etransclem4 - Mathbox for Glauco Siliprandi

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Theorem etransclem4 44940

Description: 𝐹 expressed as a finite product. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

**Hypotheses**
Ref	Expression
etransclem4.a	⊢ (𝜑 → 𝐴 ⊆ ℂ)
etransclem4.p	⊢ (𝜑 → 𝑃 ∈ ℕ)
etransclem4.M	⊢ (𝜑 → 𝑀 ∈ ℕ₀)
etransclem4.f	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥 − 𝑗)↑𝑃)))
etransclem4.h	⊢ 𝐻 = (𝑗 ∈ (0...𝑀) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))
etransclem4.e	⊢ 𝐸 = (𝑥 ∈ 𝐴 ↦ ∏𝑗 ∈ (0...𝑀)((𝐻‘𝑗)‘𝑥))

**Assertion**
Ref	Expression
etransclem4	⊢ (𝜑 → 𝐹 = 𝐸)

Distinct variable groups: 𝐴,𝑗,𝑥 𝑗,𝑀 𝑃,𝑗 𝜑,𝑗,𝑥 Allowed substitution hints: 𝑃(𝑥) 𝐸(𝑥,𝑗) 𝐹(𝑥,𝑗) 𝐻(𝑥,𝑗) 𝑀(𝑥)

**Proof of Theorem etransclem4**
Step	Hyp	Ref	Expression
1		simpr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → 𝑗 ∈ (0...𝑀))
2		etransclem4.a	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ⊆ ℂ)
3		cnex 11187	. . . . . . . . . . 11 ⊢ ℂ ∈ V
4	3	ssex 5320	. . . . . . . . . 10 ⊢ (𝐴 ⊆ ℂ → 𝐴 ∈ V)
5		mptexg 7219	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → (𝑥 ∈ 𝐴 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))) ∈ V)
6	2, 4, 5	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))) ∈ V)
7	6	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (𝑥 ∈ 𝐴 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))) ∈ V)
8		etransclem4.h	. . . . . . . . 9 ⊢ 𝐻 = (𝑗 ∈ (0...𝑀) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))
9	8	fvmpt2 7006	. . . . . . . 8 ⊢ ((𝑗 ∈ (0...𝑀) ∧ (𝑥 ∈ 𝐴 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))) ∈ V) → (𝐻‘𝑗) = (𝑥 ∈ 𝐴 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))
10	1, 7, 9	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (𝐻‘𝑗) = (𝑥 ∈ 𝐴 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))
11		ovexd 7440	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑥 ∈ 𝐴) → ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) ∈ V)
12	10, 11	fvmpt2d 7008	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑥 ∈ 𝐴) → ((𝐻‘𝑗)‘𝑥) = ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))
13	12	an32s 650	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑗 ∈ (0...𝑀)) → ((𝐻‘𝑗)‘𝑥) = ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))
14	13	prodeq2dv 15863	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∏𝑗 ∈ (0...𝑀)((𝐻‘𝑗)‘𝑥) = ∏𝑗 ∈ (0...𝑀)((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))
15		etransclem4.M	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
16		nn0uz 12860	. . . . . . 7 ⊢ ℕ₀ = (ℤ_≥‘0)
17	15, 16	eleqtrdi 2843	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘0))
18	17	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑀 ∈ (ℤ_≥‘0))
19	2	sselda 3981	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℂ)
20	19	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑗 ∈ (0...𝑀)) → 𝑥 ∈ ℂ)
21		elfzelz 13497	. . . . . . . . 9 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℤ)
22	21	zcnd 12663	. . . . . . . 8 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℂ)
23	22	adantl 482	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑗 ∈ (0...𝑀)) → 𝑗 ∈ ℂ)
24	20, 23	subcld 11567	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑗 ∈ (0...𝑀)) → (𝑥 − 𝑗) ∈ ℂ)
25		etransclem4.p	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ ℕ)
26		nnm1nn0 12509	. . . . . . . . 9 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
27	25, 26	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑃 − 1) ∈ ℕ₀)
28	25	nnnn0d 12528	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℕ₀)
29	27, 28	ifcld 4573	. . . . . . 7 ⊢ (𝜑 → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℕ₀)
30	29	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑗 ∈ (0...𝑀)) → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℕ₀)
31	24, 30	expcld 14107	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑗 ∈ (0...𝑀)) → ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) ∈ ℂ)
32		oveq2 7413	. . . . . 6 ⊢ (𝑗 = 0 → (𝑥 − 𝑗) = (𝑥 − 0))
33		iftrue 4533	. . . . . 6 ⊢ (𝑗 = 0 → if(𝑗 = 0, (𝑃 − 1), 𝑃) = (𝑃 − 1))
34	32, 33	oveq12d 7423	. . . . 5 ⊢ (𝑗 = 0 → ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) = ((𝑥 − 0)↑(𝑃 − 1)))
35	18, 31, 34	fprod1p 15908	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∏𝑗 ∈ (0...𝑀)((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) = (((𝑥 − 0)↑(𝑃 − 1)) · ∏𝑗 ∈ ((0 + 1)...𝑀)((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))
36	19	subid1d 11556	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥 − 0) = 𝑥)
37	36	oveq1d 7420	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 − 0)↑(𝑃 − 1)) = (𝑥↑(𝑃 − 1)))
38		0p1e1 12330	. . . . . . . . 9 ⊢ (0 + 1) = 1
39	38	oveq1i 7415	. . . . . . . 8 ⊢ ((0 + 1)...𝑀) = (1...𝑀)
40	39	a1i 11	. . . . . . 7 ⊢ (𝜑 → ((0 + 1)...𝑀) = (1...𝑀))
41		0red 11213	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (1...𝑀) → 0 ∈ ℝ)
42		1red 11211	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (1...𝑀) → 1 ∈ ℝ)
43		elfzelz 13497	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (1...𝑀) → 𝑗 ∈ ℤ)
44	43	zred 12662	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (1...𝑀) → 𝑗 ∈ ℝ)
45		0lt1 11732	. . . . . . . . . . . . . 14 ⊢ 0 < 1
46	45	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (1...𝑀) → 0 < 1)
47		elfzle1 13500	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (1...𝑀) → 1 ≤ 𝑗)
48	41, 42, 44, 46, 47	ltletrd 11370	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (1...𝑀) → 0 < 𝑗)
49	48	gt0ne0d 11774	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (1...𝑀) → 𝑗 ≠ 0)
50	49	neneqd 2945	. . . . . . . . . 10 ⊢ (𝑗 ∈ (1...𝑀) → ¬ 𝑗 = 0)
51	50	iffalsed 4538	. . . . . . . . 9 ⊢ (𝑗 ∈ (1...𝑀) → if(𝑗 = 0, (𝑃 − 1), 𝑃) = 𝑃)
52	51	oveq2d 7421	. . . . . . . 8 ⊢ (𝑗 ∈ (1...𝑀) → ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) = ((𝑥 − 𝑗)↑𝑃))
53	52	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) = ((𝑥 − 𝑗)↑𝑃))
54	40, 53	prodeq12rdv 15867	. . . . . 6 ⊢ (𝜑 → ∏𝑗 ∈ ((0 + 1)...𝑀)((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) = ∏𝑗 ∈ (1...𝑀)((𝑥 − 𝑗)↑𝑃))
55	54	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∏𝑗 ∈ ((0 + 1)...𝑀)((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) = ∏𝑗 ∈ (1...𝑀)((𝑥 − 𝑗)↑𝑃))
56	37, 55	oveq12d 7423	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((𝑥 − 0)↑(𝑃 − 1)) · ∏𝑗 ∈ ((0 + 1)...𝑀)((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))) = ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥 − 𝑗)↑𝑃)))
57	14, 35, 56	3eqtrrd 2777	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥 − 𝑗)↑𝑃)) = ∏𝑗 ∈ (0...𝑀)((𝐻‘𝑗)‘𝑥))
58	57	mpteq2dva 5247	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥 − 𝑗)↑𝑃))) = (𝑥 ∈ 𝐴 ↦ ∏𝑗 ∈ (0...𝑀)((𝐻‘𝑗)‘𝑥)))
59		etransclem4.f	. 2 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥 − 𝑗)↑𝑃)))
60		etransclem4.e	. 2 ⊢ 𝐸 = (𝑥 ∈ 𝐴 ↦ ∏𝑗 ∈ (0...𝑀)((𝐻‘𝑗)‘𝑥))
61	58, 59, 60	3eqtr4g 2797	1 ⊢ (𝜑 → 𝐹 = 𝐸)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ⊆ wss 3947 ifcif 4527 class class class wbr 5147 ↦ cmpt 5230 ‘cfv 6540 (class class class)co 7405 ℂcc 11104 0cc0 11106 1c1 11107 + caddc 11109 · cmul 11111 < clt 11244 − cmin 11440 ℕcn 12208 ℕ₀cn0 12468 ℤ_≥cuz 12818 ...cfz 13480 ↑cexp 14023 ∏cprod 15845

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-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

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 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-z 12555 df-uz 12819 df-rp 12971 df-fz 13481 df-fzo 13624 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-prod 15846

This theorem is referenced by: etransclem13 44949 etransclem29 44965

W3C validator