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Theorem mplidom 33712

Description: The multivariate polynomials over an integral domain form an integral domain. See ply1idom 26108. (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 6827	. . . . . 6 ⊢ (𝑒 = 𝑓 → ((((𝑗 ∪ {𝑥}) selectVars 𝑅)‘{𝑥})‘𝑒) = ((((𝑗 ∪ {𝑥}) selectVars 𝑅)‘{𝑥})‘𝑓))
5	4	fveq1d 6829	. . . . 5 ⊢ (𝑒 = 𝑓 → (((((𝑗 ∪ {𝑥}) selectVars 𝑅)‘{𝑥})‘𝑒)‘{⟨𝑥, (𝑚‘∅)⟩}) = (((((𝑗 ∪ {𝑥}) selectVars 𝑅)‘{𝑥})‘𝑓)‘{⟨𝑥, (𝑚‘∅)⟩}))
6	5	mpteq2dv 5166	. . . 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 5176	. . 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 6826	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → (𝑚‘∅) = (𝑛‘∅))
9	8	opeq2d 4811	. . . . . . 7 ⊢ (𝑚 = 𝑛 → ⟨𝑥, (𝑚‘∅)⟩ = ⟨𝑥, (𝑛‘∅)⟩)
10	9	sneqd 4567	. . . . . 6 ⊢ (𝑚 = 𝑛 → {⟨𝑥, (𝑚‘∅)⟩} = {⟨𝑥, (𝑛‘∅)⟩})
11	10	fveq2d 6831	. . . . 5 ⊢ (𝑚 = 𝑛 → (((((𝑗 ∪ {𝑥}) selectVars 𝑅)‘{𝑥})‘𝑓)‘{⟨𝑥, (𝑚‘∅)⟩}) = (((((𝑗 ∪ {𝑥}) selectVars 𝑅)‘{𝑥})‘𝑓)‘{⟨𝑥, (𝑛‘∅)⟩}))
12	11	cbvmptv 5176	. . . 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 5168	. . 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 2762	. 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 2739	. 2 ⊢ (Base‘((𝑗 ∪ {𝑥}) mPoly 𝑅)) = (Base‘((𝑗 ∪ {𝑥}) mPoly 𝑅))
16		eqid 2739	. 2 ⊢ ((𝑗 ∪ {𝑥}) mPoly 𝑅) = ((𝑗 ∪ {𝑥}) mPoly 𝑅)
17		eqid 2739	. 2 ⊢ (((𝑗 ∪ {𝑥}) ∖ {𝑥}) mPoly 𝑅) = (((𝑗 ∪ {𝑥}) ∖ {𝑥}) mPoly 𝑅)
18		eqid 2739	. 2 ⊢ (Poly₁‘(((𝑗 ∪ {𝑥}) ∖ {𝑥}) mPoly 𝑅)) = (Poly₁‘(((𝑗 ∪ {𝑥}) ∖ {𝑥}) mPoly 𝑅))
19	1, 2, 3, 14, 15, 16, 17, 18	mplidomlem 33711	1 ⊢ (𝜑 → 𝑃 ∈ IDomn)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1547 ∈ wcel 2119 ∖ cdif 3880 ∪ cun 3881 ∅c0 4261 {csn 4555 ⟨cop 4561 ↦ cmpt 5153 ‘cfv 6485 (class class class)co 7356 1_oc1o 8388 ↑_m cmap 8763 Fincfn 8883 ℕ₀cn0 12428 Basecbs 17170 IDomncidom 20665 mPoly cmpl 21881 selectVars cslv 22092 Poly₁cpl1 22162

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 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 ax-pre-sup 11107 ax-addf 11108

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-iin 4924 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-se 5572 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-isom 6494 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-ofr 7621 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8633 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-sup 9345 df-oi 9415 df-card 9854 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-fzo 13600 df-seq 13955 df-hash 14284 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-starv 17226 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ds 17233 df-unif 17234 df-hom 17235 df-cco 17236 df-0g 17395 df-gsum 17396 df-prds 17401 df-pws 17403 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18742 df-submnd 18743 df-grp 18903 df-minusg 18904 df-sbg 18905 df-mulg 19035 df-subg 19090 df-ghm 19179 df-cntz 19283 df-cmn 19748 df-abl 19749 df-mgp 20113 df-rng 20125 df-ur 20154 df-srg 20159 df-ring 20207 df-cring 20208 df-rhm 20443 df-rim 20444 df-ric 20446 df-nzr 20485 df-subrng 20518 df-subrg 20542 df-rlreg 20666 df-domn 20667 df-idom 20668 df-lmod 20852 df-lss 20922 df-lsp 20962 df-cnfld 21348 df-assa 21828 df-asp 21829 df-ascl 21830 df-psr 21884 df-mvr 21885 df-mpl 21886 df-opsr 21888 df-evls 22050 df-evl 22051 df-selv 22093 df-psr1 22165 df-vr1 22166 df-ply1 22167 df-coe1 22168 df-mdeg 26038 df-deg1 26039

This theorem is referenced by: (None)

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