1arithufd - Mathbox for Thierry Arnoux

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Theorem 1arithufd 34126

Description: Existence of a factorization into irreducible elements in a unique factorization domain. Any nonzero, non-unit element 𝑋 of a UFD 𝑅 can be written as a product of primes 𝑓. As shown in 1arithidom 34115, that factorization is unique, up to the order of the factors and multiplication by units. (Contributed by Thierry Arnoux, 3-Jun-2025.)

**Hypotheses**
Ref	Expression
1arithufd.b	⊢ 𝐵 = (Base‘𝑅)
1arithufd.0	⊢ 0 = (0_g‘𝑅)
1arithufd.u	⊢ 𝑈 = (Unit‘𝑅)
1arithufd.p	⊢ 𝑃 = (RPrime‘𝑅)
1arithufd.m	⊢ 𝑀 = (mulGrp‘𝑅)
1arithufd.r	⊢ (𝜑 → 𝑅 ∈ UFD)
1arithufd.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
1arithufd.2	⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈)
1arithufd.3	⊢ (𝜑 → 𝑋 ≠ 0 )

**Assertion**
Ref	Expression
1arithufd	⊢ (𝜑 → ∃𝑓 ∈ Word 𝑃𝑋 = (𝑀 Σ_g 𝑓))

Distinct variable groups: 0 ,𝑓 𝑓,𝑀 𝑃,𝑓 𝑅,𝑓 𝜑,𝑓 𝐵,𝑓 𝑈,𝑓 𝑓,𝑋

Proof of Theorem 1arithufd Dummy variables 𝑥 𝑔 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 490	. . . 4 ⊢ ((𝜑 ∧ 𝑅 ∈ DivRing) → 𝑅 ∈ DivRing)
2		1arithufd.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
3	2	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑅 ∈ DivRing) → 𝑋 ∈ 𝐵)
4		1arithufd.3	. . . . 5 ⊢ (𝜑 → 𝑋 ≠ 0 )
5	4	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑅 ∈ DivRing) → 𝑋 ≠ 0 )
6		1arithufd.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
7		1arithufd.u	. . . . . 6 ⊢ 𝑈 = (Unit‘𝑅)
8		1arithufd.0	. . . . . 6 ⊢ 0 = (0_g‘𝑅)
9	6, 7, 8	drngunit 20990	. . . . 5 ⊢ (𝑅 ∈ DivRing → (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 )))
10	9	biimpar 483	. . . 4 ⊢ ((𝑅 ∈ DivRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 )) → 𝑋 ∈ 𝑈)
11	1, 3, 5, 10	syl12anc 850	. . 3 ⊢ ((𝜑 ∧ 𝑅 ∈ DivRing) → 𝑋 ∈ 𝑈)
12		1arithufd.2	. . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈)
13	12	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑅 ∈ DivRing) → ¬ 𝑋 ∈ 𝑈)
14	11, 13	pm2.21dd 198	. 2 ⊢ ((𝜑 ∧ 𝑅 ∈ DivRing) → ∃𝑓 ∈ Word 𝑃𝑋 = (𝑀 Σ_g 𝑓))
15		1arithufd.p	. . . . 5 ⊢ 𝑃 = (RPrime‘𝑅)
16		1arithufd.m	. . . . 5 ⊢ 𝑀 = (mulGrp‘𝑅)
17		1arithufd.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ UFD)
18	17	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑅 ∈ DivRing) → 𝑅 ∈ UFD)
19		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑅 ∈ DivRing) → ¬ 𝑅 ∈ DivRing)
20		eqeq1 2764	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑦 = (𝑀 Σ_g 𝑓) ↔ 𝑥 = (𝑀 Σ_g 𝑓)))
21	20	rexbidv 3186	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (∃𝑓 ∈ Word 𝑃𝑦 = (𝑀 Σ_g 𝑓) ↔ ∃𝑓 ∈ Word 𝑃𝑥 = (𝑀 Σ_g 𝑓)))
22	21	cbvrabv 3422	. . . . . 6 ⊢ {𝑦 ∈ 𝐵 ∣ ∃𝑓 ∈ Word 𝑃𝑦 = (𝑀 Σ_g 𝑓)} = {𝑥 ∈ 𝐵 ∣ ∃𝑓 ∈ Word 𝑃𝑥 = (𝑀 Σ_g 𝑓)}
23		oveq2 7428	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → (𝑀 Σ_g 𝑓) = (𝑀 Σ_g 𝑔))
24	23	eqeq2d 2771	. . . . . . 7 ⊢ (𝑓 = 𝑔 → (𝑥 = (𝑀 Σ_g 𝑓) ↔ 𝑥 = (𝑀 Σ_g 𝑔)))
25	24	cbvrexvw 3241	. . . . . 6 ⊢ (∃𝑓 ∈ Word 𝑃𝑥 = (𝑀 Σ_g 𝑓) ↔ ∃𝑔 ∈ Word 𝑃𝑥 = (𝑀 Σ_g 𝑔))
26	22, 25	rabbieq 3420	. . . . 5 ⊢ {𝑦 ∈ 𝐵 ∣ ∃𝑓 ∈ Word 𝑃𝑦 = (𝑀 Σ_g 𝑓)} = {𝑥 ∈ 𝐵 ∣ ∃𝑔 ∈ Word 𝑃𝑥 = (𝑀 Σ_g 𝑔)}
27	2	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑅 ∈ DivRing) → 𝑋 ∈ 𝐵)
28	12	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑅 ∈ DivRing) → ¬ 𝑋 ∈ 𝑈)
29	4	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑅 ∈ DivRing) → 𝑋 ≠ 0 )
30	6, 8, 7, 15, 16, 18, 19, 26, 27, 28, 29	1arithufdlem4 34125	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑅 ∈ DivRing) → 𝑋 ∈ {𝑦 ∈ 𝐵 ∣ ∃𝑓 ∈ Word 𝑃𝑦 = (𝑀 Σ_g 𝑓)})
31		eqeq1 2764	. . . . . 6 ⊢ (𝑦 = 𝑋 → (𝑦 = (𝑀 Σ_g 𝑓) ↔ 𝑋 = (𝑀 Σ_g 𝑓)))
32	31	rexbidv 3186	. . . . 5 ⊢ (𝑦 = 𝑋 → (∃𝑓 ∈ Word 𝑃𝑦 = (𝑀 Σ_g 𝑓) ↔ ∃𝑓 ∈ Word 𝑃𝑋 = (𝑀 Σ_g 𝑓)))
33	32	elrab 3644	. . . 4 ⊢ (𝑋 ∈ {𝑦 ∈ 𝐵 ∣ ∃𝑓 ∈ Word 𝑃𝑦 = (𝑀 Σ_g 𝑓)} ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑓 ∈ Word 𝑃𝑋 = (𝑀 Σ_g 𝑓)))
34	30, 33	sylib 221	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ∈ DivRing) → (𝑋 ∈ 𝐵 ∧ ∃𝑓 ∈ Word 𝑃𝑋 = (𝑀 Σ_g 𝑓)))
35	34	simprd 501	. 2 ⊢ ((𝜑 ∧ ¬ 𝑅 ∈ DivRing) → ∃𝑓 ∈ Word 𝑃𝑋 = (𝑀 Σ_g 𝑓))
36	14, 35	pm2.61dan 825	1 ⊢ (𝜑 → ∃𝑓 ∈ Word 𝑃𝑋 = (𝑀 Σ_g 𝑓))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 ∃wrex 3086 {crab 3412 ‘cfv 6538 (class class class)co 7420 Word cword 14661 Basecbs 17392 0_gc0g 17615 Σ_g cgsu 17616 mulGrpcmgp 20365 Unitcui 20590 RPrimecrpm 20667 DivRingcdr 20985 UFDcufd 34116

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-ac2 10544 ax-cnex 11259 ax-resscn 11260 ax-1cn 11261 ax-icn 11262 ax-addcl 11263 ax-addrcl 11264 ax-mulcl 11265 ax-mulrcl 11266 ax-mulcom 11267 ax-addass 11268 ax-mulass 11269 ax-distr 11270 ax-i2m1 11271 ax-1ne0 11272 ax-1rid 11273 ax-rnegex 11274 ax-rrecex 11275 ax-cnre 11276 ax-pre-lttri 11277 ax-pre-lttrn 11278 ax-pre-ltadd 11279 ax-pre-mulgt0 11280

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-rpss 7739 df-om 7878 df-1st 8001 df-2nd 8002 df-supp 8180 df-tpos 8246 df-frecs 8302 df-wrecs 8333 df-recs 8382 df-rdg 8421 df-1o 8479 df-oadd 8483 df-er 8720 df-en 8977 df-dom 8978 df-sdom 8979 df-fin 8980 df-fsupp 9357 df-oi 9507 df-dju 9985 df-card 10023 df-ac 10198 df-pnf 11348 df-mnf 11349 df-xr 11350 df-ltxr 11351 df-le 11352 df-sub 11546 df-neg 11547 df-nn 12339 df-2 12408 df-3 12409 df-4 12410 df-5 12411 df-6 12412 df-7 12413 df-8 12414 df-n0 12610 df-xnn0 12683 df-z 12697 df-uz 12969 df-fz 13643 df-fzo 13792 df-seq 14148 df-hash 14478 df-word 14662 df-lsw 14711 df-concat 14719 df-s1 14746 df-substr 14793 df-pfx 14825 df-sets 17347 df-slot 17365 df-ndx 17377 df-base 17393 df-ress 17414 df-plusg 17446 df-mulr 17447 df-sca 17449 df-vsca 17450 df-ip 17451 df-0g 17617 df-gsum 17618 df-mgm 18821 df-sgrp 18913 df-mnd 18929 df-submnd 18984 df-grp 19152 df-minusg 19153 df-sbg 19154 df-subg 19338 df-cntz 19536 df-lsm 19855 df-cmn 20001 df-abl 20002 df-mgp 20366 df-rng 20380 df-ur 20413 df-ring 20466 df-cring 20467 df-oppr 20572 df-dvdsr 20592 df-unit 20593 df-invr 20623 df-rprm 20668 df-nzr 20768 df-subrg 20827 df-domn 20952 df-idom 20953 df-drng 20987 df-lmod 21142 df-lss 21212 df-lsp 21252 df-sra 21453 df-rgmod 21454 df-lidl 21491 df-rsp 21492 df-prmidl 21622 df-ufd 34117

This theorem is used by: dfufd2 34128

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