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Mirrors > Home > ILE Home > Th. List > 1arith2 | GIF version |
Description: Fundamental theorem of arithmetic, where a prime factorization is represented as a finite monotonic 1-based sequence of primes. Every positive integer has a unique prime factorization. Theorem 1.10 in [ApostolNT] p. 17. This is Metamath 100 proof #80. (Contributed by Paul Chapman, 17-Nov-2012.) (Revised by Mario Carneiro, 30-May-2014.) |
Ref | Expression |
---|---|
1arith.1 | ⊢ 𝑀 = (𝑛 ∈ ℕ ↦ (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛))) |
1arith.2 | ⊢ 𝑅 = {𝑒 ∈ (ℕ0 ↑𝑚 ℙ) ∣ (◡𝑒 “ ℕ) ∈ Fin} |
Ref | Expression |
---|---|
1arith2 | ⊢ ∀𝑧 ∈ ℕ ∃!𝑔 ∈ 𝑅 (𝑀‘𝑧) = 𝑔 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1arith.1 | . . . . . 6 ⊢ 𝑀 = (𝑛 ∈ ℕ ↦ (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛))) | |
2 | 1arith.2 | . . . . . 6 ⊢ 𝑅 = {𝑒 ∈ (ℕ0 ↑𝑚 ℙ) ∣ (◡𝑒 “ ℕ) ∈ Fin} | |
3 | 1, 2 | 1arith 12499 | . . . . 5 ⊢ 𝑀:ℕ–1-1-onto→𝑅 |
4 | f1ocnv 5509 | . . . . 5 ⊢ (𝑀:ℕ–1-1-onto→𝑅 → ◡𝑀:𝑅–1-1-onto→ℕ) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ ◡𝑀:𝑅–1-1-onto→ℕ |
6 | f1ofveu 5902 | . . . 4 ⊢ ((◡𝑀:𝑅–1-1-onto→ℕ ∧ 𝑧 ∈ ℕ) → ∃!𝑔 ∈ 𝑅 (◡𝑀‘𝑔) = 𝑧) | |
7 | 5, 6 | mpan 424 | . . 3 ⊢ (𝑧 ∈ ℕ → ∃!𝑔 ∈ 𝑅 (◡𝑀‘𝑔) = 𝑧) |
8 | f1ocnvfvb 5819 | . . . . 5 ⊢ ((𝑀:ℕ–1-1-onto→𝑅 ∧ 𝑧 ∈ ℕ ∧ 𝑔 ∈ 𝑅) → ((𝑀‘𝑧) = 𝑔 ↔ (◡𝑀‘𝑔) = 𝑧)) | |
9 | 3, 8 | mp3an1 1335 | . . . 4 ⊢ ((𝑧 ∈ ℕ ∧ 𝑔 ∈ 𝑅) → ((𝑀‘𝑧) = 𝑔 ↔ (◡𝑀‘𝑔) = 𝑧)) |
10 | 9 | reubidva 2677 | . . 3 ⊢ (𝑧 ∈ ℕ → (∃!𝑔 ∈ 𝑅 (𝑀‘𝑧) = 𝑔 ↔ ∃!𝑔 ∈ 𝑅 (◡𝑀‘𝑔) = 𝑧)) |
11 | 7, 10 | mpbird 167 | . 2 ⊢ (𝑧 ∈ ℕ → ∃!𝑔 ∈ 𝑅 (𝑀‘𝑧) = 𝑔) |
12 | 11 | rgen 2547 | 1 ⊢ ∀𝑧 ∈ ℕ ∃!𝑔 ∈ 𝑅 (𝑀‘𝑧) = 𝑔 |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 = wceq 1364 ∈ wcel 2164 ∀wral 2472 ∃!wreu 2474 {crab 2476 ↦ cmpt 4090 ◡ccnv 4656 “ cima 4660 –1-1-onto→wf1o 5249 ‘cfv 5250 (class class class)co 5914 ↑𝑚 cmap 6697 Fincfn 6789 ℕcn 8976 ℕ0cn0 9234 ℙcprime 12239 pCnt cpc 12416 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4462 ax-setind 4567 ax-iinf 4618 ax-cnex 7957 ax-resscn 7958 ax-1cn 7959 ax-1re 7960 ax-icn 7961 ax-addcl 7962 ax-addrcl 7963 ax-mulcl 7964 ax-mulrcl 7965 ax-addcom 7966 ax-mulcom 7967 ax-addass 7968 ax-mulass 7969 ax-distr 7970 ax-i2m1 7971 ax-0lt1 7972 ax-1rid 7973 ax-0id 7974 ax-rnegex 7975 ax-precex 7976 ax-cnre 7977 ax-pre-ltirr 7978 ax-pre-ltwlin 7979 ax-pre-lttrn 7980 ax-pre-apti 7981 ax-pre-ltadd 7982 ax-pre-mulgt0 7983 ax-pre-mulext 7984 ax-arch 7985 ax-caucvg 7986 |
This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-if 3558 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-tr 4128 df-id 4322 df-po 4325 df-iso 4326 df-iord 4395 df-on 4397 df-ilim 4398 df-suc 4400 df-iom 4621 df-xp 4663 df-rel 4664 df-cnv 4665 df-co 4666 df-dm 4667 df-rn 4668 df-res 4669 df-ima 4670 df-iota 5211 df-fun 5252 df-fn 5253 df-f 5254 df-f1 5255 df-fo 5256 df-f1o 5257 df-fv 5258 df-isom 5259 df-riota 5869 df-ov 5917 df-oprab 5918 df-mpo 5919 df-1st 6188 df-2nd 6189 df-recs 6353 df-frec 6439 df-1o 6464 df-2o 6465 df-er 6582 df-map 6699 df-en 6790 df-fin 6792 df-sup 7037 df-inf 7038 df-pnf 8050 df-mnf 8051 df-xr 8052 df-ltxr 8053 df-le 8054 df-sub 8186 df-neg 8187 df-reap 8588 df-ap 8595 df-div 8686 df-inn 8977 df-2 9035 df-3 9036 df-4 9037 df-n0 9235 df-xnn0 9298 df-z 9312 df-uz 9587 df-q 9679 df-rp 9714 df-fz 10069 df-fzo 10203 df-fl 10333 df-mod 10388 df-seqfrec 10513 df-exp 10604 df-cj 10980 df-re 10981 df-im 10982 df-rsqrt 11136 df-abs 11137 df-dvds 11925 df-gcd 12074 df-prm 12240 df-pc 12417 |
This theorem is referenced by: (None) |
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