1arithufd - Mathbox for Thierry Arnoux

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Theorem 1arithufd 33576

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 33565, 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 20734	. . . . 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 837	. . 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 2741	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑦 = (𝑀 Σ_g 𝑓) ↔ 𝑥 = (𝑀 Σ_g 𝑓)))
21	20	rexbidv 3179	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (∃𝑓 ∈ Word 𝑃𝑦 = (𝑀 Σ_g 𝑓) ↔ ∃𝑓 ∈ Word 𝑃𝑥 = (𝑀 Σ_g 𝑓)))
22	21	cbvrabv 3447	. . . . . 6 ⊢ {𝑦 ∈ 𝐵 ∣ ∃𝑓 ∈ Word 𝑃𝑦 = (𝑀 Σ_g 𝑓)} = {𝑥 ∈ 𝐵 ∣ ∃𝑓 ∈ Word 𝑃𝑥 = (𝑀 Σ_g 𝑓)}
23		oveq2 7439	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → (𝑀 Σ_g 𝑓) = (𝑀 Σ_g 𝑔))
24	23	eqeq2d 2748	. . . . . . 7 ⊢ (𝑓 = 𝑔 → (𝑥 = (𝑀 Σ_g 𝑓) ↔ 𝑥 = (𝑀 Σ_g 𝑔)))
25	24	cbvrexvw 3238	. . . . . 6 ⊢ (∃𝑓 ∈ Word 𝑃𝑥 = (𝑀 Σ_g 𝑓) ↔ ∃𝑔 ∈ Word 𝑃𝑥 = (𝑀 Σ_g 𝑔))
26	22, 25	rabbieq 3445	. . . . 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 33575	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑅 ∈ DivRing) → 𝑋 ∈ {𝑦 ∈ 𝐵 ∣ ∃𝑓 ∈ Word 𝑃𝑦 = (𝑀 Σ_g 𝑓)})
31		eqeq1 2741	. . . . . 6 ⊢ (𝑦 = 𝑋 → (𝑦 = (𝑀 Σ_g 𝑓) ↔ 𝑋 = (𝑀 Σ_g 𝑓)))
32	31	rexbidv 3179	. . . . 5 ⊢ (𝑦 = 𝑋 → (∃𝑓 ∈ Word 𝑃𝑦 = (𝑀 Σ_g 𝑓) ↔ ∃𝑓 ∈ Word 𝑃𝑋 = (𝑀 Σ_g 𝑓)))
33	32	elrab 3692	. . . 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 813	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 2108 ≠ wne 2940 ∃wrex 3070 {crab 3436 ‘cfv 6561 (class class class)co 7431 Word cword 14552 Basecbs 17247 0_gc0g 17484 Σ_g cgsu 17485 mulGrpcmgp 20137 Unitcui 20355 RPrimecrpm 20432 DivRingcdr 20729 UFDcufd 33566

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-ac2 10503 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-rpss 7743 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-oadd 8510 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-oi 9550 df-dju 9941 df-card 9979 df-ac 10156 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-n0 12527 df-xnn0 12600 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 df-seq 14043 df-hash 14370 df-word 14553 df-lsw 14601 df-concat 14609 df-s1 14634 df-substr 14679 df-pfx 14709 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-0g 17486 df-gsum 17487 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-subg 19141 df-cntz 19335 df-lsm 19654 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-rprm 20433 df-nzr 20513 df-subrg 20570 df-domn 20695 df-idom 20696 df-drng 20731 df-lmod 20860 df-lss 20930 df-lsp 20970 df-sra 21172 df-rgmod 21173 df-lidl 21218 df-rsp 21219 df-prmidl 33464 df-ufd 33567

This theorem is referenced by: dfufd2 33578

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