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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 2799 | . 2 ⊢ (𝜑 → (Base‘𝑆) = (Base‘𝑆)) | |
2 | eqidd 2799 | . 2 ⊢ (𝜑 → (+g‘𝑆) = (+g‘𝑆)) | |
3 | eqidd 2799 | . 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 19295 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
8 | 6, 7 | syl 17 | . . 3 ⊢ (𝜑 → 𝑅 ∈ Grp) |
9 | 4, 5, 8 | psrgrp 20636 | . 2 ⊢ (𝜑 → 𝑆 ∈ Grp) |
10 | eqid 2798 | . . 3 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
11 | eqid 2798 | . . 3 ⊢ (.r‘𝑆) = (.r‘𝑆) | |
12 | 6 | 3ad2ant1 1130 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑅 ∈ Ring) |
13 | simp2 1134 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑆)) | |
14 | simp3 1135 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑦 ∈ (Base‘𝑆)) | |
15 | 4, 10, 11, 12, 13, 14 | psrmulcl 20626 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(.r‘𝑆)𝑦) ∈ (Base‘𝑆)) |
16 | 5 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝐼 ∈ 𝑉) |
17 | 6 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑅 ∈ Ring) |
18 | eqid 2798 | . . 3 ⊢ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
19 | simpr1 1191 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑥 ∈ (Base‘𝑆)) | |
20 | simpr2 1192 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑦 ∈ (Base‘𝑆)) | |
21 | simpr3 1193 | . . 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 20643 | . 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 2798 | . . 3 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
24 | 4, 16, 17, 18, 11, 10, 19, 20, 21, 23 | psrdi 20644 | . 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 20645 | . 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 2798 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
27 | eqid 2798 | . . 3 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
28 | eqid 2798 | . . 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 20640 | . 2 ⊢ (𝜑 → (𝑟 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑟 = (𝐼 × {0}), (1r‘𝑅), (0g‘𝑅))) ∈ (Base‘𝑆)) |
30 | 5 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → 𝐼 ∈ 𝑉) |
31 | 6 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → 𝑅 ∈ Ring) |
32 | simpr 488 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑆)) | |
33 | 4, 30, 31, 18, 26, 27, 28, 10, 11, 32 | psrlidm 20641 | . 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 20642 | . 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 19331 | 1 ⊢ (𝜑 → 𝑆 ∈ Ring) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 {crab 3110 ifcif 4425 {csn 4525 ↦ cmpt 5110 × cxp 5517 ◡ccnv 5518 “ cima 5522 ‘cfv 6324 (class class class)co 7135 ↑m cmap 8389 Fincfn 8492 0cc0 10526 ℕcn 11625 ℕ0cn0 11885 Basecbs 16475 +gcplusg 16557 .rcmulr 16558 0gc0g 16705 Grpcgrp 18095 1rcur 19244 Ringcrg 19290 mPwSer cmps 20589 |
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-int 4839 df-iun 4883 df-iin 4884 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-se 5479 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-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-ofr 7390 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-oi 8958 df-card 9352 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-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-seq 13365 df-hash 13687 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-tset 16576 df-0g 16707 df-gsum 16708 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-submnd 17949 df-grp 18098 df-minusg 18099 df-mulg 18217 df-ghm 18348 df-cntz 18439 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-psr 20594 |
This theorem is referenced by: psr1 20650 psrcrng 20651 psrassa 20652 subrgpsr 20657 mplsubrg 20678 opsrring 20874 |
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