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Mirrors > Home > MPE Home > Th. List > mpfmulcl | Structured version Visualization version GIF version |
Description: The product of multivariate polynomial functions. (Contributed by Mario Carneiro, 19-Mar-2015.) |
Ref | Expression |
---|---|
mpfsubrg.q | ⊢ 𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅) |
mpfmulcl.t | ⊢ · = (.r‘𝑆) |
Ref | Expression |
---|---|
mpfmulcl | ⊢ ((𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄) → (𝐹 ∘𝑓 · 𝐺) ∈ 𝑄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2778 | . . 3 ⊢ (𝑆 ↑s ((Base‘𝑆) ↑𝑚 𝐼)) = (𝑆 ↑s ((Base‘𝑆) ↑𝑚 𝐼)) | |
2 | eqid 2778 | . . 3 ⊢ (Base‘(𝑆 ↑s ((Base‘𝑆) ↑𝑚 𝐼))) = (Base‘(𝑆 ↑s ((Base‘𝑆) ↑𝑚 𝐼))) | |
3 | mpfsubrg.q | . . . . . 6 ⊢ 𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅) | |
4 | 3 | mpfrcl 19914 | . . . . 5 ⊢ (𝐹 ∈ 𝑄 → (𝐼 ∈ V ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆))) |
5 | 4 | adantr 474 | . . . 4 ⊢ ((𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄) → (𝐼 ∈ V ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆))) |
6 | 5 | simp2d 1134 | . . 3 ⊢ ((𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄) → 𝑆 ∈ CRing) |
7 | ovexd 6956 | . . 3 ⊢ ((𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄) → ((Base‘𝑆) ↑𝑚 𝐼) ∈ V) | |
8 | 3 | mpfsubrg 19928 | . . . . . 6 ⊢ ((𝐼 ∈ V ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (SubRing‘(𝑆 ↑s ((Base‘𝑆) ↑𝑚 𝐼)))) |
9 | 5, 8 | syl 17 | . . . . 5 ⊢ ((𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄) → 𝑄 ∈ (SubRing‘(𝑆 ↑s ((Base‘𝑆) ↑𝑚 𝐼)))) |
10 | 2 | subrgss 19173 | . . . . 5 ⊢ (𝑄 ∈ (SubRing‘(𝑆 ↑s ((Base‘𝑆) ↑𝑚 𝐼))) → 𝑄 ⊆ (Base‘(𝑆 ↑s ((Base‘𝑆) ↑𝑚 𝐼)))) |
11 | 9, 10 | syl 17 | . . . 4 ⊢ ((𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄) → 𝑄 ⊆ (Base‘(𝑆 ↑s ((Base‘𝑆) ↑𝑚 𝐼)))) |
12 | simpl 476 | . . . 4 ⊢ ((𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄) → 𝐹 ∈ 𝑄) | |
13 | 11, 12 | sseldd 3822 | . . 3 ⊢ ((𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄) → 𝐹 ∈ (Base‘(𝑆 ↑s ((Base‘𝑆) ↑𝑚 𝐼)))) |
14 | simpr 479 | . . . 4 ⊢ ((𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄) → 𝐺 ∈ 𝑄) | |
15 | 11, 14 | sseldd 3822 | . . 3 ⊢ ((𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄) → 𝐺 ∈ (Base‘(𝑆 ↑s ((Base‘𝑆) ↑𝑚 𝐼)))) |
16 | mpfmulcl.t | . . 3 ⊢ · = (.r‘𝑆) | |
17 | eqid 2778 | . . 3 ⊢ (.r‘(𝑆 ↑s ((Base‘𝑆) ↑𝑚 𝐼))) = (.r‘(𝑆 ↑s ((Base‘𝑆) ↑𝑚 𝐼))) | |
18 | 1, 2, 6, 7, 13, 15, 16, 17 | pwsmulrval 16537 | . 2 ⊢ ((𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄) → (𝐹(.r‘(𝑆 ↑s ((Base‘𝑆) ↑𝑚 𝐼)))𝐺) = (𝐹 ∘𝑓 · 𝐺)) |
19 | 17 | subrgmcl 19184 | . . . 4 ⊢ ((𝑄 ∈ (SubRing‘(𝑆 ↑s ((Base‘𝑆) ↑𝑚 𝐼))) ∧ 𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄) → (𝐹(.r‘(𝑆 ↑s ((Base‘𝑆) ↑𝑚 𝐼)))𝐺) ∈ 𝑄) |
20 | 19 | 3expib 1113 | . . 3 ⊢ (𝑄 ∈ (SubRing‘(𝑆 ↑s ((Base‘𝑆) ↑𝑚 𝐼))) → ((𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄) → (𝐹(.r‘(𝑆 ↑s ((Base‘𝑆) ↑𝑚 𝐼)))𝐺) ∈ 𝑄)) |
21 | 9, 20 | mpcom 38 | . 2 ⊢ ((𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄) → (𝐹(.r‘(𝑆 ↑s ((Base‘𝑆) ↑𝑚 𝐼)))𝐺) ∈ 𝑄) |
22 | 18, 21 | eqeltrrd 2860 | 1 ⊢ ((𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄) → (𝐹 ∘𝑓 · 𝐺) ∈ 𝑄) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 Vcvv 3398 ⊆ wss 3792 ran crn 5356 ‘cfv 6135 (class class class)co 6922 ∘𝑓 cof 7172 ↑𝑚 cmap 8140 Basecbs 16255 .rcmulr 16339 ↑s cpws 16493 CRingccrg 18935 SubRingcsubrg 19168 evalSub ces 19900 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-ofr 7175 df-om 7344 df-1st 7445 df-2nd 7446 df-supp 7577 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-2o 7844 df-oadd 7847 df-er 8026 df-map 8142 df-pm 8143 df-ixp 8195 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-fsupp 8564 df-sup 8636 df-oi 8704 df-card 9098 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-fz 12644 df-fzo 12785 df-seq 13120 df-hash 13436 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-sca 16354 df-vsca 16355 df-ip 16356 df-tset 16357 df-ple 16358 df-ds 16360 df-hom 16362 df-cco 16363 df-0g 16488 df-gsum 16489 df-prds 16494 df-pws 16496 df-mre 16632 df-mrc 16633 df-acs 16635 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-mhm 17721 df-submnd 17722 df-grp 17812 df-minusg 17813 df-sbg 17814 df-mulg 17928 df-subg 17975 df-ghm 18042 df-cntz 18133 df-cmn 18581 df-abl 18582 df-mgp 18877 df-ur 18889 df-srg 18893 df-ring 18936 df-cring 18937 df-rnghom 19104 df-subrg 19170 df-lmod 19257 df-lss 19325 df-lsp 19367 df-assa 19709 df-asp 19710 df-ascl 19711 df-psr 19753 df-mvr 19754 df-mpl 19755 df-evls 19902 |
This theorem is referenced by: mzpmfp 38270 |
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