Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2610 | . . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | mplmon.o | . . . . . . . . 9 ⊢ 1 = (1r‘𝑅) | |
4 | 2, 3 | ringidcl 18391 | . . . . . . . 8 ⊢ (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅)) |
5 | mplmon.z | . . . . . . . . 9 ⊢ 0 = (0g‘𝑅) | |
6 | 2, 5 | ring0cl 18392 | . . . . . . . 8 ⊢ (𝑅 ∈ Ring → 0 ∈ (Base‘𝑅)) |
7 | 4, 6 | ifcld 4081 | . . . . . . 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 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → if(𝑦 = 𝑋, 1 , 0 ) ∈ (Base‘𝑅)) |
10 | eqid 2610 | . . . . 5 ⊢ (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) | |
11 | 9, 10 | fmptd 6292 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )):𝐷⟶(Base‘𝑅)) |
12 | fvex 6113 | . . . . 5 ⊢ (Base‘𝑅) ∈ V | |
13 | mplmon.d | . . . . . 6 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
14 | ovex 6577 | . . . . . 6 ⊢ (ℕ0 ↑𝑚 𝐼) ∈ V | |
15 | 13, 14 | rabex2 4742 | . . . . 5 ⊢ 𝐷 ∈ V |
16 | 12, 15 | elmap 7772 | . . . 4 ⊢ ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ ((Base‘𝑅) ↑𝑚 𝐷) ↔ (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )):𝐷⟶(Base‘𝑅)) |
17 | 11, 16 | sylibr 223 | . . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ ((Base‘𝑅) ↑𝑚 𝐷)) |
18 | eqid 2610 | . . . 4 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅) | |
19 | eqid 2610 | . . . 4 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅)) | |
20 | mplmon.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
21 | 18, 2, 13, 19, 20 | psrbas 19199 | . . 3 ⊢ (𝜑 → (Base‘(𝐼 mPwSer 𝑅)) = ((Base‘𝑅) ↑𝑚 𝐷)) |
22 | 17, 21 | eleqtrrd 2691 | . 2 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ (Base‘(𝐼 mPwSer 𝑅))) |
23 | 15 | mptex 6390 | . . . . 5 ⊢ (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ V |
24 | funmpt 5840 | . . . . 5 ⊢ Fun (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) | |
25 | fvex 6113 | . . . . . 6 ⊢ (0g‘𝑅) ∈ V | |
26 | 5, 25 | eqeltri 2684 | . . . . 5 ⊢ 0 ∈ V |
27 | 23, 24, 26 | 3pm3.2i 1232 | . . . 4 ⊢ ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ V ∧ Fun (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∧ 0 ∈ V) |
28 | 27 | a1i 11 | . . 3 ⊢ (𝜑 → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ V ∧ Fun (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∧ 0 ∈ V)) |
29 | snfi 7923 | . . . 4 ⊢ {𝑋} ∈ Fin | |
30 | 29 | a1i 11 | . . 3 ⊢ (𝜑 → {𝑋} ∈ Fin) |
31 | eldifsni 4261 | . . . . . . 7 ⊢ (𝑦 ∈ (𝐷 ∖ {𝑋}) → 𝑦 ≠ 𝑋) | |
32 | 31 | adantl 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → 𝑦 ≠ 𝑋) |
33 | 32 | neneqd 2787 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → ¬ 𝑦 = 𝑋) |
34 | 33 | iffalsed 4047 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → if(𝑦 = 𝑋, 1 , 0 ) = 0 ) |
35 | 15 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ V) |
36 | 34, 35 | suppss2 7216 | . . 3 ⊢ (𝜑 → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) supp 0 ) ⊆ {𝑋}) |
37 | suppssfifsupp 8173 | . . 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 ) | |
38 | 28, 30, 36, 37 | syl12anc 1316 | . 2 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) finSupp 0 ) |
39 | mplmon.s | . . 3 ⊢ 𝑃 = (𝐼 mPoly 𝑅) | |
40 | mplmon.b | . . 3 ⊢ 𝐵 = (Base‘𝑃) | |
41 | 39, 18, 19, 5, 40 | mplelbas 19251 | . 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 )) |
42 | 22, 38, 41 | sylanbrc 695 | 1 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 {crab 2900 Vcvv 3173 ∖ cdif 3537 ⊆ wss 3540 ifcif 4036 {csn 4125 class class class wbr 4583 ↦ cmpt 4643 ◡ccnv 5037 “ cima 5041 Fun wfun 5798 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 supp csupp 7182 ↑𝑚 cmap 7744 Fincfn 7841 finSupp cfsupp 8158 ℕcn 10897 ℕ0cn0 11169 Basecbs 15695 0gc0g 15923 1rcur 18324 Ringcrg 18370 mPwSer cmps 19172 mPoly cmpl 19174 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-tset 15787 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-mgp 18313 df-ur 18325 df-ring 18372 df-psr 19177 df-mpl 19179 |
This theorem is referenced by: mplmonmul 19285 mplcoe1 19286 mplbas2 19291 mplmon2 19314 mplmon2cl 19321 mplmon2mul 19322 |
Copyright terms: Public domain | W3C validator |