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Mathbox for Thierry Arnoux |
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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 33545, 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 20751 | . . . . 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 2739 | . . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑦 = (𝑀 Σg 𝑓) ↔ 𝑥 = (𝑀 Σg 𝑓))) | |
21 | 20 | rexbidv 3177 | . . . . . . 7 ⊢ (𝑦 = 𝑥 → (∃𝑓 ∈ Word 𝑃𝑦 = (𝑀 Σg 𝑓) ↔ ∃𝑓 ∈ Word 𝑃𝑥 = (𝑀 Σg 𝑓))) |
22 | 21 | cbvrabv 3444 | . . . . . 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 2746 | . . . . . . 7 ⊢ (𝑓 = 𝑔 → (𝑥 = (𝑀 Σg 𝑓) ↔ 𝑥 = (𝑀 Σg 𝑔))) |
25 | 24 | cbvrexvw 3236 | . . . . . 6 ⊢ (∃𝑓 ∈ Word 𝑃𝑥 = (𝑀 Σg 𝑓) ↔ ∃𝑔 ∈ Word 𝑃𝑥 = (𝑀 Σg 𝑔)) |
26 | 22, 25 | rabbieq 3442 | . . . . 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 33555 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝑅 ∈ DivRing) → 𝑋 ∈ {𝑦 ∈ 𝐵 ∣ ∃𝑓 ∈ Word 𝑃𝑦 = (𝑀 Σg 𝑓)}) |
31 | eqeq1 2739 | . . . . . 6 ⊢ (𝑦 = 𝑋 → (𝑦 = (𝑀 Σg 𝑓) ↔ 𝑋 = (𝑀 Σg 𝑓))) | |
32 | 31 | rexbidv 3177 | . . . . 5 ⊢ (𝑦 = 𝑋 → (∃𝑓 ∈ Word 𝑃𝑦 = (𝑀 Σg 𝑓) ↔ ∃𝑓 ∈ Word 𝑃𝑋 = (𝑀 Σg 𝑓))) |
33 | 32 | elrab 3695 | . . . 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 1537 ∈ wcel 2106 ≠ wne 2938 ∃wrex 3068 {crab 3433 ‘cfv 6563 (class class class)co 7431 Word cword 14549 Basecbs 17245 0gc0g 17486 Σg cgsu 17487 mulGrpcmgp 20152 Unitcui 20372 RPrimecrpm 20449 DivRingcdr 20746 UFDcufd 33546 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-ac2 10501 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-rpss 7742 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-oadd 8509 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-oi 9548 df-dju 9939 df-card 9977 df-ac 10154 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-n0 12525 df-xnn0 12598 df-z 12612 df-uz 12877 df-fz 13545 df-fzo 13692 df-seq 14040 df-hash 14367 df-word 14550 df-lsw 14598 df-concat 14606 df-s1 14631 df-substr 14676 df-pfx 14706 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-0g 17488 df-gsum 17489 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-cntz 19348 df-lsm 19669 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-rprm 20450 df-nzr 20530 df-subrg 20587 df-domn 20712 df-idom 20713 df-drng 20748 df-lmod 20877 df-lss 20948 df-lsp 20988 df-sra 21190 df-rgmod 21191 df-lidl 21236 df-rsp 21237 df-prmidl 33444 df-ufd 33547 |
This theorem is referenced by: dfufd2 33558 |
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