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Theorem mplneg 21965

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 2736	. . . 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 21957	. . 3 ⊢ (𝜑 → 𝐵 ∈ (SubGrp‘(𝐼 mPwSer 𝑅)))
7		mplneg.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
8	2, 1, 3	mplval2 21951	. . . 4 ⊢ 𝑃 = ((𝐼 mPwSer 𝑅) ↾_s 𝐵)
9		eqid 2736	. . . 4 ⊢ (inv_g‘(𝐼 mPwSer 𝑅)) = (inv_g‘(𝐼 mPwSer 𝑅))
10		mplneg.m	. . . 4 ⊢ 𝑀 = (inv_g‘𝑃)
11	8, 9, 10	subginv 19063	. . 3 ⊢ ((𝐵 ∈ (SubGrp‘(𝐼 mPwSer 𝑅)) ∧ 𝑋 ∈ 𝐵) → ((inv_g‘(𝐼 mPwSer 𝑅))‘𝑋) = (𝑀‘𝑋))
12	6, 7, 11	syl2anc 584	. 2 ⊢ (𝜑 → ((inv_g‘(𝐼 mPwSer 𝑅))‘𝑋) = (𝑀‘𝑋))
13		eqid 2736	. . 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 2736	. . 3 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅))
16	2, 1, 3, 15	mplbasss 21952	. . . . 5 ⊢ 𝐵 ⊆ (Base‘(𝐼 mPwSer 𝑅))
17	16	sseli 3929	. . . 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 21914	. 2 ⊢ (𝜑 → ((inv_g‘(𝐼 mPwSer 𝑅))‘𝑋) = (𝑁 ∘ 𝑋))
20	12, 19	eqtr3d 2773	1 ⊢ (𝜑 → (𝑀‘𝑋) = (𝑁 ∘ 𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2113 {crab 3399 ^◡ccnv 5623 “ cima 5627 ∘ ccom 5628 ‘cfv 6492 (class class class)co 7358 ↑_m cmap 8763 Fincfn 8883 ℕcn 12145 ℕ₀cn0 12401 Basecbs 17136 Grpcgrp 18863 inv_gcminusg 18864 SubGrpcsubg 19050 mPwSer cmps 21860 mPoly cmpl 21862

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 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 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-er 8635 df-map 8765 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-sup 9345 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-z 12489 df-dec 12608 df-uz 12752 df-fz 13424 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-sca 17193 df-vsca 17194 df-ip 17195 df-tset 17196 df-ple 17197 df-ds 17199 df-hom 17201 df-cco 17202 df-0g 17361 df-prds 17367 df-pws 17369 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-grp 18866 df-minusg 18867 df-subg 19053 df-psr 21865 df-mpl 21867

This theorem is referenced by: mhpinvcl 22095

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