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Mirrors > Home > MPE Home > Th. List > 1arithlem4 | Structured version Visualization version GIF version |
Description: Lemma for 1arith 16109. (Contributed by Mario Carneiro, 30-May-2014.) |
Ref | Expression |
---|---|
1arith.1 | ⊢ 𝑀 = (𝑛 ∈ ℕ ↦ (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛))) |
1arithlem4.2 | ⊢ 𝐺 = (𝑦 ∈ ℕ ↦ if(𝑦 ∈ ℙ, (𝑦↑(𝐹‘𝑦)), 1)) |
1arithlem4.3 | ⊢ (𝜑 → 𝐹:ℙ⟶ℕ0) |
1arithlem4.4 | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
1arithlem4.5 | ⊢ ((𝜑 ∧ (𝑞 ∈ ℙ ∧ 𝑁 ≤ 𝑞)) → (𝐹‘𝑞) = 0) |
Ref | Expression |
---|---|
1arithlem4 | ⊢ (𝜑 → ∃𝑥 ∈ ℕ 𝐹 = (𝑀‘𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1arithlem4.2 | . . . . 5 ⊢ 𝐺 = (𝑦 ∈ ℕ ↦ if(𝑦 ∈ ℙ, (𝑦↑(𝐹‘𝑦)), 1)) | |
2 | 1arithlem4.3 | . . . . . . 7 ⊢ (𝜑 → 𝐹:ℙ⟶ℕ0) | |
3 | 2 | ffvelrnda 6670 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℙ) → (𝐹‘𝑦) ∈ ℕ0) |
4 | 3 | ralrimiva 3126 | . . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ ℙ (𝐹‘𝑦) ∈ ℕ0) |
5 | 1, 4 | pcmptcl 16073 | . . . 4 ⊢ (𝜑 → (𝐺:ℕ⟶ℕ ∧ seq1( · , 𝐺):ℕ⟶ℕ)) |
6 | 5 | simprd 488 | . . 3 ⊢ (𝜑 → seq1( · , 𝐺):ℕ⟶ℕ) |
7 | 1arithlem4.4 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
8 | 6, 7 | ffvelrnd 6671 | . 2 ⊢ (𝜑 → (seq1( · , 𝐺)‘𝑁) ∈ ℕ) |
9 | 1arith.1 | . . . . . . 7 ⊢ 𝑀 = (𝑛 ∈ ℕ ↦ (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛))) | |
10 | 9 | 1arithlem2 16106 | . . . . . 6 ⊢ (((seq1( · , 𝐺)‘𝑁) ∈ ℕ ∧ 𝑞 ∈ ℙ) → ((𝑀‘(seq1( · , 𝐺)‘𝑁))‘𝑞) = (𝑞 pCnt (seq1( · , 𝐺)‘𝑁))) |
11 | 8, 10 | sylan 572 | . . . . 5 ⊢ ((𝜑 ∧ 𝑞 ∈ ℙ) → ((𝑀‘(seq1( · , 𝐺)‘𝑁))‘𝑞) = (𝑞 pCnt (seq1( · , 𝐺)‘𝑁))) |
12 | 4 | adantr 473 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ ℙ) → ∀𝑦 ∈ ℙ (𝐹‘𝑦) ∈ ℕ0) |
13 | 7 | adantr 473 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ ℙ) → 𝑁 ∈ ℕ) |
14 | simpr 477 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ ℙ) → 𝑞 ∈ ℙ) | |
15 | fveq2 6493 | . . . . . 6 ⊢ (𝑦 = 𝑞 → (𝐹‘𝑦) = (𝐹‘𝑞)) | |
16 | 1, 12, 13, 14, 15 | pcmpt 16074 | . . . . 5 ⊢ ((𝜑 ∧ 𝑞 ∈ ℙ) → (𝑞 pCnt (seq1( · , 𝐺)‘𝑁)) = if(𝑞 ≤ 𝑁, (𝐹‘𝑞), 0)) |
17 | 13 | nnred 11448 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ ℙ) → 𝑁 ∈ ℝ) |
18 | prmz 15865 | . . . . . . . 8 ⊢ (𝑞 ∈ ℙ → 𝑞 ∈ ℤ) | |
19 | 18 | zred 11893 | . . . . . . 7 ⊢ (𝑞 ∈ ℙ → 𝑞 ∈ ℝ) |
20 | 19 | adantl 474 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ ℙ) → 𝑞 ∈ ℝ) |
21 | ifid 4383 | . . . . . . 7 ⊢ if(𝑞 ≤ 𝑁, (𝐹‘𝑞), (𝐹‘𝑞)) = (𝐹‘𝑞) | |
22 | 1arithlem4.5 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑞 ∈ ℙ ∧ 𝑁 ≤ 𝑞)) → (𝐹‘𝑞) = 0) | |
23 | 22 | anassrs 460 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑁 ≤ 𝑞) → (𝐹‘𝑞) = 0) |
24 | 23 | ifeq2d 4363 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑁 ≤ 𝑞) → if(𝑞 ≤ 𝑁, (𝐹‘𝑞), (𝐹‘𝑞)) = if(𝑞 ≤ 𝑁, (𝐹‘𝑞), 0)) |
25 | 21, 24 | syl5reqr 2823 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑁 ≤ 𝑞) → if(𝑞 ≤ 𝑁, (𝐹‘𝑞), 0) = (𝐹‘𝑞)) |
26 | iftrue 4350 | . . . . . . 7 ⊢ (𝑞 ≤ 𝑁 → if(𝑞 ≤ 𝑁, (𝐹‘𝑞), 0) = (𝐹‘𝑞)) | |
27 | 26 | adantl 474 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ≤ 𝑁) → if(𝑞 ≤ 𝑁, (𝐹‘𝑞), 0) = (𝐹‘𝑞)) |
28 | 17, 20, 25, 27 | lecasei 10538 | . . . . 5 ⊢ ((𝜑 ∧ 𝑞 ∈ ℙ) → if(𝑞 ≤ 𝑁, (𝐹‘𝑞), 0) = (𝐹‘𝑞)) |
29 | 11, 16, 28 | 3eqtrrd 2813 | . . . 4 ⊢ ((𝜑 ∧ 𝑞 ∈ ℙ) → (𝐹‘𝑞) = ((𝑀‘(seq1( · , 𝐺)‘𝑁))‘𝑞)) |
30 | 29 | ralrimiva 3126 | . . 3 ⊢ (𝜑 → ∀𝑞 ∈ ℙ (𝐹‘𝑞) = ((𝑀‘(seq1( · , 𝐺)‘𝑁))‘𝑞)) |
31 | 9 | 1arithlem3 16107 | . . . . 5 ⊢ ((seq1( · , 𝐺)‘𝑁) ∈ ℕ → (𝑀‘(seq1( · , 𝐺)‘𝑁)):ℙ⟶ℕ0) |
32 | 8, 31 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑀‘(seq1( · , 𝐺)‘𝑁)):ℙ⟶ℕ0) |
33 | ffn 6338 | . . . . 5 ⊢ (𝐹:ℙ⟶ℕ0 → 𝐹 Fn ℙ) | |
34 | ffn 6338 | . . . . 5 ⊢ ((𝑀‘(seq1( · , 𝐺)‘𝑁)):ℙ⟶ℕ0 → (𝑀‘(seq1( · , 𝐺)‘𝑁)) Fn ℙ) | |
35 | eqfnfv 6621 | . . . . 5 ⊢ ((𝐹 Fn ℙ ∧ (𝑀‘(seq1( · , 𝐺)‘𝑁)) Fn ℙ) → (𝐹 = (𝑀‘(seq1( · , 𝐺)‘𝑁)) ↔ ∀𝑞 ∈ ℙ (𝐹‘𝑞) = ((𝑀‘(seq1( · , 𝐺)‘𝑁))‘𝑞))) | |
36 | 33, 34, 35 | syl2an 586 | . . . 4 ⊢ ((𝐹:ℙ⟶ℕ0 ∧ (𝑀‘(seq1( · , 𝐺)‘𝑁)):ℙ⟶ℕ0) → (𝐹 = (𝑀‘(seq1( · , 𝐺)‘𝑁)) ↔ ∀𝑞 ∈ ℙ (𝐹‘𝑞) = ((𝑀‘(seq1( · , 𝐺)‘𝑁))‘𝑞))) |
37 | 2, 32, 36 | syl2anc 576 | . . 3 ⊢ (𝜑 → (𝐹 = (𝑀‘(seq1( · , 𝐺)‘𝑁)) ↔ ∀𝑞 ∈ ℙ (𝐹‘𝑞) = ((𝑀‘(seq1( · , 𝐺)‘𝑁))‘𝑞))) |
38 | 30, 37 | mpbird 249 | . 2 ⊢ (𝜑 → 𝐹 = (𝑀‘(seq1( · , 𝐺)‘𝑁))) |
39 | fveq2 6493 | . . 3 ⊢ (𝑥 = (seq1( · , 𝐺)‘𝑁) → (𝑀‘𝑥) = (𝑀‘(seq1( · , 𝐺)‘𝑁))) | |
40 | 39 | rspceeqv 3547 | . 2 ⊢ (((seq1( · , 𝐺)‘𝑁) ∈ ℕ ∧ 𝐹 = (𝑀‘(seq1( · , 𝐺)‘𝑁))) → ∃𝑥 ∈ ℕ 𝐹 = (𝑀‘𝑥)) |
41 | 8, 38, 40 | syl2anc 576 | 1 ⊢ (𝜑 → ∃𝑥 ∈ ℕ 𝐹 = (𝑀‘𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1507 ∈ wcel 2048 ∀wral 3082 ∃wrex 3083 ifcif 4344 class class class wbr 4923 ↦ cmpt 5002 Fn wfn 6177 ⟶wf 6178 ‘cfv 6182 (class class class)co 6970 ℝcr 10326 0cc0 10327 1c1 10328 · cmul 10332 ≤ cle 10467 ℕcn 11431 ℕ0cn0 11700 seqcseq 13177 ↑cexp 13237 ℙcprime 15861 pCnt cpc 16019 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 ax-pre-sup 10405 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7494 df-2nd 7495 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-1o 7897 df-2o 7898 df-er 8081 df-en 8299 df-dom 8300 df-sdom 8301 df-fin 8302 df-sup 8693 df-inf 8694 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-div 11091 df-nn 11432 df-2 11496 df-3 11497 df-n0 11701 df-z 11787 df-uz 12052 df-q 12156 df-rp 12198 df-fz 12702 df-fl 12970 df-mod 13046 df-seq 13178 df-exp 13238 df-cj 14309 df-re 14310 df-im 14311 df-sqrt 14445 df-abs 14446 df-dvds 15458 df-gcd 15694 df-prm 15862 df-pc 16020 |
This theorem is referenced by: 1arith 16109 |
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