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Mirrors > Home > MPE Home > Th. List > opsrmulr | Structured version Visualization version GIF version |
Description: The multiplication operation of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 30-Aug-2015.) |
Ref | Expression |
---|---|
opsrbas.s | ⊢ 𝑆 = (𝐼 mPwSer 𝑅) |
opsrbas.o | ⊢ 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇) |
opsrbas.t | ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼)) |
Ref | Expression |
---|---|
opsrmulr | ⊢ (𝜑 → (.r‘𝑆) = (.r‘𝑂)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opsrbas.s | . 2 ⊢ 𝑆 = (𝐼 mPwSer 𝑅) | |
2 | opsrbas.o | . 2 ⊢ 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇) | |
3 | opsrbas.t | . 2 ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼)) | |
4 | df-mulr 16278 | . 2 ⊢ .r = Slot 3 | |
5 | 3nn 11388 | . 2 ⊢ 3 ∈ ℕ | |
6 | 3lt10 11918 | . 2 ⊢ 3 < ;10 | |
7 | 1, 2, 3, 4, 5, 6 | opsrbaslem 19797 | 1 ⊢ (𝜑 → (.r‘𝑆) = (.r‘𝑂)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1653 ⊆ wss 3767 × cxp 5308 ‘cfv 6099 (class class class)co 6876 3c3 11365 .rcmulr 16265 mPwSer cmps 19671 ordPwSer copws 19675 |
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 2354 ax-ext 2775 ax-rep 4962 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 ax-cnex 10278 ax-resscn 10279 ax-1cn 10280 ax-icn 10281 ax-addcl 10282 ax-addrcl 10283 ax-mulcl 10284 ax-mulrcl 10285 ax-mulcom 10286 ax-addass 10287 ax-mulass 10288 ax-distr 10289 ax-i2m1 10290 ax-1ne0 10291 ax-1rid 10292 ax-rnegex 10293 ax-rrecex 10294 ax-cnre 10295 ax-pre-lttri 10296 ax-pre-lttrn 10297 ax-pre-ltadd 10298 ax-pre-mulgt0 10299 |
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 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-nel 3073 df-ral 3092 df-rex 3093 df-reu 3094 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-pss 3783 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-tp 4371 df-op 4373 df-uni 4627 df-iun 4710 df-br 4842 df-opab 4904 df-mpt 4921 df-tr 4944 df-id 5218 df-eprel 5223 df-po 5231 df-so 5232 df-fr 5269 df-we 5271 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-pred 5896 df-ord 5942 df-on 5943 df-lim 5944 df-suc 5945 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-riota 6837 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-om 7298 df-wrecs 7643 df-recs 7705 df-rdg 7743 df-er 7980 df-en 8194 df-dom 8195 df-sdom 8196 df-pnf 10363 df-mnf 10364 df-xr 10365 df-ltxr 10366 df-le 10367 df-sub 10556 df-neg 10557 df-nn 11311 df-2 11372 df-3 11373 df-4 11374 df-5 11375 df-6 11376 df-7 11377 df-8 11378 df-9 11379 df-dec 11780 df-ndx 16184 df-slot 16185 df-base 16187 df-sets 16188 df-mulr 16278 df-ple 16284 df-psr 19676 df-opsr 19680 |
This theorem is referenced by: opsrcrng 19807 opsrassa 19808 ply1subrg 19886 opsr1 19908 psr1mulr 19913 opsrring 19934 |
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