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Mirrors > Home > MPE Home > Th. List > mplmon | Structured version Visualization version GIF version |
Description: A monomial is a polynomial. (Contributed by Mario Carneiro, 9-Jan-2015.) |
Ref | Expression |
---|---|
mplmon.s | ⊢ 𝑃 = (𝐼 mPoly 𝑅) |
mplmon.b | ⊢ 𝐵 = (Base‘𝑃) |
mplmon.z | ⊢ 0 = (0g‘𝑅) |
mplmon.o | ⊢ 1 = (1r‘𝑅) |
mplmon.d | ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} |
mplmon.i | ⊢ (𝜑 → 𝐼 ∈ 𝑊) |
mplmon.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
mplmon.x | ⊢ (𝜑 → 𝑋 ∈ 𝐷) |
Ref | Expression |
---|---|
mplmon | ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mplmon.r | . . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
2 | eqid 2736 | . . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | mplmon.o | . . . . . . . . 9 ⊢ 1 = (1r‘𝑅) | |
4 | 2, 3 | ringidcl 19989 | . . . . . . . 8 ⊢ (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅)) |
5 | mplmon.z | . . . . . . . . 9 ⊢ 0 = (0g‘𝑅) | |
6 | 2, 5 | ring0cl 19990 | . . . . . . . 8 ⊢ (𝑅 ∈ Ring → 0 ∈ (Base‘𝑅)) |
7 | 4, 6 | ifcld 4532 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → if(𝑦 = 𝑋, 1 , 0 ) ∈ (Base‘𝑅)) |
8 | 1, 7 | syl 17 | . . . . . 6 ⊢ (𝜑 → if(𝑦 = 𝑋, 1 , 0 ) ∈ (Base‘𝑅)) |
9 | 8 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → if(𝑦 = 𝑋, 1 , 0 ) ∈ (Base‘𝑅)) |
10 | 9 | fmpttd 7063 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )):𝐷⟶(Base‘𝑅)) |
11 | fvex 6855 | . . . . 5 ⊢ (Base‘𝑅) ∈ V | |
12 | mplmon.d | . . . . . 6 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
13 | ovex 7390 | . . . . . 6 ⊢ (ℕ0 ↑m 𝐼) ∈ V | |
14 | 12, 13 | rabex2 5291 | . . . . 5 ⊢ 𝐷 ∈ V |
15 | 11, 14 | elmap 8809 | . . . 4 ⊢ ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ ((Base‘𝑅) ↑m 𝐷) ↔ (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )):𝐷⟶(Base‘𝑅)) |
16 | 10, 15 | sylibr 233 | . . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ ((Base‘𝑅) ↑m 𝐷)) |
17 | eqid 2736 | . . . 4 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅) | |
18 | eqid 2736 | . . . 4 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅)) | |
19 | mplmon.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
20 | 17, 2, 12, 18, 19 | psrbas 21346 | . . 3 ⊢ (𝜑 → (Base‘(𝐼 mPwSer 𝑅)) = ((Base‘𝑅) ↑m 𝐷)) |
21 | 16, 20 | eleqtrrd 2841 | . 2 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ (Base‘(𝐼 mPwSer 𝑅))) |
22 | 14 | mptex 7173 | . . . . 5 ⊢ (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ V |
23 | funmpt 6539 | . . . . 5 ⊢ Fun (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) | |
24 | 5 | fvexi 6856 | . . . . 5 ⊢ 0 ∈ V |
25 | 22, 23, 24 | 3pm3.2i 1339 | . . . 4 ⊢ ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ V ∧ Fun (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∧ 0 ∈ V) |
26 | 25 | a1i 11 | . . 3 ⊢ (𝜑 → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ V ∧ Fun (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∧ 0 ∈ V)) |
27 | snfi 8988 | . . . 4 ⊢ {𝑋} ∈ Fin | |
28 | 27 | a1i 11 | . . 3 ⊢ (𝜑 → {𝑋} ∈ Fin) |
29 | eldifsni 4750 | . . . . . . 7 ⊢ (𝑦 ∈ (𝐷 ∖ {𝑋}) → 𝑦 ≠ 𝑋) | |
30 | 29 | adantl 482 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → 𝑦 ≠ 𝑋) |
31 | 30 | neneqd 2948 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → ¬ 𝑦 = 𝑋) |
32 | 31 | iffalsed 4497 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → if(𝑦 = 𝑋, 1 , 0 ) = 0 ) |
33 | 14 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ V) |
34 | 32, 33 | suppss2 8131 | . . 3 ⊢ (𝜑 → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) supp 0 ) ⊆ {𝑋}) |
35 | suppssfifsupp 9320 | . . 3 ⊢ ((((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ V ∧ Fun (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∧ 0 ∈ V) ∧ ({𝑋} ∈ Fin ∧ ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) supp 0 ) ⊆ {𝑋})) → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) finSupp 0 ) | |
36 | 26, 28, 34, 35 | syl12anc 835 | . 2 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) finSupp 0 ) |
37 | mplmon.s | . . 3 ⊢ 𝑃 = (𝐼 mPoly 𝑅) | |
38 | mplmon.b | . . 3 ⊢ 𝐵 = (Base‘𝑃) | |
39 | 37, 17, 18, 5, 38 | mplelbas 21399 | . 2 ⊢ ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ 𝐵 ↔ ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) finSupp 0 )) |
40 | 21, 36, 39 | sylanbrc 583 | 1 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 {crab 3407 Vcvv 3445 ∖ cdif 3907 ⊆ wss 3910 ifcif 4486 {csn 4586 class class class wbr 5105 ↦ cmpt 5188 ◡ccnv 5632 “ cima 5636 Fun wfun 6490 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 supp csupp 8092 ↑m cmap 8765 Fincfn 8883 finSupp cfsupp 9305 ℕcn 12153 ℕ0cn0 12413 Basecbs 17083 0gc0g 17321 1rcur 19913 Ringcrg 19964 mPwSer cmps 21306 mPoly cmpl 21308 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-map 8767 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-uz 12764 df-fz 13425 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-sca 17149 df-vsca 17150 df-tset 17152 df-0g 17323 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-grp 18751 df-mgp 19897 df-ur 19914 df-ring 19966 df-psr 21311 df-mpl 21313 |
This theorem is referenced by: mplmonmul 21437 mplcoe1 21438 mplbas2 21443 mplmon2 21469 mplmon2cl 21476 mplmon2mul 21477 |
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