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Mirrors > Home > MPE Home > Th. List > mplneg | Structured version Visualization version GIF version |
Description: The negative function on multivariate polynomials. (Contributed by SN, 25-May-2024.) |
Ref | Expression |
---|---|
mplneg.p | ⊢ 𝑃 = (𝐼 mPoly 𝑅) |
mplneg.b | ⊢ 𝐵 = (Base‘𝑃) |
mplneg.n | ⊢ 𝑁 = (invg‘𝑅) |
mplneg.m | ⊢ 𝑀 = (invg‘𝑃) |
mplneg.i | ⊢ (𝜑 → 𝐼 ∈ 𝑉) |
mplneg.r | ⊢ (𝜑 → 𝑅 ∈ Grp) |
mplneg.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
Ref | Expression |
---|---|
mplneg | ⊢ (𝜑 → (𝑀‘𝑋) = (𝑁 ∘ 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . . 4 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅) | |
2 | mplneg.p | . . . 4 ⊢ 𝑃 = (𝐼 mPoly 𝑅) | |
3 | mplneg.b | . . . 4 ⊢ 𝐵 = (Base‘𝑃) | |
4 | mplneg.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
5 | mplneg.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Grp) | |
6 | 1, 2, 3, 4, 5 | mplsubg 21543 | . . 3 ⊢ (𝜑 → 𝐵 ∈ (SubGrp‘(𝐼 mPwSer 𝑅))) |
7 | mplneg.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
8 | 2, 1, 3 | mplval2 21537 | . . . 4 ⊢ 𝑃 = ((𝐼 mPwSer 𝑅) ↾s 𝐵) |
9 | eqid 2733 | . . . 4 ⊢ (invg‘(𝐼 mPwSer 𝑅)) = (invg‘(𝐼 mPwSer 𝑅)) | |
10 | mplneg.m | . . . 4 ⊢ 𝑀 = (invg‘𝑃) | |
11 | 8, 9, 10 | subginv 19007 | . . 3 ⊢ ((𝐵 ∈ (SubGrp‘(𝐼 mPwSer 𝑅)) ∧ 𝑋 ∈ 𝐵) → ((invg‘(𝐼 mPwSer 𝑅))‘𝑋) = (𝑀‘𝑋)) |
12 | 6, 7, 11 | syl2anc 585 | . 2 ⊢ (𝜑 → ((invg‘(𝐼 mPwSer 𝑅))‘𝑋) = (𝑀‘𝑋)) |
13 | eqid 2733 | . . 3 ⊢ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
14 | mplneg.n | . . 3 ⊢ 𝑁 = (invg‘𝑅) | |
15 | eqid 2733 | . . 3 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅)) | |
16 | 2, 1, 3, 15 | mplbasss 21538 | . . . . 5 ⊢ 𝐵 ⊆ (Base‘(𝐼 mPwSer 𝑅)) |
17 | 16 | sseli 3977 | . . . 4 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ (Base‘(𝐼 mPwSer 𝑅))) |
18 | 7, 17 | syl 17 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘(𝐼 mPwSer 𝑅))) |
19 | 1, 4, 5, 13, 14, 15, 9, 18 | psrneg 21502 | . 2 ⊢ (𝜑 → ((invg‘(𝐼 mPwSer 𝑅))‘𝑋) = (𝑁 ∘ 𝑋)) |
20 | 12, 19 | eqtr3d 2775 | 1 ⊢ (𝜑 → (𝑀‘𝑋) = (𝑁 ∘ 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 {crab 3433 ◡ccnv 5674 “ cima 5678 ∘ ccom 5679 ‘cfv 6540 (class class class)co 7404 ↑m cmap 8816 Fincfn 8935 ℕcn 12208 ℕ0cn0 12468 Basecbs 17140 Grpcgrp 18815 invgcminusg 18816 SubGrpcsubg 18994 mPwSer cmps 21439 mPoly cmpl 21441 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7665 df-om 7851 df-1st 7970 df-2nd 7971 df-supp 8142 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-map 8818 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-sup 9433 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-fz 13481 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-hom 17217 df-cco 17218 df-0g 17383 df-prds 17389 df-pws 17391 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-grp 18818 df-minusg 18819 df-subg 18997 df-psr 21444 df-mpl 21446 |
This theorem is referenced by: mhpinvcl 21677 |
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