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Theorem gausslemma2dlem1 15951

Description: Lemma 1 for gausslemma2d 15959. (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 15940	. . . 4 ⊢ (𝜑 → 𝐻 ∈ ℕ)
4	3	nnnn0d 9555	. . 3 ⊢ (𝜑 → 𝐻 ∈ ℕ₀)
5		fprodfac 12305	. . 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 9606	. . . 4 ⊢ (𝜑 → 1 ∈ ℤ)
9	3	nnzd 9702	. . . 4 ⊢ (𝜑 → 𝐻 ∈ ℤ)
10	8, 9	fzfigd 10797	. . 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 15950	. . 3 ⊢ (𝜑 → 𝑅:(1...𝐻)–1-1-onto→(1...𝐻))
13		eqidd 2235	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → (𝑅‘𝑘) = (𝑅‘𝑘))
14		elfzelz 10362	. . . . 5 ⊢ (𝑙 ∈ (1...𝐻) → 𝑙 ∈ ℤ)
15	14	zcnd 9704	. . . 4 ⊢ (𝑙 ∈ (1...𝐻) → 𝑙 ∈ ℂ)
16	15	adantl 277	. . 3 ⊢ ((𝜑 ∧ 𝑙 ∈ (1...𝐻)) → 𝑙 ∈ ℂ)
17	7, 10, 12, 13, 16	fprodf1o 12278	. 2 ⊢ (𝜑 → ∏𝑙 ∈ (1...𝐻)𝑙 = ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘))
18	6, 17	eqtrd 2267	1 ⊢ (𝜑 → (!‘𝐻) = ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1398 ∈ wcel 2205 ∖ cdif 3210 ifcif 3622 {csn 3691 class class class wbr 4111 ↦ cmpt 4173 ‘cfv 5354 (class class class)co 6052 ℂcc 8127 1c1 8130 · cmul 8134 < clt 8310 − cmin 8446 / cdiv 8948 2c2 9290 ℕ₀cn0 9498 ...cfz 10345 !cfa 11091 ∏cprod 12240 ℙcprime 12808

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 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-coll 4227 ax-sep 4230 ax-nul 4238 ax-pow 4289 ax-pr 4324 ax-un 4556 ax-setind 4661 ax-iinf 4712 ax-cnex 8220 ax-resscn 8221 ax-1cn 8222 ax-1re 8223 ax-icn 8224 ax-addcl 8225 ax-addrcl 8226 ax-mulcl 8227 ax-mulrcl 8228 ax-addcom 8229 ax-mulcom 8230 ax-addass 8231 ax-mulass 8232 ax-distr 8233 ax-i2m1 8234 ax-0lt1 8235 ax-1rid 8236 ax-0id 8237 ax-rnegex 8238 ax-precex 8239 ax-cnre 8240 ax-pre-ltirr 8241 ax-pre-ltwlin 8242 ax-pre-lttrn 8243 ax-pre-apti 8244 ax-pre-ltadd 8245 ax-pre-mulgt0 8246 ax-pre-mulext 8247 ax-arch 8248 ax-caucvg 8249

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-xor 1421 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rmo 2530 df-rab 2531 df-v 2817 df-sbc 3045 df-csb 3141 df-dif 3215 df-un 3217 df-in 3219 df-ss 3226 df-nul 3511 df-if 3623 df-pw 3673 df-sn 3697 df-pr 3698 df-op 3700 df-uni 3917 df-int 3952 df-iun 3995 df-br 4112 df-opab 4174 df-mpt 4175 df-tr 4211 df-id 4416 df-po 4419 df-iso 4420 df-iord 4489 df-on 4491 df-ilim 4492 df-suc 4494 df-iom 4715 df-xp 4757 df-rel 4758 df-cnv 4759 df-co 4760 df-dm 4761 df-rn 4762 df-res 4763 df-ima 4764 df-iota 5314 df-fun 5356 df-fn 5357 df-f 5358 df-f1 5359 df-fo 5360 df-f1o 5361 df-fv 5362 df-isom 5363 df-riota 6005 df-ov 6055 df-oprab 6056 df-mpo 6057 df-1st 6336 df-2nd 6337 df-recs 6538 df-irdg 6603 df-frec 6624 df-1o 6649 df-2o 6650 df-oadd 6653 df-er 6769 df-en 6978 df-dom 6979 df-fin 6980 df-pnf 8312 df-mnf 8313 df-xr 8314 df-ltxr 8315 df-le 8316 df-sub 8448 df-neg 8449 df-reap 8851 df-ap 8858 df-div 8949 df-inn 9240 df-2 9298 df-3 9299 df-4 9300 df-n0 9499 df-z 9580 df-uz 9857 df-q 9955 df-rp 9990 df-ioo 10228 df-fz 10346 df-fzo 10481 df-fl 10634 df-mod 10689 df-seqfrec 10814 df-exp 10905 df-fac 11092 df-ihash 11143 df-cj 11531 df-re 11532 df-im 11533 df-rsqrt 11687 df-abs 11688 df-clim 11968 df-proddc 12241 df-dvds 12478 df-prm 12809

This theorem is referenced by: gausslemma2dlem4 15954

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