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Theorem gausslemma2dlem1 15119

Description: Lemma 1 for gausslemma2d 15127. (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 15108	. . . 4 ⊢ (𝜑 → 𝐻 ∈ ℕ)
4	3	nnnn0d 9283	. . 3 ⊢ (𝜑 → 𝐻 ∈ ℕ₀)
5		fprodfac 11745	. . 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 9334	. . . 4 ⊢ (𝜑 → 1 ∈ ℤ)
9	3	nnzd 9428	. . . 4 ⊢ (𝜑 → 𝐻 ∈ ℤ)
10	8, 9	fzfigd 10492	. . 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 15118	. . 3 ⊢ (𝜑 → 𝑅:(1...𝐻)–1-1-onto→(1...𝐻))
13		eqidd 2194	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → (𝑅‘𝑘) = (𝑅‘𝑘))
14		elfzelz 10081	. . . . 5 ⊢ (𝑙 ∈ (1...𝐻) → 𝑙 ∈ ℤ)
15	14	zcnd 9430	. . . 4 ⊢ (𝑙 ∈ (1...𝐻) → 𝑙 ∈ ℂ)
16	15	adantl 277	. . 3 ⊢ ((𝜑 ∧ 𝑙 ∈ (1...𝐻)) → 𝑙 ∈ ℂ)
17	7, 10, 12, 13, 16	fprodf1o 11718	. 2 ⊢ (𝜑 → ∏𝑙 ∈ (1...𝐻)𝑙 = ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘))
18	6, 17	eqtrd 2226	1 ⊢ (𝜑 → (!‘𝐻) = ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2164 ∖ cdif 3150 ifcif 3557 {csn 3618 class class class wbr 4029 ↦ cmpt 4090 ‘cfv 5246 (class class class)co 5910 ℂcc 7860 1c1 7863 · cmul 7867 < clt 8044 − cmin 8180 / cdiv 8681 2c2 9023 ℕ₀cn0 9230 ...cfz 10064 !cfa 10783 ∏cprod 11680 ℙcprime 12232

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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4462 ax-setind 4565 ax-iinf 4616 ax-cnex 7953 ax-resscn 7954 ax-1cn 7955 ax-1re 7956 ax-icn 7957 ax-addcl 7958 ax-addrcl 7959 ax-mulcl 7960 ax-mulrcl 7961 ax-addcom 7962 ax-mulcom 7963 ax-addass 7964 ax-mulass 7965 ax-distr 7966 ax-i2m1 7967 ax-0lt1 7968 ax-1rid 7969 ax-0id 7970 ax-rnegex 7971 ax-precex 7972 ax-cnre 7973 ax-pre-ltirr 7974 ax-pre-ltwlin 7975 ax-pre-lttrn 7976 ax-pre-apti 7977 ax-pre-ltadd 7978 ax-pre-mulgt0 7979 ax-pre-mulext 7980 ax-arch 7981 ax-caucvg 7982

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-xor 1387 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-if 3558 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-tr 4128 df-id 4322 df-po 4325 df-iso 4326 df-iord 4395 df-on 4397 df-ilim 4398 df-suc 4400 df-iom 4619 df-xp 4661 df-rel 4662 df-cnv 4663 df-co 4664 df-dm 4665 df-rn 4666 df-res 4667 df-ima 4668 df-iota 5207 df-fun 5248 df-fn 5249 df-f 5250 df-f1 5251 df-fo 5252 df-f1o 5253 df-fv 5254 df-isom 5255 df-riota 5865 df-ov 5913 df-oprab 5914 df-mpo 5915 df-1st 6184 df-2nd 6185 df-recs 6349 df-irdg 6414 df-frec 6435 df-1o 6460 df-2o 6461 df-oadd 6464 df-er 6578 df-en 6786 df-dom 6787 df-fin 6788 df-pnf 8046 df-mnf 8047 df-xr 8048 df-ltxr 8049 df-le 8050 df-sub 8182 df-neg 8183 df-reap 8584 df-ap 8591 df-div 8682 df-inn 8973 df-2 9031 df-3 9032 df-4 9033 df-n0 9231 df-z 9308 df-uz 9583 df-q 9675 df-rp 9710 df-ioo 9948 df-fz 10065 df-fzo 10199 df-fl 10329 df-mod 10384 df-seqfrec 10509 df-exp 10597 df-fac 10784 df-ihash 10834 df-cj 10973 df-re 10974 df-im 10975 df-rsqrt 11129 df-abs 11130 df-clim 11409 df-proddc 11681 df-dvds 11918 df-prm 12233

This theorem is referenced by: gausslemma2dlem4 15122

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