1arithufd - Mathbox for Thierry Arnoux

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Theorem 1arithufd 33869

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 33858, 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 20869	. . . . 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 2770	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑦 = (𝑀 Σ_g 𝑓) ↔ 𝑥 = (𝑀 Σ_g 𝑓)))
21	20	rexbidv 3192	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (∃𝑓 ∈ Word 𝑃𝑦 = (𝑀 Σ_g 𝑓) ↔ ∃𝑓 ∈ Word 𝑃𝑥 = (𝑀 Σ_g 𝑓)))
22	21	cbvrabv 3429	. . . . . 6 ⊢ {𝑦 ∈ 𝐵 ∣ ∃𝑓 ∈ Word 𝑃𝑦 = (𝑀 Σ_g 𝑓)} = {𝑥 ∈ 𝐵 ∣ ∃𝑓 ∈ Word 𝑃𝑥 = (𝑀 Σ_g 𝑓)}
23		oveq2 7431	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → (𝑀 Σ_g 𝑓) = (𝑀 Σ_g 𝑔))
24	23	eqeq2d 2777	. . . . . . 7 ⊢ (𝑓 = 𝑔 → (𝑥 = (𝑀 Σ_g 𝑓) ↔ 𝑥 = (𝑀 Σ_g 𝑔)))
25	24	cbvrexvw 3247	. . . . . 6 ⊢ (∃𝑓 ∈ Word 𝑃𝑥 = (𝑀 Σ_g 𝑓) ↔ ∃𝑔 ∈ Word 𝑃𝑥 = (𝑀 Σ_g 𝑔))
26	22, 25	rabbieq 3427	. . . . 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 33868	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑅 ∈ DivRing) → 𝑋 ∈ {𝑦 ∈ 𝐵 ∣ ∃𝑓 ∈ Word 𝑃𝑦 = (𝑀 Σ_g 𝑓)})
31		eqeq1 2770	. . . . . 6 ⊢ (𝑦 = 𝑋 → (𝑦 = (𝑀 Σ_g 𝑓) ↔ 𝑋 = (𝑀 Σ_g 𝑓)))
32	31	rexbidv 3192	. . . . 5 ⊢ (𝑦 = 𝑋 → (∃𝑓 ∈ Word 𝑃𝑦 = (𝑀 Σ_g 𝑓) ↔ ∃𝑓 ∈ Word 𝑃𝑋 = (𝑀 Σ_g 𝑓)))
33	32	elrab 3653	. . . 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 1570 ∈ wcel 2146 ≠ wne 2961 ∃wrex 3092 {crab 3419 ‘cfv 6543 (class class class)co 7423 Word cword 14570 Basecbs 17294 0_gc0g 17517 Σ_g cgsu 17518 mulGrpcmgp 20247 Unitcui 20470 RPrimecrpm 20547 DivRingcdr 20864 UFDcufd 33859

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-ac2 10465 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195

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 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4878 df-int 4918 df-iun 4963 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-se 5620 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-rpss 7733 df-om 7872 df-1st 7995 df-2nd 7996 df-supp 8166 df-tpos 8231 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-1o 8462 df-oadd 8466 df-er 8703 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-fsupp 9332 df-oi 9482 df-dju 9906 df-card 9944 df-ac 10119 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-nn 12252 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-n0 12523 df-xnn0 12596 df-z 12610 df-uz 12881 df-fz 13554 df-fzo 13702 df-seq 14058 df-hash 14387 df-word 14571 df-lsw 14620 df-concat 14628 df-s1 14655 df-substr 14701 df-pfx 14733 df-sets 17249 df-slot 17267 df-ndx 17279 df-base 17295 df-ress 17316 df-plusg 17348 df-mulr 17349 df-sca 17351 df-vsca 17352 df-ip 17353 df-0g 17519 df-gsum 17520 df-mgm 18723 df-sgrp 18806 df-mnd 18822 df-submnd 18873 df-grp 19034 df-minusg 19035 df-sbg 19036 df-subg 19220 df-cntz 19418 df-lsm 19737 df-cmn 19883 df-abl 19884 df-mgp 20248 df-rng 20262 df-ur 20295 df-ring 20348 df-cring 20349 df-oppr 20452 df-dvdsr 20472 df-unit 20473 df-invr 20503 df-rprm 20548 df-nzr 20647 df-subrg 20706 df-domn 20831 df-idom 20832 df-drng 20866 df-lmod 21020 df-lss 21090 df-lsp 21130 df-sra 21331 df-rgmod 21332 df-lidl 21369 df-rsp 21370 df-prmidl 21498 df-ufd 33860

This theorem is used by: dfufd2 33871

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