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Theorem mplneg 21998

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 2737	. . . 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 21990	. . 3 ⊢ (𝜑 → 𝐵 ∈ (SubGrp‘(𝐼 mPwSer 𝑅)))
7		mplneg.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
8	2, 1, 3	mplval2 21984	. . . 4 ⊢ 𝑃 = ((𝐼 mPwSer 𝑅) ↾_s 𝐵)
9		eqid 2737	. . . 4 ⊢ (inv_g‘(𝐼 mPwSer 𝑅)) = (inv_g‘(𝐼 mPwSer 𝑅))
10		mplneg.m	. . . 4 ⊢ 𝑀 = (inv_g‘𝑃)
11	8, 9, 10	subginv 19100	. . 3 ⊢ ((𝐵 ∈ (SubGrp‘(𝐼 mPwSer 𝑅)) ∧ 𝑋 ∈ 𝐵) → ((inv_g‘(𝐼 mPwSer 𝑅))‘𝑋) = (𝑀‘𝑋))
12	6, 7, 11	syl2anc 585	. 2 ⊢ (𝜑 → ((inv_g‘(𝐼 mPwSer 𝑅))‘𝑋) = (𝑀‘𝑋))
13		eqid 2737	. . 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 2737	. . 3 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅))
16	2, 1, 3, 15	mplbasss 21985	. . . . 5 ⊢ 𝐵 ⊆ (Base‘(𝐼 mPwSer 𝑅))
17	16	sseli 3918	. . . 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 21947	. 2 ⊢ (𝜑 → ((inv_g‘(𝐼 mPwSer 𝑅))‘𝑋) = (𝑁 ∘ 𝑋))
20	12, 19	eqtr3d 2774	1 ⊢ (𝜑 → (𝑀‘𝑋) = (𝑁 ∘ 𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 {crab 3390 ^◡ccnv 5623 “ cima 5627 ∘ ccom 5628 ‘cfv 6492 (class class class)co 7360 ↑_m cmap 8766 Fincfn 8886 ℕcn 12165 ℕ₀cn0 12428 Basecbs 17170 Grpcgrp 18900 inv_gcminusg 18901 SubGrpcsubg 19087 mPwSer cmps 21894 mPoly cmpl 21896

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8104 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-er 8636 df-map 8768 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9268 df-sup 9348 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-fz 13453 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ds 17233 df-hom 17235 df-cco 17236 df-0g 17395 df-prds 17401 df-pws 17403 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18903 df-minusg 18904 df-subg 19090 df-psr 21899 df-mpl 21901

This theorem is referenced by: mhpinvcl 22128

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