![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > gausslemma2dlem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for gausslemma2d 27221. (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 27204 | . . . 4 ⊢ (𝜑 → 𝐻 ∈ ℕ) |
4 | 3 | nnnn0d 12539 | . . 3 ⊢ (𝜑 → 𝐻 ∈ ℕ0) |
5 | fprodfac 15924 | . . 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 13945 | . . 3 ⊢ (𝜑 → (1...𝐻) ∈ Fin) | |
9 | fzfi 13944 | . . . 4 ⊢ (1...𝐻) ∈ Fin | |
10 | ovex 7445 | . . . . . 6 ⊢ (𝑥 · 2) ∈ V | |
11 | ovex 7445 | . . . . . 6 ⊢ (𝑃 − (𝑥 · 2)) ∈ V | |
12 | 10, 11 | ifex 4578 | . . . . 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 6693 | . . . 4 ⊢ 𝑅 Fn (1...𝐻) |
15 | 1, 2, 13 | gausslemma2dlem1a 27212 | . . . 4 ⊢ (𝜑 → ran 𝑅 = (1...𝐻)) |
16 | rneqdmfinf1o 9334 | . . . 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 1464 | . . 3 ⊢ (𝜑 → 𝑅:(1...𝐻)–1-1-onto→(1...𝐻)) |
18 | eqidd 2732 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → (𝑅‘𝑘) = (𝑅‘𝑘)) | |
19 | elfzelz 13508 | . . . . 5 ⊢ (𝑙 ∈ (1...𝐻) → 𝑙 ∈ ℤ) | |
20 | 19 | zcnd 12674 | . . . 4 ⊢ (𝑙 ∈ (1...𝐻) → 𝑙 ∈ ℂ) |
21 | 20 | adantl 481 | . . 3 ⊢ ((𝜑 ∧ 𝑙 ∈ (1...𝐻)) → 𝑙 ∈ ℂ) |
22 | 7, 8, 17, 18, 21 | fprodf1o 15897 | . 2 ⊢ (𝜑 → ∏𝑙 ∈ (1...𝐻)𝑙 = ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘)) |
23 | 6, 22 | eqtrd 2771 | 1 ⊢ (𝜑 → (!‘𝐻) = ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ∖ cdif 3945 ifcif 4528 {csn 4628 class class class wbr 5148 ↦ cmpt 5231 ran crn 5677 Fn wfn 6538 –1-1-onto→wf1o 6542 ‘cfv 6543 (class class class)co 7412 Fincfn 8945 ℂcc 11114 1c1 11117 · cmul 11121 < clt 11255 − cmin 11451 / cdiv 11878 2c2 12274 ℕ0cn0 12479 ...cfz 13491 !cfa 14240 ∏cprod 15856 ℙcprime 16615 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9642 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-2o 8473 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-sup 9443 df-oi 9511 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-n0 12480 df-z 12566 df-uz 12830 df-rp 12982 df-ioo 13335 df-fz 13492 df-fzo 13635 df-seq 13974 df-exp 14035 df-fac 14241 df-hash 14298 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-clim 15439 df-prod 15857 df-dvds 16205 df-prm 16616 |
This theorem is referenced by: gausslemma2dlem4 27216 |
Copyright terms: Public domain | W3C validator |