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Mirrors > Home > MPE Home > Th. List > opsrring | Structured version Visualization version GIF version |
Description: Ordered power series form a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.) |
Ref | Expression |
---|---|
opsrring.o | ⊢ 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇) |
opsrring.i | ⊢ (𝜑 → 𝐼 ∈ 𝑉) |
opsrring.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
opsrring.t | ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼)) |
Ref | Expression |
---|---|
opsrring | ⊢ (𝜑 → 𝑂 ∈ Ring) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2738 | . . 3 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅) | |
2 | opsrring.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
3 | opsrring.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
4 | 1, 2, 3 | psrring 20791 | . 2 ⊢ (𝜑 → (𝐼 mPwSer 𝑅) ∈ Ring) |
5 | eqidd 2739 | . . 3 ⊢ (𝜑 → (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅))) | |
6 | opsrring.o | . . . 4 ⊢ 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇) | |
7 | opsrring.t | . . . 4 ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼)) | |
8 | 1, 6, 7 | opsrbas 20862 | . . 3 ⊢ (𝜑 → (Base‘(𝐼 mPwSer 𝑅)) = (Base‘𝑂)) |
9 | 1, 6, 7 | opsrplusg 20863 | . . . 4 ⊢ (𝜑 → (+g‘(𝐼 mPwSer 𝑅)) = (+g‘𝑂)) |
10 | 9 | oveqdr 7199 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑦 ∈ (Base‘(𝐼 mPwSer 𝑅)))) → (𝑥(+g‘(𝐼 mPwSer 𝑅))𝑦) = (𝑥(+g‘𝑂)𝑦)) |
11 | 1, 6, 7 | opsrmulr 20864 | . . . 4 ⊢ (𝜑 → (.r‘(𝐼 mPwSer 𝑅)) = (.r‘𝑂)) |
12 | 11 | oveqdr 7199 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑦 ∈ (Base‘(𝐼 mPwSer 𝑅)))) → (𝑥(.r‘(𝐼 mPwSer 𝑅))𝑦) = (𝑥(.r‘𝑂)𝑦)) |
13 | 5, 8, 10, 12 | ringpropd 19455 | . 2 ⊢ (𝜑 → ((𝐼 mPwSer 𝑅) ∈ Ring ↔ 𝑂 ∈ Ring)) |
14 | 4, 13 | mpbid 235 | 1 ⊢ (𝜑 → 𝑂 ∈ Ring) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2113 ⊆ wss 3844 × cxp 5524 ‘cfv 6340 (class class class)co 7171 Basecbs 16587 +gcplusg 16669 .rcmulr 16670 Ringcrg 19417 mPwSer cmps 20718 ordPwSer copws 20722 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5155 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7480 ax-cnex 10672 ax-resscn 10673 ax-1cn 10674 ax-icn 10675 ax-addcl 10676 ax-addrcl 10677 ax-mulcl 10678 ax-mulrcl 10679 ax-mulcom 10680 ax-addass 10681 ax-mulass 10682 ax-distr 10683 ax-i2m1 10684 ax-1ne0 10685 ax-1rid 10686 ax-rnegex 10687 ax-rrecex 10688 ax-cnre 10689 ax-pre-lttri 10690 ax-pre-lttrn 10691 ax-pre-ltadd 10692 ax-pre-mulgt0 10693 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3683 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-int 4838 df-iun 4884 df-iin 4885 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-se 5485 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-isom 6349 df-riota 7128 df-ov 7174 df-oprab 7175 df-mpo 7176 df-of 7426 df-ofr 7427 df-om 7601 df-1st 7715 df-2nd 7716 df-supp 7858 df-wrecs 7977 df-recs 8038 df-rdg 8076 df-1o 8132 df-er 8321 df-map 8440 df-pm 8441 df-ixp 8509 df-en 8557 df-dom 8558 df-sdom 8559 df-fin 8560 df-fsupp 8908 df-oi 9048 df-card 9442 df-pnf 10756 df-mnf 10757 df-xr 10758 df-ltxr 10759 df-le 10760 df-sub 10951 df-neg 10952 df-nn 11718 df-2 11780 df-3 11781 df-4 11782 df-5 11783 df-6 11784 df-7 11785 df-8 11786 df-9 11787 df-n0 11978 df-z 12064 df-dec 12181 df-uz 12326 df-fz 12983 df-fzo 13126 df-seq 13462 df-hash 13784 df-struct 16589 df-ndx 16590 df-slot 16591 df-base 16593 df-sets 16594 df-ress 16595 df-plusg 16682 df-mulr 16683 df-sca 16685 df-vsca 16686 df-tset 16688 df-ple 16689 df-0g 16819 df-gsum 16820 df-mre 16961 df-mrc 16962 df-acs 16964 df-mgm 17969 df-sgrp 18018 df-mnd 18029 df-mhm 18073 df-submnd 18074 df-grp 18223 df-minusg 18224 df-mulg 18344 df-ghm 18475 df-cntz 18566 df-cmn 19027 df-abl 19028 df-mgp 19360 df-ur 19372 df-ring 19419 df-psr 20723 df-opsr 20727 |
This theorem is referenced by: psr1ring 21023 |
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