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Mirrors > Home > MPE Home > Th. List > psrring | Structured version Visualization version GIF version |
Description: The ring of power series is a ring. (Contributed by Mario Carneiro, 29-Dec-2014.) |
Ref | Expression |
---|---|
psrring.s | ⊢ 𝑆 = (𝐼 mPwSer 𝑅) |
psrring.i | ⊢ (𝜑 → 𝐼 ∈ 𝑉) |
psrring.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
Ref | Expression |
---|---|
psrring | ⊢ (𝜑 → 𝑆 ∈ Ring) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2822 | . 2 ⊢ (𝜑 → (Base‘𝑆) = (Base‘𝑆)) | |
2 | eqidd 2822 | . 2 ⊢ (𝜑 → (+g‘𝑆) = (+g‘𝑆)) | |
3 | eqidd 2822 | . 2 ⊢ (𝜑 → (.r‘𝑆) = (.r‘𝑆)) | |
4 | psrring.s | . . 3 ⊢ 𝑆 = (𝐼 mPwSer 𝑅) | |
5 | psrring.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
6 | psrring.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
7 | ringgrp 19296 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
8 | 6, 7 | syl 17 | . . 3 ⊢ (𝜑 → 𝑅 ∈ Grp) |
9 | 4, 5, 8 | psrgrp 20172 | . 2 ⊢ (𝜑 → 𝑆 ∈ Grp) |
10 | eqid 2821 | . . 3 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
11 | eqid 2821 | . . 3 ⊢ (.r‘𝑆) = (.r‘𝑆) | |
12 | 6 | 3ad2ant1 1129 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑅 ∈ Ring) |
13 | simp2 1133 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑆)) | |
14 | simp3 1134 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑦 ∈ (Base‘𝑆)) | |
15 | 4, 10, 11, 12, 13, 14 | psrmulcl 20162 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(.r‘𝑆)𝑦) ∈ (Base‘𝑆)) |
16 | 5 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝐼 ∈ 𝑉) |
17 | 6 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑅 ∈ Ring) |
18 | eqid 2821 | . . 3 ⊢ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
19 | simpr1 1190 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑥 ∈ (Base‘𝑆)) | |
20 | simpr2 1191 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑦 ∈ (Base‘𝑆)) | |
21 | simpr3 1192 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑧 ∈ (Base‘𝑆)) | |
22 | 4, 16, 17, 18, 11, 10, 19, 20, 21 | psrass1 20179 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥(.r‘𝑆)𝑦)(.r‘𝑆)𝑧) = (𝑥(.r‘𝑆)(𝑦(.r‘𝑆)𝑧))) |
23 | eqid 2821 | . . 3 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
24 | 4, 16, 17, 18, 11, 10, 19, 20, 21, 23 | psrdi 20180 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥(.r‘𝑆)(𝑦(+g‘𝑆)𝑧)) = ((𝑥(.r‘𝑆)𝑦)(+g‘𝑆)(𝑥(.r‘𝑆)𝑧))) |
25 | 4, 16, 17, 18, 11, 10, 19, 20, 21, 23 | psrdir 20181 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥(+g‘𝑆)𝑦)(.r‘𝑆)𝑧) = ((𝑥(.r‘𝑆)𝑧)(+g‘𝑆)(𝑦(.r‘𝑆)𝑧))) |
26 | eqid 2821 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
27 | eqid 2821 | . . 3 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
28 | eqid 2821 | . . 3 ⊢ (𝑟 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑟 = (𝐼 × {0}), (1r‘𝑅), (0g‘𝑅))) = (𝑟 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑟 = (𝐼 × {0}), (1r‘𝑅), (0g‘𝑅))) | |
29 | 4, 5, 6, 18, 26, 27, 28, 10 | psr1cl 20176 | . 2 ⊢ (𝜑 → (𝑟 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑟 = (𝐼 × {0}), (1r‘𝑅), (0g‘𝑅))) ∈ (Base‘𝑆)) |
30 | 5 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → 𝐼 ∈ 𝑉) |
31 | 6 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → 𝑅 ∈ Ring) |
32 | simpr 487 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑆)) | |
33 | 4, 30, 31, 18, 26, 27, 28, 10, 11, 32 | psrlidm 20177 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → ((𝑟 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑟 = (𝐼 × {0}), (1r‘𝑅), (0g‘𝑅)))(.r‘𝑆)𝑥) = 𝑥) |
34 | 4, 30, 31, 18, 26, 27, 28, 10, 11, 32 | psrridm 20178 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → (𝑥(.r‘𝑆)(𝑟 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑟 = (𝐼 × {0}), (1r‘𝑅), (0g‘𝑅)))) = 𝑥) |
35 | 1, 2, 3, 9, 15, 22, 24, 25, 29, 33, 34 | isringd 19329 | 1 ⊢ (𝜑 → 𝑆 ∈ Ring) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 {crab 3142 ifcif 4466 {csn 4560 ↦ cmpt 5138 × cxp 5547 ◡ccnv 5548 “ cima 5552 ‘cfv 6349 (class class class)co 7150 ↑m cmap 8400 Fincfn 8503 0cc0 10531 ℕcn 11632 ℕ0cn0 11891 Basecbs 16477 +gcplusg 16559 .rcmulr 16560 0gc0g 16707 Grpcgrp 18097 1rcur 19245 Ringcrg 19291 mPwSer cmps 20125 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-iin 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-ofr 7404 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-fzo 13028 df-seq 13364 df-hash 13685 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-sca 16575 df-vsca 16576 df-tset 16578 df-0g 16709 df-gsum 16710 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-mhm 17950 df-submnd 17951 df-grp 18100 df-minusg 18101 df-mulg 18219 df-ghm 18350 df-cntz 18441 df-cmn 18902 df-abl 18903 df-mgp 19234 df-ur 19246 df-ring 19293 df-psr 20130 |
This theorem is referenced by: psr1 20186 psrcrng 20187 psrassa 20188 subrgpsr 20193 mplsubrg 20214 opsrring 20407 |
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