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Mirrors > Home > MPE Home > Th. List > gausslemma2dlem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for gausslemma2d 26110. (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 26093 | . . . 4 ⊢ (𝜑 → 𝐻 ∈ ℕ) |
4 | 3 | nnnn0d 12036 | . . 3 ⊢ (𝜑 → 𝐻 ∈ ℕ0) |
5 | fprodfac 15419 | . . 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 13432 | . . 3 ⊢ (𝜑 → (1...𝐻) ∈ Fin) | |
9 | fzfi 13431 | . . . 4 ⊢ (1...𝐻) ∈ Fin | |
10 | ovex 7203 | . . . . . 6 ⊢ (𝑥 · 2) ∈ V | |
11 | ovex 7203 | . . . . . 6 ⊢ (𝑃 − (𝑥 · 2)) ∈ V | |
12 | 10, 11 | ifex 4464 | . . . . 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 6480 | . . . 4 ⊢ 𝑅 Fn (1...𝐻) |
15 | 1, 2, 13 | gausslemma2dlem1a 26101 | . . . 4 ⊢ (𝜑 → ran 𝑅 = (1...𝐻)) |
16 | rneqdmfinf1o 8873 | . . . 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 1466 | . . 3 ⊢ (𝜑 → 𝑅:(1...𝐻)–1-1-onto→(1...𝐻)) |
18 | eqidd 2739 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → (𝑅‘𝑘) = (𝑅‘𝑘)) | |
19 | elfzelz 12998 | . . . . 5 ⊢ (𝑙 ∈ (1...𝐻) → 𝑙 ∈ ℤ) | |
20 | 19 | zcnd 12169 | . . . 4 ⊢ (𝑙 ∈ (1...𝐻) → 𝑙 ∈ ℂ) |
21 | 20 | adantl 485 | . . 3 ⊢ ((𝜑 ∧ 𝑙 ∈ (1...𝐻)) → 𝑙 ∈ ℂ) |
22 | 7, 8, 17, 18, 21 | fprodf1o 15392 | . 2 ⊢ (𝜑 → ∏𝑙 ∈ (1...𝐻)𝑙 = ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘)) |
23 | 6, 22 | eqtrd 2773 | 1 ⊢ (𝜑 → (!‘𝐻) = ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ∖ cdif 3840 ifcif 4414 {csn 4516 class class class wbr 5030 ↦ cmpt 5110 ran crn 5526 Fn wfn 6334 –1-1-onto→wf1o 6338 ‘cfv 6339 (class class class)co 7170 Fincfn 8555 ℂcc 10613 1c1 10616 · cmul 10620 < clt 10753 − cmin 10948 / cdiv 11375 2c2 11771 ℕ0cn0 11976 ...cfz 12981 !cfa 13725 ∏cprod 15351 ℙcprime 16112 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-inf2 9177 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 ax-pre-sup 10693 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-se 5484 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-isom 6348 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-1st 7714 df-2nd 7715 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-2o 8132 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-sup 8979 df-oi 9047 df-card 9441 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-div 11376 df-nn 11717 df-2 11779 df-3 11780 df-4 11781 df-n0 11977 df-z 12063 df-uz 12325 df-rp 12473 df-ioo 12825 df-fz 12982 df-fzo 13125 df-seq 13461 df-exp 13522 df-fac 13726 df-hash 13783 df-cj 14548 df-re 14549 df-im 14550 df-sqrt 14684 df-abs 14685 df-clim 14935 df-prod 15352 df-dvds 15700 df-prm 16113 |
This theorem is referenced by: gausslemma2dlem4 26105 |
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