1arithufd - Mathbox for Thierry Arnoux

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Theorem 1arithufd 33705

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 33694, 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 488	. . . 4 ⊢ ((𝜑 ∧ 𝑅 ∈ DivRing) → 𝑅 ∈ DivRing)
2		1arithufd.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
3	2	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑅 ∈ DivRing) → 𝑋 ∈ 𝐵)
4		1arithufd.3	. . . . 5 ⊢ (𝜑 → 𝑋 ≠ 0 )
5	4	adantr 484	. . . 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 20763	. . . . 5 ⊢ (𝑅 ∈ DivRing → (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 )))
10	9	biimpar 481	. . . 4 ⊢ ((𝑅 ∈ DivRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 )) → 𝑋 ∈ 𝑈)
11	1, 3, 5, 10	syl12anc 847	. . 3 ⊢ ((𝜑 ∧ 𝑅 ∈ DivRing) → 𝑋 ∈ 𝑈)
12		1arithufd.2	. . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈)
13	12	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝑅 ∈ DivRing) → ¬ 𝑋 ∈ 𝑈)
14	11, 13	pm2.21dd 197	. 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 484	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑅 ∈ DivRing) → 𝑅 ∈ UFD)
19		simpr 488	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑅 ∈ DivRing) → ¬ 𝑅 ∈ DivRing)
20		eqeq1 2765	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑦 = (𝑀 Σ_g 𝑓) ↔ 𝑥 = (𝑀 Σ_g 𝑓)))
21	20	rexbidv 3185	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (∃𝑓 ∈ Word 𝑃𝑦 = (𝑀 Σ_g 𝑓) ↔ ∃𝑓 ∈ Word 𝑃𝑥 = (𝑀 Σ_g 𝑓)))
22	21	cbvrabv 3423	. . . . . 6 ⊢ {𝑦 ∈ 𝐵 ∣ ∃𝑓 ∈ Word 𝑃𝑦 = (𝑀 Σ_g 𝑓)} = {𝑥 ∈ 𝐵 ∣ ∃𝑓 ∈ Word 𝑃𝑥 = (𝑀 Σ_g 𝑓)}
23		oveq2 7400	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → (𝑀 Σ_g 𝑓) = (𝑀 Σ_g 𝑔))
24	23	eqeq2d 2772	. . . . . . 7 ⊢ (𝑓 = 𝑔 → (𝑥 = (𝑀 Σ_g 𝑓) ↔ 𝑥 = (𝑀 Σ_g 𝑔)))
25	24	cbvrexvw 3240	. . . . . 6 ⊢ (∃𝑓 ∈ Word 𝑃𝑥 = (𝑀 Σ_g 𝑓) ↔ ∃𝑔 ∈ Word 𝑃𝑥 = (𝑀 Σ_g 𝑔))
26	22, 25	rabbieq 3421	. . . . 5 ⊢ {𝑦 ∈ 𝐵 ∣ ∃𝑓 ∈ Word 𝑃𝑦 = (𝑀 Σ_g 𝑓)} = {𝑥 ∈ 𝐵 ∣ ∃𝑔 ∈ Word 𝑃𝑥 = (𝑀 Σ_g 𝑔)}
27	2	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑅 ∈ DivRing) → 𝑋 ∈ 𝐵)
28	12	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑅 ∈ DivRing) → ¬ 𝑋 ∈ 𝑈)
29	4	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑅 ∈ DivRing) → 𝑋 ≠ 0 )
30	6, 8, 7, 15, 16, 18, 19, 26, 27, 28, 29	1arithufdlem4 33704	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑅 ∈ DivRing) → 𝑋 ∈ {𝑦 ∈ 𝐵 ∣ ∃𝑓 ∈ Word 𝑃𝑦 = (𝑀 Σ_g 𝑓)})
31		eqeq1 2765	. . . . . 6 ⊢ (𝑦 = 𝑋 → (𝑦 = (𝑀 Σ_g 𝑓) ↔ 𝑋 = (𝑀 Σ_g 𝑓)))
32	31	rexbidv 3185	. . . . 5 ⊢ (𝑦 = 𝑋 → (∃𝑓 ∈ Word 𝑃𝑦 = (𝑀 Σ_g 𝑓) ↔ ∃𝑓 ∈ Word 𝑃𝑋 = (𝑀 Σ_g 𝑓)))
33	32	elrab 3650	. . . 4 ⊢ (𝑋 ∈ {𝑦 ∈ 𝐵 ∣ ∃𝑓 ∈ Word 𝑃𝑦 = (𝑀 Σ_g 𝑓)} ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑓 ∈ Word 𝑃𝑋 = (𝑀 Σ_g 𝑓)))
34	30, 33	sylib 220	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ∈ DivRing) → (𝑋 ∈ 𝐵 ∧ ∃𝑓 ∈ Word 𝑃𝑋 = (𝑀 Σ_g 𝑓)))
35	34	simprd 499	. 2 ⊢ ((𝜑 ∧ ¬ 𝑅 ∈ DivRing) → ∃𝑓 ∈ Word 𝑃𝑋 = (𝑀 Σ_g 𝑓))
36	14, 35	pm2.61dan 822	1 ⊢ (𝜑 → ∃𝑓 ∈ Word 𝑃𝑋 = (𝑀 Σ_g 𝑓))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1559 ∈ wcel 2141 ≠ wne 2956 ∃wrex 3085 {crab 3413 ‘cfv 6517 (class class class)co 7392 Word cword 14523 Basecbs 17228 0_gc0g 17451 Σ_g cgsu 17452 mulGrpcmgp 20169 Unitcui 20383 RPrimecrpm 20460 DivRingcdr 20758 UFDcufd 33695

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714 ax-ac2 10417 ax-cnex 11126 ax-resscn 11127 ax-1cn 11128 ax-icn 11129 ax-addcl 11130 ax-addrcl 11131 ax-mulcl 11132 ax-mulrcl 11133 ax-mulcom 11134 ax-addass 11135 ax-mulass 11136 ax-distr 11137 ax-i2m1 11138 ax-1ne0 11139 ax-1rid 11140 ax-rnegex 11141 ax-rrecex 11142 ax-cnre 11143 ax-pre-lttri 11144 ax-pre-lttrn 11145 ax-pre-ltadd 11146 ax-pre-mulgt0 11147

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-int 4905 df-iun 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5540 df-eprel 5545 df-po 5553 df-so 5554 df-fr 5598 df-se 5599 df-we 5600 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-pred 6284 df-ord 6345 df-on 6346 df-lim 6347 df-suc 6348 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-f1 6522 df-fo 6523 df-f1o 6524 df-fv 6525 df-isom 6526 df-riota 7349 df-ov 7395 df-oprab 7396 df-mpo 7397 df-rpss 7702 df-om 7843 df-1st 7966 df-2nd 7967 df-supp 8136 df-tpos 8201 df-frecs 8257 df-wrecs 8288 df-recs 8337 df-rdg 8376 df-1o 8432 df-oadd 8436 df-er 8673 df-en 8924 df-dom 8925 df-sdom 8926 df-fin 8927 df-fsupp 9305 df-oi 9455 df-dju 9856 df-card 9894 df-ac 10069 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11413 df-neg 11414 df-nn 12208 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-n0 12479 df-xnn0 12552 df-z 12566 df-uz 12837 df-fz 13510 df-fzo 13657 df-seq 14012 df-hash 14341 df-word 14524 df-lsw 14573 df-concat 14581 df-s1 14607 df-substr 14652 df-pfx 14682 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17250 df-plusg 17282 df-mulr 17283 df-sca 17285 df-vsca 17286 df-ip 17287 df-0g 17453 df-gsum 17454 df-mgm 18657 df-sgrp 18736 df-mnd 18752 df-submnd 18801 df-grp 18961 df-minusg 18962 df-sbg 18963 df-subg 19148 df-cntz 19340 df-lsm 19659 df-cmn 19805 df-abl 19806 df-mgp 20170 df-rng 20182 df-ur 20211 df-ring 20264 df-cring 20265 df-oppr 20365 df-dvdsr 20385 df-unit 20386 df-invr 20416 df-rprm 20461 df-nzr 20542 df-subrg 20599 df-domn 20724 df-idom 20725 df-drng 20760 df-lmod 20909 df-lss 20979 df-lsp 21019 df-sra 21220 df-rgmod 21221 df-lidl 21258 df-rsp 21259 df-prmidl 33583 df-ufd 33696

This theorem is referenced by: dfufd2 33707

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