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Mirrors > Home > MPE Home > Th. List > psr1ring | Structured version Visualization version GIF version |
Description: Univariate power series form a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.) |
Ref | Expression |
---|---|
psr1ring.s | ⊢ 𝑆 = (PwSer1‘𝑅) |
Ref | Expression |
---|---|
psr1ring | ⊢ (𝑅 ∈ Ring → 𝑆 ∈ Ring) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psr1ring.s | . . 3 ⊢ 𝑆 = (PwSer1‘𝑅) | |
2 | 1 | psr1val 19877 | . 2 ⊢ 𝑆 = ((1𝑜 ordPwSer 𝑅)‘∅) |
3 | 1on 7807 | . . 3 ⊢ 1𝑜 ∈ On | |
4 | 3 | a1i 11 | . 2 ⊢ (𝑅 ∈ Ring → 1𝑜 ∈ On) |
5 | id 22 | . 2 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Ring) | |
6 | 0ss 4169 | . . 3 ⊢ ∅ ⊆ (1𝑜 × 1𝑜) | |
7 | 6 | a1i 11 | . 2 ⊢ (𝑅 ∈ Ring → ∅ ⊆ (1𝑜 × 1𝑜)) |
8 | 2, 4, 5, 7 | opsrring 19936 | 1 ⊢ (𝑅 ∈ Ring → 𝑆 ∈ Ring) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1653 ∈ wcel 2157 ⊆ wss 3770 ∅c0 4116 × cxp 5311 Oncon0 5942 ‘cfv 6102 1𝑜c1o 7793 Ringcrg 18862 PwSer1cps1 19866 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2378 ax-ext 2778 ax-rep 4965 ax-sep 4976 ax-nul 4984 ax-pow 5036 ax-pr 5098 ax-un 7184 ax-inf2 8789 ax-cnex 10281 ax-resscn 10282 ax-1cn 10283 ax-icn 10284 ax-addcl 10285 ax-addrcl 10286 ax-mulcl 10287 ax-mulrcl 10288 ax-mulcom 10289 ax-addass 10290 ax-mulass 10291 ax-distr 10292 ax-i2m1 10293 ax-1ne0 10294 ax-1rid 10295 ax-rnegex 10296 ax-rrecex 10297 ax-cnre 10298 ax-pre-lttri 10299 ax-pre-lttrn 10300 ax-pre-ltadd 10301 ax-pre-mulgt0 10302 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2592 df-eu 2610 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-ne 2973 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3388 df-sbc 3635 df-csb 3730 df-dif 3773 df-un 3775 df-in 3777 df-ss 3784 df-pss 3786 df-nul 4117 df-if 4279 df-pw 4352 df-sn 4370 df-pr 4372 df-tp 4374 df-op 4376 df-uni 4630 df-int 4669 df-iun 4713 df-iin 4714 df-br 4845 df-opab 4907 df-mpt 4924 df-tr 4947 df-id 5221 df-eprel 5226 df-po 5234 df-so 5235 df-fr 5272 df-se 5273 df-we 5274 df-xp 5319 df-rel 5320 df-cnv 5321 df-co 5322 df-dm 5323 df-rn 5324 df-res 5325 df-ima 5326 df-pred 5899 df-ord 5945 df-on 5946 df-lim 5947 df-suc 5948 df-iota 6065 df-fun 6104 df-fn 6105 df-f 6106 df-f1 6107 df-fo 6108 df-f1o 6109 df-fv 6110 df-isom 6111 df-riota 6840 df-ov 6882 df-oprab 6883 df-mpt2 6884 df-of 7132 df-ofr 7133 df-om 7301 df-1st 7402 df-2nd 7403 df-supp 7534 df-wrecs 7646 df-recs 7708 df-rdg 7746 df-1o 7800 df-2o 7801 df-oadd 7804 df-er 7983 df-map 8098 df-pm 8099 df-ixp 8150 df-en 8197 df-dom 8198 df-sdom 8199 df-fin 8200 df-fsupp 8519 df-oi 8658 df-card 9052 df-pnf 10366 df-mnf 10367 df-xr 10368 df-ltxr 10369 df-le 10370 df-sub 10559 df-neg 10560 df-nn 11314 df-2 11375 df-3 11376 df-4 11377 df-5 11378 df-6 11379 df-7 11380 df-8 11381 df-9 11382 df-n0 11580 df-z 11666 df-dec 11783 df-uz 11930 df-fz 12580 df-fzo 12720 df-seq 13055 df-hash 13370 df-struct 16185 df-ndx 16186 df-slot 16187 df-base 16189 df-sets 16190 df-ress 16191 df-plusg 16279 df-mulr 16280 df-sca 16282 df-vsca 16283 df-tset 16285 df-ple 16286 df-0g 16416 df-gsum 16417 df-mre 16560 df-mrc 16561 df-acs 16563 df-mgm 17556 df-sgrp 17598 df-mnd 17609 df-mhm 17649 df-submnd 17650 df-grp 17740 df-minusg 17741 df-mulg 17856 df-ghm 17970 df-cntz 18061 df-cmn 18509 df-abl 18510 df-mgp 18805 df-ur 18817 df-ring 18864 df-psr 19678 df-opsr 19682 df-psr1 19871 |
This theorem is referenced by: coe1mul2 19960 |
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