etransclem4 - Mathbox for Glauco Siliprandi

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Theorem etransclem4 45253

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 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → 𝑗 ∈ (0...𝑀))
2		etransclem4.a	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ⊆ ℂ)
3		cnex 11195	. . . . . . . . . . 11 ⊢ ℂ ∈ V
4	3	ssex 5321	. . . . . . . . . 10 ⊢ (𝐴 ⊆ ℂ → 𝐴 ∈ V)
5		mptexg 7225	. . . . . . . . . 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 480	. . . . . . . 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 7009	. . . . . . . 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 583	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (𝐻‘𝑗) = (𝑥 ∈ 𝐴 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))
11		ovexd 7447	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑥 ∈ 𝐴) → ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) ∈ V)
12	10, 11	fvmpt2d 7011	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑥 ∈ 𝐴) → ((𝐻‘𝑗)‘𝑥) = ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))
13	12	an32s 649	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑗 ∈ (0...𝑀)) → ((𝐻‘𝑗)‘𝑥) = ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))
14	13	prodeq2dv 15872	. . . 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 12869	. . . . . . 7 ⊢ ℕ₀ = (ℤ_≥‘0)
17	15, 16	eleqtrdi 2842	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘0))
18	17	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑀 ∈ (ℤ_≥‘0))
19	2	sselda 3982	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℂ)
20	19	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑗 ∈ (0...𝑀)) → 𝑥 ∈ ℂ)
21		elfzelz 13506	. . . . . . . . 9 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℤ)
22	21	zcnd 12672	. . . . . . . 8 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℂ)
23	22	adantl 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑗 ∈ (0...𝑀)) → 𝑗 ∈ ℂ)
24	20, 23	subcld 11576	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑗 ∈ (0...𝑀)) → (𝑥 − 𝑗) ∈ ℂ)
25		etransclem4.p	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ ℕ)
26		nnm1nn0 12518	. . . . . . . . 9 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
27	25, 26	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑃 − 1) ∈ ℕ₀)
28	25	nnnn0d 12537	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℕ₀)
29	27, 28	ifcld 4574	. . . . . . 7 ⊢ (𝜑 → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℕ₀)
30	29	ad2antrr 723	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑗 ∈ (0...𝑀)) → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℕ₀)
31	24, 30	expcld 14116	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑗 ∈ (0...𝑀)) → ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) ∈ ℂ)
32		oveq2 7420	. . . . . 6 ⊢ (𝑗 = 0 → (𝑥 − 𝑗) = (𝑥 − 0))
33		iftrue 4534	. . . . . 6 ⊢ (𝑗 = 0 → if(𝑗 = 0, (𝑃 − 1), 𝑃) = (𝑃 − 1))
34	32, 33	oveq12d 7430	. . . . 5 ⊢ (𝑗 = 0 → ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) = ((𝑥 − 0)↑(𝑃 − 1)))
35	18, 31, 34	fprod1p 15917	. . . 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 11565	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥 − 0) = 𝑥)
37	36	oveq1d 7427	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 − 0)↑(𝑃 − 1)) = (𝑥↑(𝑃 − 1)))
38		0p1e1 12339	. . . . . . . . 9 ⊢ (0 + 1) = 1
39	38	oveq1i 7422	. . . . . . . 8 ⊢ ((0 + 1)...𝑀) = (1...𝑀)
40	39	a1i 11	. . . . . . 7 ⊢ (𝜑 → ((0 + 1)...𝑀) = (1...𝑀))
41		0red 11222	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (1...𝑀) → 0 ∈ ℝ)
42		1red 11220	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (1...𝑀) → 1 ∈ ℝ)
43		elfzelz 13506	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (1...𝑀) → 𝑗 ∈ ℤ)
44	43	zred 12671	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (1...𝑀) → 𝑗 ∈ ℝ)
45		0lt1 11741	. . . . . . . . . . . . . 14 ⊢ 0 < 1
46	45	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (1...𝑀) → 0 < 1)
47		elfzle1 13509	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (1...𝑀) → 1 ≤ 𝑗)
48	41, 42, 44, 46, 47	ltletrd 11379	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (1...𝑀) → 0 < 𝑗)
49	48	gt0ne0d 11783	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (1...𝑀) → 𝑗 ≠ 0)
50	49	neneqd 2944	. . . . . . . . . 10 ⊢ (𝑗 ∈ (1...𝑀) → ¬ 𝑗 = 0)
51	50	iffalsed 4539	. . . . . . . . 9 ⊢ (𝑗 ∈ (1...𝑀) → if(𝑗 = 0, (𝑃 − 1), 𝑃) = 𝑃)
52	51	oveq2d 7428	. . . . . . . 8 ⊢ (𝑗 ∈ (1...𝑀) → ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) = ((𝑥 − 𝑗)↑𝑃))
53	52	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) = ((𝑥 − 𝑗)↑𝑃))
54	40, 53	prodeq12rdv 15876	. . . . . 6 ⊢ (𝜑 → ∏𝑗 ∈ ((0 + 1)...𝑀)((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) = ∏𝑗 ∈ (1...𝑀)((𝑥 − 𝑗)↑𝑃))
55	54	adantr 480	. . . . 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 7430	. . . 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 2776	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥 − 𝑗)↑𝑃)) = ∏𝑗 ∈ (0...𝑀)((𝐻‘𝑗)‘𝑥))
58	57	mpteq2dva 5248	. 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 2796	1 ⊢ (𝜑 → 𝐹 = 𝐸)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 Vcvv 3473 ⊆ wss 3948 ifcif 4528 class class class wbr 5148 ↦ cmpt 5231 ‘cfv 6543 (class class class)co 7412 ℂcc 11112 0cc0 11114 1c1 11115 + caddc 11117 · cmul 11119 < clt 11253 − cmin 11449 ℕcn 12217 ℕ₀cn0 12477 ℤ_≥cuz 12827 ...cfz 13489 ↑cexp 14032 ∏cprod 15854

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9640 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 ax-pre-sup 11192

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 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-int 4951 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-se 5632 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-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-er 8707 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-sup 9441 df-oi 9509 df-card 9938 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-n0 12478 df-z 12564 df-uz 12828 df-rp 12980 df-fz 13490 df-fzo 13633 df-seq 13972 df-exp 14033 df-hash 14296 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-clim 15437 df-prod 15855

This theorem is referenced by: etransclem13 45262 etransclem29 45278

W3C validator