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Theorem gausslemma2dlem1 15588

Description: Lemma 1 for gausslemma2d 15596. (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 15577	. . . 4 ⊢ (𝜑 → 𝐻 ∈ ℕ)
4	3	nnnn0d 9361	. . 3 ⊢ (𝜑 → 𝐻 ∈ ℕ₀)
5		fprodfac 11976	. . 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 9412	. . . 4 ⊢ (𝜑 → 1 ∈ ℤ)
9	3	nnzd 9507	. . . 4 ⊢ (𝜑 → 𝐻 ∈ ℤ)
10	8, 9	fzfigd 10589	. . 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 15587	. . 3 ⊢ (𝜑 → 𝑅:(1...𝐻)–1-1-onto→(1...𝐻))
13		eqidd 2207	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → (𝑅‘𝑘) = (𝑅‘𝑘))
14		elfzelz 10160	. . . . 5 ⊢ (𝑙 ∈ (1...𝐻) → 𝑙 ∈ ℤ)
15	14	zcnd 9509	. . . 4 ⊢ (𝑙 ∈ (1...𝐻) → 𝑙 ∈ ℂ)
16	15	adantl 277	. . 3 ⊢ ((𝜑 ∧ 𝑙 ∈ (1...𝐻)) → 𝑙 ∈ ℂ)
17	7, 10, 12, 13, 16	fprodf1o 11949	. 2 ⊢ (𝜑 → ∏𝑙 ∈ (1...𝐻)𝑙 = ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘))
18	6, 17	eqtrd 2239	1 ⊢ (𝜑 → (!‘𝐻) = ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1373 ∈ wcel 2177 ∖ cdif 3165 ifcif 3573 {csn 3635 class class class wbr 4048 ↦ cmpt 4110 ‘cfv 5277 (class class class)co 5954 ℂcc 7936 1c1 7939 · cmul 7943 < clt 8120 − cmin 8256 / cdiv 8758 2c2 9100 ℕ₀cn0 9308 ...cfz 10143 !cfa 10883 ∏cprod 11911 ℙcprime 12479

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 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2179 ax-14 2180 ax-ext 2188 ax-coll 4164 ax-sep 4167 ax-nul 4175 ax-pow 4223 ax-pr 4258 ax-un 4485 ax-setind 4590 ax-iinf 4641 ax-cnex 8029 ax-resscn 8030 ax-1cn 8031 ax-1re 8032 ax-icn 8033 ax-addcl 8034 ax-addrcl 8035 ax-mulcl 8036 ax-mulrcl 8037 ax-addcom 8038 ax-mulcom 8039 ax-addass 8040 ax-mulass 8041 ax-distr 8042 ax-i2m1 8043 ax-0lt1 8044 ax-1rid 8045 ax-0id 8046 ax-rnegex 8047 ax-precex 8048 ax-cnre 8049 ax-pre-ltirr 8050 ax-pre-ltwlin 8051 ax-pre-lttrn 8052 ax-pre-apti 8053 ax-pre-ltadd 8054 ax-pre-mulgt0 8055 ax-pre-mulext 8056 ax-arch 8057 ax-caucvg 8058

This theorem depends on definitions: df-bi 117 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-xor 1396 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2193 df-cleq 2199 df-clel 2202 df-nfc 2338 df-ne 2378 df-nel 2473 df-ral 2490 df-rex 2491 df-reu 2492 df-rmo 2493 df-rab 2494 df-v 2775 df-sbc 3001 df-csb 3096 df-dif 3170 df-un 3172 df-in 3174 df-ss 3181 df-nul 3463 df-if 3574 df-pw 3620 df-sn 3641 df-pr 3642 df-op 3644 df-uni 3854 df-int 3889 df-iun 3932 df-br 4049 df-opab 4111 df-mpt 4112 df-tr 4148 df-id 4345 df-po 4348 df-iso 4349 df-iord 4418 df-on 4420 df-ilim 4421 df-suc 4423 df-iom 4644 df-xp 4686 df-rel 4687 df-cnv 4688 df-co 4689 df-dm 4690 df-rn 4691 df-res 4692 df-ima 4693 df-iota 5238 df-fun 5279 df-fn 5280 df-f 5281 df-f1 5282 df-fo 5283 df-f1o 5284 df-fv 5285 df-isom 5286 df-riota 5909 df-ov 5957 df-oprab 5958 df-mpo 5959 df-1st 6236 df-2nd 6237 df-recs 6401 df-irdg 6466 df-frec 6487 df-1o 6512 df-2o 6513 df-oadd 6516 df-er 6630 df-en 6838 df-dom 6839 df-fin 6840 df-pnf 8122 df-mnf 8123 df-xr 8124 df-ltxr 8125 df-le 8126 df-sub 8258 df-neg 8259 df-reap 8661 df-ap 8668 df-div 8759 df-inn 9050 df-2 9108 df-3 9109 df-4 9110 df-n0 9309 df-z 9386 df-uz 9662 df-q 9754 df-rp 9789 df-ioo 10027 df-fz 10144 df-fzo 10278 df-fl 10426 df-mod 10481 df-seqfrec 10606 df-exp 10697 df-fac 10884 df-ihash 10934 df-cj 11203 df-re 11204 df-im 11205 df-rsqrt 11359 df-abs 11360 df-clim 11640 df-proddc 11912 df-dvds 12149 df-prm 12480

This theorem is referenced by: gausslemma2dlem4 15591

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