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Mirrors > Home > MPE Home > Th. List > gausslemma2dlem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for gausslemma2d 25950. (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 25933 | . . . 4 ⊢ (𝜑 → 𝐻 ∈ ℕ) |
4 | 3 | nnnn0d 11956 | . . 3 ⊢ (𝜑 → 𝐻 ∈ ℕ0) |
5 | fprodfac 15327 | . . 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 13342 | . . 3 ⊢ (𝜑 → (1...𝐻) ∈ Fin) | |
9 | fzfi 13341 | . . . 4 ⊢ (1...𝐻) ∈ Fin | |
10 | ovex 7189 | . . . . . 6 ⊢ (𝑥 · 2) ∈ V | |
11 | ovex 7189 | . . . . . 6 ⊢ (𝑃 − (𝑥 · 2)) ∈ V | |
12 | 10, 11 | ifex 4515 | . . . . 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 6491 | . . . 4 ⊢ 𝑅 Fn (1...𝐻) |
15 | 1, 2, 13 | gausslemma2dlem1a 25941 | . . . 4 ⊢ (𝜑 → ran 𝑅 = (1...𝐻)) |
16 | rneqdmfinf1o 8800 | . . . 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 1461 | . . 3 ⊢ (𝜑 → 𝑅:(1...𝐻)–1-1-onto→(1...𝐻)) |
18 | eqidd 2822 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → (𝑅‘𝑘) = (𝑅‘𝑘)) | |
19 | elfzelz 12909 | . . . . 5 ⊢ (𝑙 ∈ (1...𝐻) → 𝑙 ∈ ℤ) | |
20 | 19 | zcnd 12089 | . . . 4 ⊢ (𝑙 ∈ (1...𝐻) → 𝑙 ∈ ℂ) |
21 | 20 | adantl 484 | . . 3 ⊢ ((𝜑 ∧ 𝑙 ∈ (1...𝐻)) → 𝑙 ∈ ℂ) |
22 | 7, 8, 17, 18, 21 | fprodf1o 15300 | . 2 ⊢ (𝜑 → ∏𝑙 ∈ (1...𝐻)𝑙 = ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘)) |
23 | 6, 22 | eqtrd 2856 | 1 ⊢ (𝜑 → (!‘𝐻) = ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∖ cdif 3933 ifcif 4467 {csn 4567 class class class wbr 5066 ↦ cmpt 5146 ran crn 5556 Fn wfn 6350 –1-1-onto→wf1o 6354 ‘cfv 6355 (class class class)co 7156 Fincfn 8509 ℂcc 10535 1c1 10538 · cmul 10542 < clt 10675 − cmin 10870 / cdiv 11297 2c2 11693 ℕ0cn0 11898 ...cfz 12893 !cfa 13634 ∏cprod 15259 ℙcprime 16015 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-ioo 12743 df-fz 12894 df-fzo 13035 df-seq 13371 df-exp 13431 df-fac 13635 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-clim 14845 df-prod 15260 df-dvds 15608 df-prm 16016 |
This theorem is referenced by: gausslemma2dlem4 25945 |
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