1arithufd - Mathbox for Thierry Arnoux

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Theorem 1arithufd 33495

Description: Existence of a factorization into irreducible elements in a unique factorization domain. Any non-zero, non-unit element 𝑋 of a UFD 𝑅 can be written as a product of primes 𝑓. As shown in 1arithidom 33484, 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 484	. . . 4 ⊢ ((𝜑 ∧ 𝑅 ∈ DivRing) → 𝑅 ∈ DivRing)
2		1arithufd.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
3	2	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑅 ∈ DivRing) → 𝑋 ∈ 𝐵)
4		1arithufd.3	. . . . 5 ⊢ (𝜑 → 𝑋 ≠ 0 )
5	4	adantr 480	. . . 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 20637	. . . . 5 ⊢ (𝑅 ∈ DivRing → (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 )))
10	9	biimpar 477	. . . 4 ⊢ ((𝑅 ∈ DivRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 )) → 𝑋 ∈ 𝑈)
11	1, 3, 5, 10	syl12anc 836	. . 3 ⊢ ((𝜑 ∧ 𝑅 ∈ DivRing) → 𝑋 ∈ 𝑈)
12		1arithufd.2	. . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈)
13	12	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑅 ∈ DivRing) → ¬ 𝑋 ∈ 𝑈)
14	11, 13	pm2.21dd 195	. 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 480	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑅 ∈ DivRing) → 𝑅 ∈ UFD)
19		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑅 ∈ DivRing) → ¬ 𝑅 ∈ DivRing)
20		eqeq1 2733	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑦 = (𝑀 Σ_g 𝑓) ↔ 𝑥 = (𝑀 Σ_g 𝑓)))
21	20	rexbidv 3153	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (∃𝑓 ∈ Word 𝑃𝑦 = (𝑀 Σ_g 𝑓) ↔ ∃𝑓 ∈ Word 𝑃𝑥 = (𝑀 Σ_g 𝑓)))
22	21	cbvrabv 3407	. . . . . 6 ⊢ {𝑦 ∈ 𝐵 ∣ ∃𝑓 ∈ Word 𝑃𝑦 = (𝑀 Σ_g 𝑓)} = {𝑥 ∈ 𝐵 ∣ ∃𝑓 ∈ Word 𝑃𝑥 = (𝑀 Σ_g 𝑓)}
23		oveq2 7361	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → (𝑀 Σ_g 𝑓) = (𝑀 Σ_g 𝑔))
24	23	eqeq2d 2740	. . . . . . 7 ⊢ (𝑓 = 𝑔 → (𝑥 = (𝑀 Σ_g 𝑓) ↔ 𝑥 = (𝑀 Σ_g 𝑔)))
25	24	cbvrexvw 3208	. . . . . 6 ⊢ (∃𝑓 ∈ Word 𝑃𝑥 = (𝑀 Σ_g 𝑓) ↔ ∃𝑔 ∈ Word 𝑃𝑥 = (𝑀 Σ_g 𝑔))
26	22, 25	rabbieq 3405	. . . . 5 ⊢ {𝑦 ∈ 𝐵 ∣ ∃𝑓 ∈ Word 𝑃𝑦 = (𝑀 Σ_g 𝑓)} = {𝑥 ∈ 𝐵 ∣ ∃𝑔 ∈ Word 𝑃𝑥 = (𝑀 Σ_g 𝑔)}
27	2	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑅 ∈ DivRing) → 𝑋 ∈ 𝐵)
28	12	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑅 ∈ DivRing) → ¬ 𝑋 ∈ 𝑈)
29	4	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑅 ∈ DivRing) → 𝑋 ≠ 0 )
30	6, 8, 7, 15, 16, 18, 19, 26, 27, 28, 29	1arithufdlem4 33494	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑅 ∈ DivRing) → 𝑋 ∈ {𝑦 ∈ 𝐵 ∣ ∃𝑓 ∈ Word 𝑃𝑦 = (𝑀 Σ_g 𝑓)})
31		eqeq1 2733	. . . . . 6 ⊢ (𝑦 = 𝑋 → (𝑦 = (𝑀 Σ_g 𝑓) ↔ 𝑋 = (𝑀 Σ_g 𝑓)))
32	31	rexbidv 3153	. . . . 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 218	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ∈ DivRing) → (𝑋 ∈ 𝐵 ∧ ∃𝑓 ∈ Word 𝑃𝑋 = (𝑀 Σ_g 𝑓)))
35	34	simprd 495	. 2 ⊢ ((𝜑 ∧ ¬ 𝑅 ∈ DivRing) → ∃𝑓 ∈ Word 𝑃𝑋 = (𝑀 Σ_g 𝑓))
36	14, 35	pm2.61dan 812	1 ⊢ (𝜑 → ∃𝑓 ∈ Word 𝑃𝑋 = (𝑀 Σ_g 𝑓))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∃wrex 3053 {crab 3396 ‘cfv 6486 (class class class)co 7353 Word cword 14438 Basecbs 17138 0_gc0g 17361 Σ_g cgsu 17362 mulGrpcmgp 20043 Unitcui 20258 RPrimecrpm 20335 DivRingcdr 20632 UFDcufd 33485

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-ac2 10376 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-int 4900 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-se 5577 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-rpss 7663 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-tpos 8166 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-oadd 8399 df-er 8632 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9271 df-oi 9421 df-dju 9816 df-card 9854 df-ac 10029 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-nn 12147 df-2 12209 df-3 12210 df-4 12211 df-5 12212 df-6 12213 df-7 12214 df-8 12215 df-n0 12403 df-xnn0 12476 df-z 12490 df-uz 12754 df-fz 13429 df-fzo 13576 df-seq 13927 df-hash 14256 df-word 14439 df-lsw 14488 df-concat 14496 df-s1 14521 df-substr 14566 df-pfx 14596 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-ress 17160 df-plusg 17192 df-mulr 17193 df-sca 17195 df-vsca 17196 df-ip 17197 df-0g 17363 df-gsum 17364 df-mgm 18532 df-sgrp 18611 df-mnd 18627 df-submnd 18676 df-grp 18833 df-minusg 18834 df-sbg 18835 df-subg 19020 df-cntz 19214 df-lsm 19533 df-cmn 19679 df-abl 19680 df-mgp 20044 df-rng 20056 df-ur 20085 df-ring 20138 df-cring 20139 df-oppr 20240 df-dvdsr 20260 df-unit 20261 df-invr 20291 df-rprm 20336 df-nzr 20416 df-subrg 20473 df-domn 20598 df-idom 20599 df-drng 20634 df-lmod 20783 df-lss 20853 df-lsp 20893 df-sra 21095 df-rgmod 21096 df-lidl 21133 df-rsp 21134 df-prmidl 33383 df-ufd 33486

This theorem is referenced by: dfufd2 33497

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