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Mirrors > Home > MPE Home > Th. List > ply1moncl | Structured version Visualization version GIF version |
Description: Closure of the expression for a univariate primitive monomial. (Contributed by AV, 14-Aug-2019.) |
Ref | Expression |
---|---|
ply1moncl.p | ⊢ 𝑃 = (Poly1‘𝑅) |
ply1moncl.x | ⊢ 𝑋 = (var1‘𝑅) |
ply1moncl.n | ⊢ 𝑁 = (mulGrp‘𝑃) |
ply1moncl.e | ⊢ ↑ = (.g‘𝑁) |
ply1moncl.b | ⊢ 𝐵 = (Base‘𝑃) |
Ref | Expression |
---|---|
ply1moncl | ⊢ ((𝑅 ∈ Ring ∧ 𝐷 ∈ ℕ0) → (𝐷 ↑ 𝑋) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ply1moncl.p | . . . . 5 ⊢ 𝑃 = (Poly1‘𝑅) | |
2 | 1 | ply1ring 20087 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) |
3 | ply1moncl.n | . . . . 5 ⊢ 𝑁 = (mulGrp‘𝑃) | |
4 | 3 | ringmgp 18981 | . . . 4 ⊢ (𝑃 ∈ Ring → 𝑁 ∈ Mnd) |
5 | 2, 4 | syl 17 | . . 3 ⊢ (𝑅 ∈ Ring → 𝑁 ∈ Mnd) |
6 | 5 | adantr 481 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐷 ∈ ℕ0) → 𝑁 ∈ Mnd) |
7 | simpr 485 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐷 ∈ ℕ0) → 𝐷 ∈ ℕ0) | |
8 | ply1moncl.x | . . . 4 ⊢ 𝑋 = (var1‘𝑅) | |
9 | ply1moncl.b | . . . 4 ⊢ 𝐵 = (Base‘𝑃) | |
10 | 8, 1, 9 | vr1cl 20056 | . . 3 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ 𝐵) |
11 | 10 | adantr 481 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐷 ∈ ℕ0) → 𝑋 ∈ 𝐵) |
12 | 3, 9 | mgpbas 18923 | . . 3 ⊢ 𝐵 = (Base‘𝑁) |
13 | ply1moncl.e | . . 3 ⊢ ↑ = (.g‘𝑁) | |
14 | 12, 13 | mulgnn0cl 17987 | . 2 ⊢ ((𝑁 ∈ Mnd ∧ 𝐷 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (𝐷 ↑ 𝑋) ∈ 𝐵) |
15 | 6, 7, 11, 14 | syl3anc 1362 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐷 ∈ ℕ0) → (𝐷 ↑ 𝑋) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1520 ∈ wcel 2079 ‘cfv 6217 (class class class)co 7007 ℕ0cn0 11734 Basecbs 16300 Mndcmnd 17721 .gcmg 17969 mulGrpcmgp 18917 Ringcrg 18975 var1cv1 20015 Poly1cpl1 20016 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-rep 5075 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 ax-cnex 10428 ax-resscn 10429 ax-1cn 10430 ax-icn 10431 ax-addcl 10432 ax-addrcl 10433 ax-mulcl 10434 ax-mulrcl 10435 ax-mulcom 10436 ax-addass 10437 ax-mulass 10438 ax-distr 10439 ax-i2m1 10440 ax-1ne0 10441 ax-1rid 10442 ax-rnegex 10443 ax-rrecex 10444 ax-cnre 10445 ax-pre-lttri 10446 ax-pre-lttrn 10447 ax-pre-ltadd 10448 ax-pre-mulgt0 10449 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-nel 3089 df-ral 3108 df-rex 3109 df-reu 3110 df-rmo 3111 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-pss 3871 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-tp 4471 df-op 4473 df-uni 4740 df-int 4777 df-iun 4821 df-iin 4822 df-br 4957 df-opab 5019 df-mpt 5036 df-tr 5058 df-id 5340 df-eprel 5345 df-po 5354 df-so 5355 df-fr 5394 df-se 5395 df-we 5396 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-pred 6015 df-ord 6061 df-on 6062 df-lim 6063 df-suc 6064 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-isom 6226 df-riota 6968 df-ov 7010 df-oprab 7011 df-mpo 7012 df-of 7258 df-ofr 7259 df-om 7428 df-1st 7536 df-2nd 7537 df-supp 7673 df-wrecs 7789 df-recs 7851 df-rdg 7889 df-1o 7944 df-2o 7945 df-oadd 7948 df-er 8130 df-map 8249 df-pm 8250 df-ixp 8301 df-en 8348 df-dom 8349 df-sdom 8350 df-fin 8351 df-fsupp 8670 df-oi 8810 df-card 9203 df-pnf 10512 df-mnf 10513 df-xr 10514 df-ltxr 10515 df-le 10516 df-sub 10708 df-neg 10709 df-nn 11476 df-2 11537 df-3 11538 df-4 11539 df-5 11540 df-6 11541 df-7 11542 df-8 11543 df-9 11544 df-n0 11735 df-z 11819 df-dec 11937 df-uz 12083 df-fz 12732 df-fzo 12873 df-seq 13208 df-hash 13529 df-struct 16302 df-ndx 16303 df-slot 16304 df-base 16306 df-sets 16307 df-ress 16308 df-plusg 16395 df-mulr 16396 df-sca 16398 df-vsca 16399 df-tset 16401 df-ple 16402 df-0g 16532 df-gsum 16533 df-mre 16674 df-mrc 16675 df-acs 16677 df-mgm 17669 df-sgrp 17711 df-mnd 17722 df-mhm 17762 df-submnd 17763 df-grp 17852 df-minusg 17853 df-mulg 17970 df-subg 18018 df-ghm 18085 df-cntz 18176 df-cmn 18623 df-abl 18624 df-mgp 18918 df-ur 18930 df-ring 18977 df-subrg 19211 df-psr 19812 df-mvr 19813 df-mpl 19814 df-opsr 19816 df-psr1 20019 df-vr1 20020 df-ply1 20021 |
This theorem is referenced by: ply1tmcl 20111 ply1idvr1 20132 gsumsmonply1 20142 mat2pmatscmxcl 21020 m2pmfzgsumcl 21028 decpmatid 21050 pmatcollpw1lem1 21054 pmatcollpw2lem 21057 monmatcollpw 21059 pmatcollpwlem 21060 pmatcollpw 21061 pmatcollpwfi 21062 pmatcollpw3fi1lem1 21066 pm2mpcl 21077 idpm2idmp 21081 mp2pm2mplem4 21089 mp2pm2mplem5 21090 pm2mpghmlem2 21092 pm2mpmhmlem1 21098 pm2mpmhmlem2 21099 deg1pwle 24384 |
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