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Theorem mplidom 34025

Description: The multivariate polynomials over an integral domain form an integral domain. See ply1idom 26352. (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 6882	. . . . . 6 ⊢ (𝑒 = 𝑓 → ((((𝑗 ∪ {𝑥}) selectVars 𝑅)‘{𝑥})‘𝑒) = ((((𝑗 ∪ {𝑥}) selectVars 𝑅)‘{𝑥})‘𝑓))
5	4	fveq1d 6884	. . . . 5 ⊢ (𝑒 = 𝑓 → (((((𝑗 ∪ {𝑥}) selectVars 𝑅)‘{𝑥})‘𝑒)‘{⟨𝑥, (𝑚‘∅)⟩}) = (((((𝑗 ∪ {𝑥}) selectVars 𝑅)‘{𝑥})‘𝑓)‘{⟨𝑥, (𝑚‘∅)⟩}))
6	5	mpteq2dv 5203	. . . 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 5213	. . 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 6881	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → (𝑚‘∅) = (𝑛‘∅))
9	8	opeq2d 4843	. . . . . . 7 ⊢ (𝑚 = 𝑛 → ⟨𝑥, (𝑚‘∅)⟩ = ⟨𝑥, (𝑛‘∅)⟩)
10	9	sneqd 4599	. . . . . 6 ⊢ (𝑚 = 𝑛 → {⟨𝑥, (𝑚‘∅)⟩} = {⟨𝑥, (𝑛‘∅)⟩})
11	10	fveq2d 6886	. . . . 5 ⊢ (𝑚 = 𝑛 → (((((𝑗 ∪ {𝑥}) selectVars 𝑅)‘{𝑥})‘𝑓)‘{⟨𝑥, (𝑚‘∅)⟩}) = (((((𝑗 ∪ {𝑥}) selectVars 𝑅)‘{𝑥})‘𝑓)‘{⟨𝑥, (𝑛‘∅)⟩}))
12	11	cbvmptv 5213	. . . 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 5205	. . 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 2785	. 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 2762	. 2 ⊢ (Base‘((𝑗 ∪ {𝑥}) mPoly 𝑅)) = (Base‘((𝑗 ∪ {𝑥}) mPoly 𝑅))
16		eqid 2762	. 2 ⊢ ((𝑗 ∪ {𝑥}) mPoly 𝑅) = ((𝑗 ∪ {𝑥}) mPoly 𝑅)
17		eqid 2762	. 2 ⊢ (((𝑗 ∪ {𝑥}) ∖ {𝑥}) mPoly 𝑅) = (((𝑗 ∪ {𝑥}) ∖ {𝑥}) mPoly 𝑅)
18		eqid 2762	. 2 ⊢ (Poly₁‘(((𝑗 ∪ {𝑥}) ∖ {𝑥}) mPoly 𝑅)) = (Poly₁‘(((𝑗 ∪ {𝑥}) ∖ {𝑥}) mPoly 𝑅))
19	1, 2, 3, 14, 15, 16, 17, 18	mplidomlem 34024	1 ⊢ (𝜑 → 𝑃 ∈ IDomn)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2145 ∖ cdif 3899 ∪ cun 3900 ∅c0 4282 {csn 4587 ⟨cop 4593 ↦ cmpt 5190 ‘cfv 6537 (class class class)co 7416 1_oc1o 8451 ↑_m cmap 8829 Fincfn 8955 ℕ₀cn0 12531 Basecbs 17305 IDomncidom 20856 mPoly cmpl 22122 selectVars cslv 22333 Poly₁cpl1 22403

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205 ax-addf 11206

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-se 5613 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7681 df-ofr 7682 df-om 7866 df-1st 7989 df-2nd 7990 df-supp 8162 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8458 df-2o 8459 df-er 8699 df-map 8831 df-pm 8832 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9335 df-sup 9415 df-oi 9485 df-card 9947 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-nn 12261 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12532 df-z 12619 df-dec 12740 df-uz 12891 df-fz 13564 df-fzo 13712 df-seq 14068 df-hash 14397 df-struct 17243 df-sets 17260 df-slot 17278 df-ndx 17290 df-base 17306 df-ress 17327 df-plusg 17359 df-mulr 17360 df-starv 17361 df-sca 17362 df-vsca 17363 df-ip 17364 df-tset 17365 df-ple 17366 df-ds 17368 df-unif 17369 df-hom 17370 df-cco 17371 df-0g 17530 df-gsum 17531 df-prds 17536 df-pws 17538 df-mre 17674 df-mrc 17675 df-acs 17677 df-mgm 18734 df-sgrp 18823 df-mnd 18839 df-mhm 18892 df-submnd 18893 df-grp 19061 df-minusg 19062 df-sbg 19063 df-mulg 19192 df-subg 19247 df-ghm 19342 df-cntz 19445 df-cmn 19910 df-abl 19911 df-mgp 20275 df-rng 20289 df-ur 20322 df-srg 20327 df-ring 20375 df-cring 20376 df-rhm 20614 df-rim 20615 df-ric 20655 df-nzr 20674 df-subrng 20709 df-subrg 20733 df-rlreg 20857 df-domn 20858 df-idom 20859 df-lmod 21047 df-lss 21117 df-lsp 21157 df-cnfld 21587 df-assa 22069 df-asp 22070 df-ascl 22071 df-psr 22125 df-mvr 22126 df-mpl 22127 df-opsr 22129 df-evls 22291 df-evl 22292 df-selv 22334 df-psr1 22406 df-vr1 22407 df-ply1 22408 df-coe1 22409 df-mdeg 26282 df-deg1 26283

This theorem is used by: (None)

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