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Theorem mplneg 20758

Description: The negative function on multivariate polynomials. (Contributed by SN, 25-May-2024.)

**Hypotheses**
Ref	Expression
mplneg.p	⊢ 𝑃 = (𝐼 mPoly 𝑅)
mplneg.b	⊢ 𝐵 = (Base‘𝑃)
mplneg.n	⊢ 𝑁 = (inv_g‘𝑅)
mplneg.m	⊢ 𝑀 = (inv_g‘𝑃)
mplneg.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
mplneg.r	⊢ (𝜑 → 𝑅 ∈ Grp)
mplneg.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)

**Assertion**
Ref	Expression
mplneg	⊢ (𝜑 → (𝑀‘𝑋) = (𝑁 ∘ 𝑋))

Proof of Theorem mplneg Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2759	. . . 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 20752	. . 3 ⊢ (𝜑 → 𝐵 ∈ (SubGrp‘(𝐼 mPwSer 𝑅)))
7		mplneg.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
8	2, 1, 3	mplval2 20746	. . . 4 ⊢ 𝑃 = ((𝐼 mPwSer 𝑅) ↾_s 𝐵)
9		eqid 2759	. . . 4 ⊢ (inv_g‘(𝐼 mPwSer 𝑅)) = (inv_g‘(𝐼 mPwSer 𝑅))
10		mplneg.m	. . . 4 ⊢ 𝑀 = (inv_g‘𝑃)
11	8, 9, 10	subginv 18338	. . 3 ⊢ ((𝐵 ∈ (SubGrp‘(𝐼 mPwSer 𝑅)) ∧ 𝑋 ∈ 𝐵) → ((inv_g‘(𝐼 mPwSer 𝑅))‘𝑋) = (𝑀‘𝑋))
12	6, 7, 11	syl2anc 588	. 2 ⊢ (𝜑 → ((inv_g‘(𝐼 mPwSer 𝑅))‘𝑋) = (𝑀‘𝑋))
13		eqid 2759	. . 3 ⊢ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}
14		mplneg.n	. . 3 ⊢ 𝑁 = (inv_g‘𝑅)
15		eqid 2759	. . 3 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅))
16	2, 1, 3, 15	mplbasss 20747	. . . . 5 ⊢ 𝐵 ⊆ (Base‘(𝐼 mPwSer 𝑅))
17	16	sseli 3884	. . . 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 20713	. 2 ⊢ (𝜑 → ((inv_g‘(𝐼 mPwSer 𝑅))‘𝑋) = (𝑁 ∘ 𝑋))
20	12, 19	eqtr3d 2796	1 ⊢ (𝜑 → (𝑀‘𝑋) = (𝑁 ∘ 𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2112 {crab 3072 ^◡ccnv 5516 “ cima 5520 ∘ ccom 5521 ‘cfv 6328 (class class class)co 7143 ↑_m cmap 8409 Fincfn 8520 ℕcn 11659 ℕ₀cn0 11919 Basecbs 16526 Grpcgrp 18154 inv_gcminusg 18155 SubGrpcsubg 18325 mPwSer cmps 20651 mPoly cmpl 20653

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 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-rep 5149 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 ax-cnex 10616 ax-resscn 10617 ax-1cn 10618 ax-icn 10619 ax-addcl 10620 ax-addrcl 10621 ax-mulcl 10622 ax-mulrcl 10623 ax-mulcom 10624 ax-addass 10625 ax-mulass 10626 ax-distr 10627 ax-i2m1 10628 ax-1ne0 10629 ax-1rid 10630 ax-rnegex 10631 ax-rrecex 10632 ax-cnre 10633 ax-pre-lttri 10634 ax-pre-lttrn 10635 ax-pre-ltadd 10636 ax-pre-mulgt0 10637

This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-nel 3054 df-ral 3073 df-rex 3074 df-reu 3075 df-rmo 3076 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-pss 3873 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-tp 4520 df-op 4522 df-uni 4792 df-int 4832 df-iun 4878 df-br 5026 df-opab 5088 df-mpt 5106 df-tr 5132 df-id 5423 df-eprel 5428 df-po 5436 df-so 5437 df-fr 5476 df-we 5478 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-pred 6119 df-ord 6165 df-on 6166 df-lim 6167 df-suc 6168 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-riota 7101 df-ov 7146 df-oprab 7147 df-mpo 7148 df-of 7398 df-om 7573 df-1st 7686 df-2nd 7687 df-supp 7829 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-map 8411 df-en 8521 df-dom 8522 df-sdom 8523 df-fin 8524 df-fsupp 8852 df-pnf 10700 df-mnf 10701 df-xr 10702 df-ltxr 10703 df-le 10704 df-sub 10895 df-neg 10896 df-nn 11660 df-2 11722 df-3 11723 df-4 11724 df-5 11725 df-6 11726 df-7 11727 df-8 11728 df-9 11729 df-n0 11920 df-z 12006 df-uz 12268 df-fz 12925 df-struct 16528 df-ndx 16529 df-slot 16530 df-base 16532 df-sets 16533 df-ress 16534 df-plusg 16621 df-mulr 16622 df-sca 16624 df-vsca 16625 df-tset 16627 df-0g 16758 df-mgm 17903 df-sgrp 17952 df-mnd 17963 df-grp 18157 df-minusg 18158 df-subg 18328 df-psr 20656 df-mpl 20658

This theorem is referenced by: mhpinvcl 20880

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