gausslemma2dlem1 - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > gausslemma2dlem1		GIF version

Theorem gausslemma2dlem1 15796

Description: Lemma 1 for gausslemma2d 15804. (Contributed by AV, 5-Jul-2021.)

**Hypotheses**
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))))

**Assertion**
Ref	Expression
gausslemma2dlem1	⊢ (𝜑 → (!‘𝐻) = ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘))

Distinct variable groups: 𝑥,𝐻 𝑥,𝑃 𝜑,𝑥 𝑘,𝐻 𝑅,𝑘 𝜑,𝑘 Allowed substitution hints: 𝑃(𝑘) 𝑅(𝑥)

Proof of Theorem gausslemma2dlem1 Dummy variable 𝑙 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		gausslemma2d.p	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))
2		gausslemma2d.h	. . . . 5 ⊢ 𝐻 = ((𝑃 − 1) / 2)
3	1, 2	gausslemma2dlem0b 15785	. . . 4 ⊢ (𝜑 → 𝐻 ∈ ℕ)
4	3	nnnn0d 9455	. . 3 ⊢ (𝜑 → 𝐻 ∈ ℕ₀)
5		fprodfac 12181	. . 3 ⊢ (𝐻 ∈ ℕ₀ → (!‘𝐻) = ∏𝑙 ∈ (1...𝐻)𝑙)
6	4, 5	syl 14	. 2 ⊢ (𝜑 → (!‘𝐻) = ∏𝑙 ∈ (1...𝐻)𝑙)
7		id 19	. . 3 ⊢ (𝑙 = (𝑅‘𝑘) → 𝑙 = (𝑅‘𝑘))
8		1zzd 9506	. . . 4 ⊢ (𝜑 → 1 ∈ ℤ)
9	3	nnzd 9601	. . . 4 ⊢ (𝜑 → 𝐻 ∈ ℤ)
10	8, 9	fzfigd 10694	. . 3 ⊢ (𝜑 → (1...𝐻) ∈ Fin)
11		gausslemma2d.r	. . . 4 ⊢ 𝑅 = (𝑥 ∈ (1...𝐻) ↦ if((𝑥 · 2) < (𝑃 / 2), (𝑥 · 2), (𝑃 − (𝑥 · 2))))
12	1, 2, 11	gausslemma2dlem1f1o 15795	. . 3 ⊢ (𝜑 → 𝑅:(1...𝐻)–1-1-onto→(1...𝐻))
13		eqidd 2232	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → (𝑅‘𝑘) = (𝑅‘𝑘))
14		elfzelz 10260	. . . . 5 ⊢ (𝑙 ∈ (1...𝐻) → 𝑙 ∈ ℤ)
15	14	zcnd 9603	. . . 4 ⊢ (𝑙 ∈ (1...𝐻) → 𝑙 ∈ ℂ)
16	15	adantl 277	. . 3 ⊢ ((𝜑 ∧ 𝑙 ∈ (1...𝐻)) → 𝑙 ∈ ℂ)
17	7, 10, 12, 13, 16	fprodf1o 12154	. 2 ⊢ (𝜑 → ∏𝑙 ∈ (1...𝐻)𝑙 = ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘))
18	6, 17	eqtrd 2264	1 ⊢ (𝜑 → (!‘𝐻) = ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1397 ∈ wcel 2202 ∖ cdif 3197 ifcif 3605 {csn 3669 class class class wbr 4088 ↦ cmpt 4150 ‘cfv 5326 (class class class)co 6018 ℂcc 8030 1c1 8033 · cmul 8037 < clt 8214 − cmin 8350 / cdiv 8852 2c2 9194 ℕ₀cn0 9402 ...cfz 10243 !cfa 10988 ∏cprod 12116 ℙcprime 12684

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4204 ax-sep 4207 ax-nul 4215 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-iinf 4686 ax-cnex 8123 ax-resscn 8124 ax-1cn 8125 ax-1re 8126 ax-icn 8127 ax-addcl 8128 ax-addrcl 8129 ax-mulcl 8130 ax-mulrcl 8131 ax-addcom 8132 ax-mulcom 8133 ax-addass 8134 ax-mulass 8135 ax-distr 8136 ax-i2m1 8137 ax-0lt1 8138 ax-1rid 8139 ax-0id 8140 ax-rnegex 8141 ax-precex 8142 ax-cnre 8143 ax-pre-ltirr 8144 ax-pre-ltwlin 8145 ax-pre-lttrn 8146 ax-pre-apti 8147 ax-pre-ltadd 8148 ax-pre-mulgt0 8149 ax-pre-mulext 8150 ax-arch 8151 ax-caucvg 8152

This theorem depends on definitions: df-bi 117 df-dc 842 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-xor 1420 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rmo 2518 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-nul 3495 df-if 3606 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-br 4089 df-opab 4151 df-mpt 4152 df-tr 4188 df-id 4390 df-po 4393 df-iso 4394 df-iord 4463 df-on 4465 df-ilim 4466 df-suc 4468 df-iom 4689 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-isom 5335 df-riota 5971 df-ov 6021 df-oprab 6022 df-mpo 6023 df-1st 6303 df-2nd 6304 df-recs 6471 df-irdg 6536 df-frec 6557 df-1o 6582 df-2o 6583 df-oadd 6586 df-er 6702 df-en 6910 df-dom 6911 df-fin 6912 df-pnf 8216 df-mnf 8217 df-xr 8218 df-ltxr 8219 df-le 8220 df-sub 8352 df-neg 8353 df-reap 8755 df-ap 8762 df-div 8853 df-inn 9144 df-2 9202 df-3 9203 df-4 9204 df-n0 9403 df-z 9480 df-uz 9756 df-q 9854 df-rp 9889 df-ioo 10127 df-fz 10244 df-fzo 10378 df-fl 10531 df-mod 10586 df-seqfrec 10711 df-exp 10802 df-fac 10989 df-ihash 11039 df-cj 11407 df-re 11408 df-im 11409 df-rsqrt 11563 df-abs 11564 df-clim 11844 df-proddc 12117 df-dvds 12354 df-prm 12685

This theorem is referenced by: gausslemma2dlem4 15799

W3C validator