etransclem14 - Mathbox for Glauco Siliprandi

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Theorem etransclem14 45049

Description: Value of the term 𝑇, when 𝐽 = 0 and (𝐶‘0) = 𝑃 − 1 (Contributed by Glauco Siliprandi, 5-Apr-2020.)

**Hypotheses**
Ref	Expression
etransclem14.n	⊢ (𝜑 → 𝑃 ∈ ℕ)
etransclem14.m	⊢ (𝜑 → 𝑀 ∈ ℕ₀)
etransclem14.c	⊢ (𝜑 → 𝐶:(0...𝑀)⟶(0...𝑁))
etransclem14.t	⊢ 𝑇 = (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐶‘𝑗))) · (if((𝑃 − 1) < (𝐶‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐶‘0)))) · (𝐽↑((𝑃 − 1) − (𝐶‘0))))) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐶‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐶‘𝑗)))))))
etransclem14.j	⊢ (𝜑 → 𝐽 = 0)
etransclem14.cpm1	⊢ (𝜑 → (𝐶‘0) = (𝑃 − 1))

**Assertion**
Ref	Expression
etransclem14	⊢ (𝜑 → 𝑇 = (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐶‘𝑗))) · ((!‘(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐶‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝑗)))) · (-𝑗↑(𝑃 − (𝐶‘𝑗))))))))

Distinct variable groups: 𝑗,𝑀 𝜑,𝑗 Allowed substitution hints: 𝐶(𝑗) 𝑃(𝑗) 𝑇(𝑗) 𝐽(𝑗) 𝑁(𝑗)

**Proof of Theorem etransclem14**
Step	Hyp	Ref	Expression
1		etransclem14.t	. 2 ⊢ 𝑇 = (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐶‘𝑗))) · (if((𝑃 − 1) < (𝐶‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐶‘0)))) · (𝐽↑((𝑃 − 1) − (𝐶‘0))))) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐶‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐶‘𝑗)))))))
2		etransclem14.cpm1	. . . . . . . 8 ⊢ (𝜑 → (𝐶‘0) = (𝑃 − 1))
3		fzssre 44109	. . . . . . . . . 10 ⊢ (0...𝑁) ⊆ ℝ
4		etransclem14.c	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶:(0...𝑀)⟶(0...𝑁))
5		etransclem14.m	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
6		nn0uz 12866	. . . . . . . . . . . . 13 ⊢ ℕ₀ = (ℤ_≥‘0)
7	5, 6	eleqtrdi 2843	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘0))
8		eluzfz1 13510	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ (ℤ_≥‘0) → 0 ∈ (0...𝑀))
9	7, 8	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ∈ (0...𝑀))
10	4, 9	ffvelcdmd 7087	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶‘0) ∈ (0...𝑁))
11	3, 10	sselid 3980	. . . . . . . . 9 ⊢ (𝜑 → (𝐶‘0) ∈ ℝ)
12	2, 11	eqeltrrd 2834	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 − 1) ∈ ℝ)
13	11, 12	lttri3d 11356	. . . . . . . 8 ⊢ (𝜑 → ((𝐶‘0) = (𝑃 − 1) ↔ (¬ (𝐶‘0) < (𝑃 − 1) ∧ ¬ (𝑃 − 1) < (𝐶‘0))))
14	2, 13	mpbid 231	. . . . . . 7 ⊢ (𝜑 → (¬ (𝐶‘0) < (𝑃 − 1) ∧ ¬ (𝑃 − 1) < (𝐶‘0)))
15	14	simprd 496	. . . . . 6 ⊢ (𝜑 → ¬ (𝑃 − 1) < (𝐶‘0))
16	15	iffalsed 4539	. . . . 5 ⊢ (𝜑 → if((𝑃 − 1) < (𝐶‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐶‘0)))) · (𝐽↑((𝑃 − 1) − (𝐶‘0))))) = (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐶‘0)))) · (𝐽↑((𝑃 − 1) − (𝐶‘0)))))
17	12	recnd 11244	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃 − 1) ∈ ℂ)
18	2	eqcomd 2738	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃 − 1) = (𝐶‘0))
19	17, 18	subeq0bd 11642	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑃 − 1) − (𝐶‘0)) = 0)
20	19	fveq2d 6895	. . . . . . . . 9 ⊢ (𝜑 → (!‘((𝑃 − 1) − (𝐶‘0))) = (!‘0))
21		fac0 14238	. . . . . . . . 9 ⊢ (!‘0) = 1
22	20, 21	eqtrdi 2788	. . . . . . . 8 ⊢ (𝜑 → (!‘((𝑃 − 1) − (𝐶‘0))) = 1)
23	22	oveq2d 7427	. . . . . . 7 ⊢ (𝜑 → ((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐶‘0)))) = ((!‘(𝑃 − 1)) / 1))
24		etransclem14.n	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ ℕ)
25		nnm1nn0 12515	. . . . . . . . . . 11 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
26	24, 25	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 − 1) ∈ ℕ₀)
27	26	faccld 14246	. . . . . . . . 9 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℕ)
28	27	nncnd 12230	. . . . . . . 8 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℂ)
29	28	div1d 11984	. . . . . . 7 ⊢ (𝜑 → ((!‘(𝑃 − 1)) / 1) = (!‘(𝑃 − 1)))
30	23, 29	eqtrd 2772	. . . . . 6 ⊢ (𝜑 → ((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐶‘0)))) = (!‘(𝑃 − 1)))
31		etransclem14.j	. . . . . . . 8 ⊢ (𝜑 → 𝐽 = 0)
32	31, 19	oveq12d 7429	. . . . . . 7 ⊢ (𝜑 → (𝐽↑((𝑃 − 1) − (𝐶‘0))) = (0↑0))
33		0exp0e1 14034	. . . . . . 7 ⊢ (0↑0) = 1
34	32, 33	eqtrdi 2788	. . . . . 6 ⊢ (𝜑 → (𝐽↑((𝑃 − 1) − (𝐶‘0))) = 1)
35	30, 34	oveq12d 7429	. . . . 5 ⊢ (𝜑 → (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐶‘0)))) · (𝐽↑((𝑃 − 1) − (𝐶‘0)))) = ((!‘(𝑃 − 1)) · 1))
36	28	mulridd 11233	. . . . 5 ⊢ (𝜑 → ((!‘(𝑃 − 1)) · 1) = (!‘(𝑃 − 1)))
37	16, 35, 36	3eqtrd 2776	. . . 4 ⊢ (𝜑 → if((𝑃 − 1) < (𝐶‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐶‘0)))) · (𝐽↑((𝑃 − 1) − (𝐶‘0))))) = (!‘(𝑃 − 1)))
38	31	oveq1d 7426	. . . . . . . . 9 ⊢ (𝜑 → (𝐽 − 𝑗) = (0 − 𝑗))
39		df-neg 11449	. . . . . . . . 9 ⊢ -𝑗 = (0 − 𝑗)
40	38, 39	eqtr4di 2790	. . . . . . . 8 ⊢ (𝜑 → (𝐽 − 𝑗) = -𝑗)
41	40	oveq1d 7426	. . . . . . 7 ⊢ (𝜑 → ((𝐽 − 𝑗)↑(𝑃 − (𝐶‘𝑗))) = (-𝑗↑(𝑃 − (𝐶‘𝑗))))
42	41	oveq2d 7427	. . . . . 6 ⊢ (𝜑 → (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐶‘𝑗)))) = (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝑗)))) · (-𝑗↑(𝑃 − (𝐶‘𝑗)))))
43	42	ifeq2d 4548	. . . . 5 ⊢ (𝜑 → if(𝑃 < (𝐶‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐶‘𝑗))))) = if(𝑃 < (𝐶‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝑗)))) · (-𝑗↑(𝑃 − (𝐶‘𝑗))))))
44	43	prodeq2ad 44393	. . . 4 ⊢ (𝜑 → ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐶‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐶‘𝑗))))) = ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐶‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝑗)))) · (-𝑗↑(𝑃 − (𝐶‘𝑗))))))
45	37, 44	oveq12d 7429	. . 3 ⊢ (𝜑 → (if((𝑃 − 1) < (𝐶‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐶‘0)))) · (𝐽↑((𝑃 − 1) − (𝐶‘0))))) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐶‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐶‘𝑗)))))) = ((!‘(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐶‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝑗)))) · (-𝑗↑(𝑃 − (𝐶‘𝑗)))))))
46	45	oveq2d 7427	. 2 ⊢ (𝜑 → (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐶‘𝑗))) · (if((𝑃 − 1) < (𝐶‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐶‘0)))) · (𝐽↑((𝑃 − 1) − (𝐶‘0))))) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐶‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐶‘𝑗))))))) = (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐶‘𝑗))) · ((!‘(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐶‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝑗)))) · (-𝑗↑(𝑃 − (𝐶‘𝑗))))))))
47	1, 46	eqtrid 2784	1 ⊢ (𝜑 → 𝑇 = (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐶‘𝑗))) · ((!‘(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐶‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝑗)))) · (-𝑗↑(𝑃 − (𝐶‘𝑗))))))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ifcif 4528 class class class wbr 5148 ⟶wf 6539 ‘cfv 6543 (class class class)co 7411 ℝcr 11111 0cc0 11112 1c1 11113 · cmul 11117 < clt 11250 − cmin 11446 -cneg 11447 / cdiv 11873 ℕcn 12214 ℕ₀cn0 12474 ℤ_≥cuz 12824 ...cfz 13486 ↑cexp 14029 !cfa 14235 ∏cprod 15851

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 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-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-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-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-div 11874 df-nn 12215 df-n0 12475 df-z 12561 df-uz 12825 df-fz 13487 df-seq 13969 df-exp 14030 df-fac 14236 df-prod 15852

This theorem is referenced by: etransclem28 45063

W3C validator