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| Mirrors > Home > MPE Home > Th. List > Mathboxes > 1arithufd | Structured version Visualization version GIF version | ||
| 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.) |
| Ref | Expression |
|---|---|
| 1arithufd.b | ⊢ 𝐵 = (Base‘𝑅) |
| 1arithufd.0 | ⊢ 0 = (0g‘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 ) |
| Ref | Expression |
|---|---|
| 1arithufd | ⊢ (𝜑 → ∃𝑓 ∈ Word 𝑃𝑋 = (𝑀 Σg 𝑓)) |
| 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 = (0g‘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 Σg 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 Σg 𝑓) ↔ 𝑥 = (𝑀 Σg 𝑓))) | |
| 21 | 20 | rexbidv 3179 | . . . . . . 7 ⊢ (𝑦 = 𝑥 → (∃𝑓 ∈ Word 𝑃𝑦 = (𝑀 Σg 𝑓) ↔ ∃𝑓 ∈ Word 𝑃𝑥 = (𝑀 Σg 𝑓))) |
| 22 | 21 | cbvrabv 3447 | . . . . . 6 ⊢ {𝑦 ∈ 𝐵 ∣ ∃𝑓 ∈ Word 𝑃𝑦 = (𝑀 Σg 𝑓)} = {𝑥 ∈ 𝐵 ∣ ∃𝑓 ∈ Word 𝑃𝑥 = (𝑀 Σg 𝑓)} |
| 23 | oveq2 7439 | . . . . . . . 8 ⊢ (𝑓 = 𝑔 → (𝑀 Σg 𝑓) = (𝑀 Σg 𝑔)) | |
| 24 | 23 | eqeq2d 2748 | . . . . . . 7 ⊢ (𝑓 = 𝑔 → (𝑥 = (𝑀 Σg 𝑓) ↔ 𝑥 = (𝑀 Σg 𝑔))) |
| 25 | 24 | cbvrexvw 3238 | . . . . . 6 ⊢ (∃𝑓 ∈ Word 𝑃𝑥 = (𝑀 Σg 𝑓) ↔ ∃𝑔 ∈ Word 𝑃𝑥 = (𝑀 Σg 𝑔)) |
| 26 | 22, 25 | rabbieq 3445 | . . . . 5 ⊢ {𝑦 ∈ 𝐵 ∣ ∃𝑓 ∈ Word 𝑃𝑦 = (𝑀 Σg 𝑓)} = {𝑥 ∈ 𝐵 ∣ ∃𝑔 ∈ Word 𝑃𝑥 = (𝑀 Σg 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 Σg 𝑓)}) |
| 31 | eqeq1 2741 | . . . . . 6 ⊢ (𝑦 = 𝑋 → (𝑦 = (𝑀 Σg 𝑓) ↔ 𝑋 = (𝑀 Σg 𝑓))) | |
| 32 | 31 | rexbidv 3179 | . . . . 5 ⊢ (𝑦 = 𝑋 → (∃𝑓 ∈ Word 𝑃𝑦 = (𝑀 Σg 𝑓) ↔ ∃𝑓 ∈ Word 𝑃𝑋 = (𝑀 Σg 𝑓))) |
| 33 | 32 | elrab 3692 | . . . 4 ⊢ (𝑋 ∈ {𝑦 ∈ 𝐵 ∣ ∃𝑓 ∈ Word 𝑃𝑦 = (𝑀 Σg 𝑓)} ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑓 ∈ Word 𝑃𝑋 = (𝑀 Σg 𝑓))) |
| 34 | 30, 33 | sylib 218 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ∈ DivRing) → (𝑋 ∈ 𝐵 ∧ ∃𝑓 ∈ Word 𝑃𝑋 = (𝑀 Σg 𝑓))) |
| 35 | 34 | simprd 495 | . 2 ⊢ ((𝜑 ∧ ¬ 𝑅 ∈ DivRing) → ∃𝑓 ∈ Word 𝑃𝑋 = (𝑀 Σg 𝑓)) |
| 36 | 14, 35 | pm2.61dan 813 | 1 ⊢ (𝜑 → ∃𝑓 ∈ Word 𝑃𝑋 = (𝑀 Σg 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 0gc0g 17484 Σg 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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