y | ⊢ 𝑌 = (𝑅 ↑s 𝐼) |
pwsdiagrhm.b | ⊢ 𝐵 = (Base‘𝑅) |
pwsdiagrhm.f | ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝐼 × {𝑥})) |
Ref | Expression |
---|---|
pwsdiagrhm | ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝐹 ∈ (𝑅 RingHom 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 476 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝑅 ∈ Ring) | |
2 | pwsdiagrhm.y | . . . 4 ⊢ 𝑌 = (𝑅 ↑s 𝐼) | |
3 | 2 | pwsring 19002 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝑌 ∈ Ring) |
4 | 1, 3 | jca 507 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → (𝑅 ∈ Ring ∧ 𝑌 ∈ Ring)) |
5 | ringgrp 18939 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
6 | pwsdiagrhm.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
7 | pwsdiagrhm.f | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝐼 × {𝑥})) | |
8 | 2, 6, 7 | pwsdiagghm 18072 | . . . 4 ⊢ ((𝑅 ∈ Grp ∧ 𝐼 ∈ 𝑊) → 𝐹 ∈ (𝑅 GrpHom 𝑌)) |
9 | 5, 8 | sylan 575 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝐹 ∈ (𝑅 GrpHom 𝑌)) |
10 | eqid 2777 | . . . . . 6 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
11 | 10 | ringmgp 18940 | . . . . 5 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑅) ∈ Mnd) |
12 | eqid 2777 | . . . . . 6 ⊢ ((mulGrp‘𝑅) ↑s 𝐼) = ((mulGrp‘𝑅) ↑s 𝐼) | |
13 | 10, 6 | mgpbas 18882 | . . . . . 6 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅)) |
14 | 12, 13, 7 | pwsdiagmhm 17755 | . . . . 5 ⊢ (((mulGrp‘𝑅) ∈ Mnd ∧ 𝐼 ∈ 𝑊) → 𝐹 ∈ ((mulGrp‘𝑅) MndHom ((mulGrp‘𝑅) ↑s 𝐼))) |
15 | 11, 14 | sylan 575 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝐹 ∈ ((mulGrp‘𝑅) MndHom ((mulGrp‘𝑅) ↑s 𝐼))) |
16 | eqidd 2778 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → (Base‘(mulGrp‘𝑅)) = (Base‘(mulGrp‘𝑅))) | |
17 | eqidd 2778 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → (Base‘(mulGrp‘𝑌)) = (Base‘(mulGrp‘𝑌))) | |
18 | eqid 2777 | . . . . . . 7 ⊢ (mulGrp‘𝑌) = (mulGrp‘𝑌) | |
19 | eqid 2777 | . . . . . . 7 ⊢ (Base‘(mulGrp‘𝑌)) = (Base‘(mulGrp‘𝑌)) | |
20 | eqid 2777 | . . . . . . 7 ⊢ (Base‘((mulGrp‘𝑅) ↑s 𝐼)) = (Base‘((mulGrp‘𝑅) ↑s 𝐼)) | |
21 | eqid 2777 | . . . . . . 7 ⊢ (+g‘(mulGrp‘𝑌)) = (+g‘(mulGrp‘𝑌)) | |
22 | eqid 2777 | . . . . . . 7 ⊢ (+g‘((mulGrp‘𝑅) ↑s 𝐼)) = (+g‘((mulGrp‘𝑅) ↑s 𝐼)) | |
23 | 2, 10, 12, 18, 19, 20, 21, 22 | pwsmgp 19005 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → ((Base‘(mulGrp‘𝑌)) = (Base‘((mulGrp‘𝑅) ↑s 𝐼)) ∧ (+g‘(mulGrp‘𝑌)) = (+g‘((mulGrp‘𝑅) ↑s 𝐼)))) |
24 | 23 | simpld 490 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → (Base‘(mulGrp‘𝑌)) = (Base‘((mulGrp‘𝑅) ↑s 𝐼))) |
25 | eqidd 2778 | . . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) ∧ (𝑦 ∈ (Base‘(mulGrp‘𝑅)) ∧ 𝑧 ∈ (Base‘(mulGrp‘𝑅)))) → (𝑦(+g‘(mulGrp‘𝑅))𝑧) = (𝑦(+g‘(mulGrp‘𝑅))𝑧)) | |
26 | 23 | simprd 491 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → (+g‘(mulGrp‘𝑌)) = (+g‘((mulGrp‘𝑅) ↑s 𝐼))) |
27 | 26 | oveqdr 6950 | . . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) ∧ (𝑦 ∈ (Base‘(mulGrp‘𝑌)) ∧ 𝑧 ∈ (Base‘(mulGrp‘𝑌)))) → (𝑦(+g‘(mulGrp‘𝑌))𝑧) = (𝑦(+g‘((mulGrp‘𝑅) ↑s 𝐼))𝑧)) |
28 | 16, 17, 16, 24, 25, 27 | mhmpropd 17727 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → ((mulGrp‘𝑅) MndHom (mulGrp‘𝑌)) = ((mulGrp‘𝑅) MndHom ((mulGrp‘𝑅) ↑s 𝐼))) |
29 | 15, 28 | eleqtrrd 2861 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝐹 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑌))) |
30 | 9, 29 | jca 507 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → (𝐹 ∈ (𝑅 GrpHom 𝑌) ∧ 𝐹 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑌)))) |
31 | 10, 18 | isrhm 19110 | . 2 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑌) ↔ ((𝑅 ∈ Ring ∧ 𝑌 ∈ Ring) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑌) ∧ 𝐹 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑌))))) |
32 | 4, 30, 31 | sylanbrc 578 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝐹 ∈ (𝑅 RingHom 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2106 {csn 4397 ↦ cmpt 4965 × cxp 5353 ‘cfv 6135 (class class class)co 6922 Basecbs 16255 +gcplusg 16338 ↑s cpws 16493 Mndcmnd 17680 MndHom cmhm 17719 Grpcgrp 17809 GrpHom cghm 18041 mulGrpcmgp 18876 Ringcrg 18934 RingHom crh 19101 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-ixp 8195 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-sup 8636 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-fz 12644 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-plusg 16351 df-mulr 16352 df-sca 16354 df-vsca 16355 df-ip 16356 df-tset 16357 df-ple 16358 df-ds 16360 df-hom 16362 df-cco 16363 df-0g 16488 df-prds 16494 df-pws 16496 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-mhm 17721 df-grp 17812 df-minusg 17813 df-ghm 18042 df-mgp 18877 df-ur 18889 df-ring 18936 df-rnghom 19104 |
This theorem is referenced by: evlsval2 19916 |
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