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Theorem mplidom 33769

Description: The multivariate polynomials over an integral domain form an integral domain. See ply1idom 26154. (Contributed by Thierry Arnoux, 4-May-2026.)

**Hypotheses**
Ref	Expression
mplidom.p	⊢ 𝑃 = (𝐼 mPoly 𝑅)
mplidom.i	⊢ (𝜑 → 𝐼 ∈ Fin)
mplidom.r	⊢ (𝜑 → 𝑅 ∈ IDomn)

**Assertion**
Ref	Expression
mplidom	⊢ (𝜑 → 𝑃 ∈ IDomn)

Proof of Theorem mplidom Dummy variables 𝑓 𝑗 𝑛 𝑥 𝑒 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mplidom.p	. 2 ⊢ 𝑃 = (𝐼 mPoly 𝑅)
2		mplidom.i	. 2 ⊢ (𝜑 → 𝐼 ∈ Fin)
3		mplidom.r	. 2 ⊢ (𝜑 → 𝑅 ∈ IDomn)
4		fveq2 6852	. . . . . 6 ⊢ (𝑒 = 𝑓 → ((((𝑗 ∪ {𝑥}) selectVars 𝑅)‘{𝑥})‘𝑒) = ((((𝑗 ∪ {𝑥}) selectVars 𝑅)‘{𝑥})‘𝑓))
5	4	fveq1d 6854	. . . . 5 ⊢ (𝑒 = 𝑓 → (((((𝑗 ∪ {𝑥}) selectVars 𝑅)‘{𝑥})‘𝑒)‘{⟨𝑥, (𝑚‘∅)⟩}) = (((((𝑗 ∪ {𝑥}) selectVars 𝑅)‘{𝑥})‘𝑓)‘{⟨𝑥, (𝑚‘∅)⟩}))
6	5	mpteq2dv 5184	. . . 4 ⊢ (𝑒 = 𝑓 → (𝑚 ∈ (ℕ₀ ↑_m 1_o) ↦ (((((𝑗 ∪ {𝑥}) selectVars 𝑅)‘{𝑥})‘𝑒)‘{⟨𝑥, (𝑚‘∅)⟩})) = (𝑚 ∈ (ℕ₀ ↑_m 1_o) ↦ (((((𝑗 ∪ {𝑥}) selectVars 𝑅)‘{𝑥})‘𝑓)‘{⟨𝑥, (𝑚‘∅)⟩})))
7	6	cbvmptv 5194	. . 3 ⊢ (𝑒 ∈ (Base‘((𝑗 ∪ {𝑥}) mPoly 𝑅)) ↦ (𝑚 ∈ (ℕ₀ ↑_m 1_o) ↦ (((((𝑗 ∪ {𝑥}) selectVars 𝑅)‘{𝑥})‘𝑒)‘{⟨𝑥, (𝑚‘∅)⟩}))) = (𝑓 ∈ (Base‘((𝑗 ∪ {𝑥}) mPoly 𝑅)) ↦ (𝑚 ∈ (ℕ₀ ↑_m 1_o) ↦ (((((𝑗 ∪ {𝑥}) selectVars 𝑅)‘{𝑥})‘𝑓)‘{⟨𝑥, (𝑚‘∅)⟩})))
8		fveq1 6851	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → (𝑚‘∅) = (𝑛‘∅))
9	8	opeq2d 4828	. . . . . . 7 ⊢ (𝑚 = 𝑛 → ⟨𝑥, (𝑚‘∅)⟩ = ⟨𝑥, (𝑛‘∅)⟩)
10	9	sneqd 4584	. . . . . 6 ⊢ (𝑚 = 𝑛 → {⟨𝑥, (𝑚‘∅)⟩} = {⟨𝑥, (𝑛‘∅)⟩})
11	10	fveq2d 6856	. . . . 5 ⊢ (𝑚 = 𝑛 → (((((𝑗 ∪ {𝑥}) selectVars 𝑅)‘{𝑥})‘𝑓)‘{⟨𝑥, (𝑚‘∅)⟩}) = (((((𝑗 ∪ {𝑥}) selectVars 𝑅)‘{𝑥})‘𝑓)‘{⟨𝑥, (𝑛‘∅)⟩}))
12	11	cbvmptv 5194	. . . 4 ⊢ (𝑚 ∈ (ℕ₀ ↑_m 1_o) ↦ (((((𝑗 ∪ {𝑥}) selectVars 𝑅)‘{𝑥})‘𝑓)‘{⟨𝑥, (𝑚‘∅)⟩})) = (𝑛 ∈ (ℕ₀ ↑_m 1_o) ↦ (((((𝑗 ∪ {𝑥}) selectVars 𝑅)‘{𝑥})‘𝑓)‘{⟨𝑥, (𝑛‘∅)⟩}))
13	12	mpteq2i 5186	. . 3 ⊢ (𝑓 ∈ (Base‘((𝑗 ∪ {𝑥}) mPoly 𝑅)) ↦ (𝑚 ∈ (ℕ₀ ↑_m 1_o) ↦ (((((𝑗 ∪ {𝑥}) selectVars 𝑅)‘{𝑥})‘𝑓)‘{⟨𝑥, (𝑚‘∅)⟩}))) = (𝑓 ∈ (Base‘((𝑗 ∪ {𝑥}) mPoly 𝑅)) ↦ (𝑛 ∈ (ℕ₀ ↑_m 1_o) ↦ (((((𝑗 ∪ {𝑥}) selectVars 𝑅)‘{𝑥})‘𝑓)‘{⟨𝑥, (𝑛‘∅)⟩})))
14	7, 13	eqtri 2775	. 2 ⊢ (𝑒 ∈ (Base‘((𝑗 ∪ {𝑥}) mPoly 𝑅)) ↦ (𝑚 ∈ (ℕ₀ ↑_m 1_o) ↦ (((((𝑗 ∪ {𝑥}) selectVars 𝑅)‘{𝑥})‘𝑒)‘{⟨𝑥, (𝑚‘∅)⟩}))) = (𝑓 ∈ (Base‘((𝑗 ∪ {𝑥}) mPoly 𝑅)) ↦ (𝑛 ∈ (ℕ₀ ↑_m 1_o) ↦ (((((𝑗 ∪ {𝑥}) selectVars 𝑅)‘{𝑥})‘𝑓)‘{⟨𝑥, (𝑛‘∅)⟩})))
15		eqid 2752	. 2 ⊢ (Base‘((𝑗 ∪ {𝑥}) mPoly 𝑅)) = (Base‘((𝑗 ∪ {𝑥}) mPoly 𝑅))
16		eqid 2752	. 2 ⊢ ((𝑗 ∪ {𝑥}) mPoly 𝑅) = ((𝑗 ∪ {𝑥}) mPoly 𝑅)
17		eqid 2752	. 2 ⊢ (((𝑗 ∪ {𝑥}) ∖ {𝑥}) mPoly 𝑅) = (((𝑗 ∪ {𝑥}) ∖ {𝑥}) mPoly 𝑅)
18		eqid 2752	. 2 ⊢ (Poly₁‘(((𝑗 ∪ {𝑥}) ∖ {𝑥}) mPoly 𝑅)) = (Poly₁‘(((𝑗 ∪ {𝑥}) ∖ {𝑥}) mPoly 𝑅))
19	1, 2, 3, 14, 15, 16, 17, 18	mplidomlem 33768	1 ⊢ (𝜑 → 𝑃 ∈ IDomn)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1550 ∈ wcel 2132 ∖ cdif 3892 ∪ cun 3893 ∅c0 4276 {csn 4572 ⟨cop 4578 ↦ cmpt 5171 ‘cfv 6506 (class class class)co 7381 1_oc1o 8414 ↑_m cmap 8792 Fincfn 8912 ℕ₀cn0 12467 Basecbs 17217 IDomncidom 20711 mPoly cmpl 21927 selectVars cslv 22138 Poly₁cpl1 22208

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1805 ax-4 1819 ax-5 1920 ax-6 1977 ax-7 2018 ax-8 2134 ax-9 2142 ax-10 2165 ax-11 2181 ax-12 2202 ax-ext 2724 ax-rep 5217 ax-sep 5236 ax-nul 5246 ax-pow 5312 ax-pr 5380 ax-un 7703 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 ax-addf 11138

This theorem depends on definitions: df-bi 209 df-an 399 df-or 857 df-3or 1096 df-3an 1097 df-tru 1553 df-fal 1563 df-ex 1790 df-nf 1794 df-sb 2081 df-mo 2556 df-eu 2586 df-clab 2731 df-cleq 2744 df-clel 2827 df-nfc 2901 df-ne 2948 df-nel 3052 df-ral 3067 df-rex 3077 df-rmo 3357 df-reu 3358 df-rab 3405 df-v 3446 df-sbc 3736 df-csb 3844 df-dif 3898 df-un 3900 df-in 3902 df-ss 3912 df-pss 3915 df-nul 4277 df-if 4471 df-pw 4547 df-sn 4573 df-pr 4575 df-tp 4577 df-op 4579 df-uni 4856 df-int 4896 df-iun 4941 df-iin 4942 df-br 5091 df-opab 5153 df-mpt 5172 df-tr 5198 df-id 5531 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5589 df-se 5590 df-we 5591 df-xp 5642 df-rel 5643 df-cnv 5644 df-co 5645 df-dm 5646 df-rn 5647 df-res 5648 df-ima 5649 df-pred 6273 df-ord 6334 df-on 6335 df-lim 6336 df-suc 6337 df-iota 6462 df-fun 6508 df-fn 6509 df-f 6510 df-f1 6511 df-fo 6512 df-f1o 6513 df-fv 6514 df-isom 6515 df-riota 7338 df-ov 7384 df-oprab 7385 df-mpo 7386 df-of 7645 df-ofr 7646 df-om 7832 df-1st 7955 df-2nd 7956 df-supp 8125 df-frecs 8246 df-wrecs 8277 df-recs 8326 df-rdg 8365 df-1o 8421 df-2o 8422 df-er 8662 df-map 8794 df-pm 8795 df-ixp 8865 df-en 8913 df-dom 8914 df-sdom 8915 df-fin 8916 df-fsupp 9294 df-sup 9374 df-oi 9444 df-card 9883 df-pnf 11204 df-mnf 11205 df-xr 11206 df-ltxr 11207 df-le 11208 df-sub 11402 df-neg 11403 df-nn 12197 df-2 12266 df-3 12267 df-4 12268 df-5 12269 df-6 12270 df-7 12271 df-8 12272 df-9 12273 df-n0 12468 df-z 12555 df-dec 12675 df-uz 12826 df-fz 13499 df-fzo 13646 df-seq 14001 df-hash 14330 df-struct 17155 df-sets 17172 df-slot 17190 df-ndx 17202 df-base 17218 df-ress 17239 df-plusg 17271 df-mulr 17272 df-starv 17273 df-sca 17274 df-vsca 17275 df-ip 17276 df-tset 17277 df-ple 17278 df-ds 17280 df-unif 17281 df-hom 17282 df-cco 17283 df-0g 17442 df-gsum 17443 df-prds 17448 df-pws 17450 df-mre 17586 df-mrc 17587 df-acs 17589 df-mgm 18646 df-sgrp 18725 df-mnd 18741 df-mhm 18789 df-submnd 18790 df-grp 18950 df-minusg 18951 df-sbg 18952 df-mulg 19082 df-subg 19137 df-ghm 19226 df-cntz 19329 df-cmn 19794 df-abl 19795 df-mgp 20159 df-rng 20171 df-ur 20200 df-srg 20205 df-ring 20253 df-cring 20254 df-rhm 20489 df-rim 20490 df-ric 20492 df-nzr 20531 df-subrng 20564 df-subrg 20588 df-rlreg 20712 df-domn 20713 df-idom 20714 df-lmod 20898 df-lss 20968 df-lsp 21008 df-cnfld 21394 df-assa 21874 df-asp 21875 df-ascl 21876 df-psr 21930 df-mvr 21931 df-mpl 21932 df-opsr 21934 df-evls 22096 df-evl 22097 df-selv 22139 df-psr1 22211 df-vr1 22212 df-ply1 22213 df-coe1 22214 df-mdeg 26084 df-deg1 26085

This theorem is referenced by: (None)

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