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Mirrors > Home > MPE Home > Th. List > opsrassa | Structured version Visualization version GIF version |
Description: The ring of ordered power series is an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.) |
Ref | Expression |
---|---|
opsrcrng.o | ⊢ 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇) |
opsrcrng.i | ⊢ (𝜑 → 𝐼 ∈ 𝑉) |
opsrcrng.r | ⊢ (𝜑 → 𝑅 ∈ CRing) |
opsrcrng.t | ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼)) |
Ref | Expression |
---|---|
opsrassa | ⊢ (𝜑 → 𝑂 ∈ AssAlg) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . 3 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅) | |
2 | opsrcrng.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
3 | opsrcrng.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
4 | 1, 2, 3 | psrassa 20194 | . 2 ⊢ (𝜑 → (𝐼 mPwSer 𝑅) ∈ AssAlg) |
5 | eqidd 2822 | . . 3 ⊢ (𝜑 → (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅))) | |
6 | opsrcrng.o | . . . 4 ⊢ 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇) | |
7 | opsrcrng.t | . . . 4 ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼)) | |
8 | 1, 6, 7 | opsrbas 20259 | . . 3 ⊢ (𝜑 → (Base‘(𝐼 mPwSer 𝑅)) = (Base‘𝑂)) |
9 | 1, 6, 7 | opsrplusg 20260 | . . . 4 ⊢ (𝜑 → (+g‘(𝐼 mPwSer 𝑅)) = (+g‘𝑂)) |
10 | 9 | oveqdr 7184 | . . 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 20261 | . . . 4 ⊢ (𝜑 → (.r‘(𝐼 mPwSer 𝑅)) = (.r‘𝑂)) |
12 | 11 | oveqdr 7184 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑦 ∈ (Base‘(𝐼 mPwSer 𝑅)))) → (𝑥(.r‘(𝐼 mPwSer 𝑅))𝑦) = (𝑥(.r‘𝑂)𝑦)) |
13 | 1, 2, 3 | psrsca 20169 | . . 3 ⊢ (𝜑 → 𝑅 = (Scalar‘(𝐼 mPwSer 𝑅))) |
14 | 1, 6, 7, 2, 3 | opsrsca 20263 | . . 3 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑂)) |
15 | eqid 2821 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
16 | 1, 6, 7 | opsrvsca 20262 | . . . 4 ⊢ (𝜑 → ( ·𝑠 ‘(𝐼 mPwSer 𝑅)) = ( ·𝑠 ‘𝑂)) |
17 | 16 | oveqdr 7184 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘(𝐼 mPwSer 𝑅)))) → (𝑥( ·𝑠 ‘(𝐼 mPwSer 𝑅))𝑦) = (𝑥( ·𝑠 ‘𝑂)𝑦)) |
18 | 5, 8, 10, 12, 13, 14, 15, 17 | assapropd 20101 | . 2 ⊢ (𝜑 → ((𝐼 mPwSer 𝑅) ∈ AssAlg ↔ 𝑂 ∈ AssAlg)) |
19 | 4, 18 | mpbid 234 | 1 ⊢ (𝜑 → 𝑂 ∈ AssAlg) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ⊆ wss 3936 × cxp 5553 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 +gcplusg 16565 .rcmulr 16566 ·𝑠 cvsca 16569 CRingccrg 19298 AssAlgcasa 20082 mPwSer cmps 20131 ordPwSer copws 20135 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-ofr 7410 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-fz 12894 df-fzo 13035 df-seq 13371 df-hash 13692 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-sca 16581 df-vsca 16582 df-tset 16584 df-ple 16585 df-0g 16715 df-gsum 16716 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-mhm 17956 df-submnd 17957 df-grp 18106 df-minusg 18107 df-mulg 18225 df-ghm 18356 df-cntz 18447 df-cmn 18908 df-abl 18909 df-mgp 19240 df-ur 19252 df-ring 19299 df-cring 19300 df-lmod 19636 df-assa 20085 df-psr 20136 df-opsr 20140 |
This theorem is referenced by: psr1assa 20356 |
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