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Theorem mplneg 21790

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 2730	. . . 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 21782	. . 3 ⊢ (𝜑 → 𝐵 ∈ (SubGrp‘(𝐼 mPwSer 𝑅)))
7		mplneg.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
8	2, 1, 3	mplval2 21776	. . . 4 ⊢ 𝑃 = ((𝐼 mPwSer 𝑅) ↾_s 𝐵)
9		eqid 2730	. . . 4 ⊢ (inv_g‘(𝐼 mPwSer 𝑅)) = (inv_g‘(𝐼 mPwSer 𝑅))
10		mplneg.m	. . . 4 ⊢ 𝑀 = (inv_g‘𝑃)
11	8, 9, 10	subginv 19051	. . 3 ⊢ ((𝐵 ∈ (SubGrp‘(𝐼 mPwSer 𝑅)) ∧ 𝑋 ∈ 𝐵) → ((inv_g‘(𝐼 mPwSer 𝑅))‘𝑋) = (𝑀‘𝑋))
12	6, 7, 11	syl2anc 582	. 2 ⊢ (𝜑 → ((inv_g‘(𝐼 mPwSer 𝑅))‘𝑋) = (𝑀‘𝑋))
13		eqid 2730	. . 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 2730	. . 3 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅))
16	2, 1, 3, 15	mplbasss 21777	. . . . 5 ⊢ 𝐵 ⊆ (Base‘(𝐼 mPwSer 𝑅))
17	16	sseli 3979	. . . 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 21741	. 2 ⊢ (𝜑 → ((inv_g‘(𝐼 mPwSer 𝑅))‘𝑋) = (𝑁 ∘ 𝑋))
20	12, 19	eqtr3d 2772	1 ⊢ (𝜑 → (𝑀‘𝑋) = (𝑁 ∘ 𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2104 {crab 3430 ^◡ccnv 5676 “ cima 5680 ∘ ccom 5681 ‘cfv 6544 (class class class)co 7413 ↑_m cmap 8824 Fincfn 8943 ℕcn 12218 ℕ₀cn0 12478 Basecbs 17150 Grpcgrp 18857 inv_gcminusg 18858 SubGrpcsubg 19038 mPwSer cmps 21678 mPoly cmpl 21680

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8151 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-er 8707 df-map 8826 df-ixp 8896 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fsupp 9366 df-sup 9441 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11452 df-neg 11453 df-nn 12219 df-2 12281 df-3 12282 df-4 12283 df-5 12284 df-6 12285 df-7 12286 df-8 12287 df-9 12288 df-n0 12479 df-z 12565 df-dec 12684 df-uz 12829 df-fz 13491 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17151 df-ress 17180 df-plusg 17216 df-mulr 17217 df-sca 17219 df-vsca 17220 df-ip 17221 df-tset 17222 df-ple 17223 df-ds 17225 df-hom 17227 df-cco 17228 df-0g 17393 df-prds 17399 df-pws 17401 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-grp 18860 df-minusg 18861 df-subg 19041 df-psr 21683 df-mpl 21685

This theorem is referenced by: mhpinvcl 21916

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