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Mirrors > Home > MPE Home > Th. List > gausslemma2dlem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for gausslemma2d 25099. (Contributed by AV, 5-Jul-2021.) |
Ref | Expression |
---|---|
gausslemma2d.p | ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2})) |
gausslemma2d.h | ⊢ 𝐻 = ((𝑃 − 1) / 2) |
gausslemma2d.r | ⊢ 𝑅 = (𝑥 ∈ (1...𝐻) ↦ if((𝑥 · 2) < (𝑃 / 2), (𝑥 · 2), (𝑃 − (𝑥 · 2)))) |
Ref | Expression |
---|---|
gausslemma2dlem1 | ⊢ (𝜑 → (!‘𝐻) = ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gausslemma2d.p | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2})) | |
2 | gausslemma2d.h | . . . . 5 ⊢ 𝐻 = ((𝑃 − 1) / 2) | |
3 | 1, 2 | gausslemma2dlem0b 25082 | . . . 4 ⊢ (𝜑 → 𝐻 ∈ ℕ) |
4 | 3 | nnnn0d 11351 | . . 3 ⊢ (𝜑 → 𝐻 ∈ ℕ0) |
5 | fprodfac 14703 | . . 3 ⊢ (𝐻 ∈ ℕ0 → (!‘𝐻) = ∏𝑙 ∈ (1...𝐻)𝑙) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝜑 → (!‘𝐻) = ∏𝑙 ∈ (1...𝐻)𝑙) |
7 | id 22 | . . 3 ⊢ (𝑙 = (𝑅‘𝑘) → 𝑙 = (𝑅‘𝑘)) | |
8 | fzfid 12772 | . . 3 ⊢ (𝜑 → (1...𝐻) ∈ Fin) | |
9 | fzfi 12771 | . . . 4 ⊢ (1...𝐻) ∈ Fin | |
10 | ovex 6678 | . . . . . 6 ⊢ (𝑥 · 2) ∈ V | |
11 | ovex 6678 | . . . . . 6 ⊢ (𝑃 − (𝑥 · 2)) ∈ V | |
12 | 10, 11 | ifex 4156 | . . . . 5 ⊢ if((𝑥 · 2) < (𝑃 / 2), (𝑥 · 2), (𝑃 − (𝑥 · 2))) ∈ V |
13 | gausslemma2d.r | . . . . 5 ⊢ 𝑅 = (𝑥 ∈ (1...𝐻) ↦ if((𝑥 · 2) < (𝑃 / 2), (𝑥 · 2), (𝑃 − (𝑥 · 2)))) | |
14 | 12, 13 | fnmpti 6022 | . . . 4 ⊢ 𝑅 Fn (1...𝐻) |
15 | 1, 2, 13 | gausslemma2dlem1a 25090 | . . . 4 ⊢ (𝜑 → ran 𝑅 = (1...𝐻)) |
16 | rneqdmfinf1o 8242 | . . . 4 ⊢ (((1...𝐻) ∈ Fin ∧ 𝑅 Fn (1...𝐻) ∧ ran 𝑅 = (1...𝐻)) → 𝑅:(1...𝐻)–1-1-onto→(1...𝐻)) | |
17 | 9, 14, 15, 16 | mp3an12i 1428 | . . 3 ⊢ (𝜑 → 𝑅:(1...𝐻)–1-1-onto→(1...𝐻)) |
18 | eqidd 2623 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → (𝑅‘𝑘) = (𝑅‘𝑘)) | |
19 | elfzelz 12342 | . . . . 5 ⊢ (𝑙 ∈ (1...𝐻) → 𝑙 ∈ ℤ) | |
20 | 19 | zcnd 11483 | . . . 4 ⊢ (𝑙 ∈ (1...𝐻) → 𝑙 ∈ ℂ) |
21 | 20 | adantl 482 | . . 3 ⊢ ((𝜑 ∧ 𝑙 ∈ (1...𝐻)) → 𝑙 ∈ ℂ) |
22 | 7, 8, 17, 18, 21 | fprodf1o 14676 | . 2 ⊢ (𝜑 → ∏𝑙 ∈ (1...𝐻)𝑙 = ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘)) |
23 | 6, 22 | eqtrd 2656 | 1 ⊢ (𝜑 → (!‘𝐻) = ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∖ cdif 3571 ifcif 4086 {csn 4177 class class class wbr 4653 ↦ cmpt 4729 ran crn 5115 Fn wfn 5883 –1-1-onto→wf1o 5887 ‘cfv 5888 (class class class)co 6650 Fincfn 7955 ℂcc 9934 1c1 9937 · cmul 9941 < clt 10074 − cmin 10266 / cdiv 10684 2c2 11070 ℕ0cn0 11292 ...cfz 12326 !cfa 13060 ∏cprod 14635 ℙcprime 15385 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-ioo 12179 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-fac 13061 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-prod 14636 df-dvds 14984 df-prm 15386 |
This theorem is referenced by: gausslemma2dlem4 25094 |
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