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Theorem stoweidlem19 39573
Description: If a set of real functions is closed under multiplication and it contains constants, then it is closed under finite exponentiation. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
stoweidlem19.1 𝑡𝐹
stoweidlem19.2 𝑡𝜑
stoweidlem19.3 ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)
stoweidlem19.4 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
stoweidlem19.5 ((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴)
stoweidlem19.6 (𝜑𝐹𝐴)
stoweidlem19.7 (𝜑𝑁 ∈ ℕ0)
Assertion
Ref Expression
stoweidlem19 (𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑁)) ∈ 𝐴)
Distinct variable groups:   𝑓,𝑔,𝑡,𝐴   𝑓,𝐹,𝑔   𝑇,𝑓,𝑔,𝑡   𝜑,𝑓,𝑔   𝑡,𝑁   𝑥,𝑡,𝐴   𝑥,𝑇   𝜑,𝑥
Allowed substitution hints:   𝜑(𝑡)   𝐹(𝑥,𝑡)   𝑁(𝑥,𝑓,𝑔)

Proof of Theorem stoweidlem19
Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 stoweidlem19.7 . 2 (𝜑𝑁 ∈ ℕ0)
2 oveq2 6623 . . . . . 6 (𝑛 = 0 → ((𝐹𝑡)↑𝑛) = ((𝐹𝑡)↑0))
32mpteq2dv 4715 . . . . 5 (𝑛 = 0 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑛)) = (𝑡𝑇 ↦ ((𝐹𝑡)↑0)))
43eleq1d 2683 . . . 4 (𝑛 = 0 → ((𝑡𝑇 ↦ ((𝐹𝑡)↑𝑛)) ∈ 𝐴 ↔ (𝑡𝑇 ↦ ((𝐹𝑡)↑0)) ∈ 𝐴))
54imbi2d 330 . . 3 (𝑛 = 0 → ((𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑛)) ∈ 𝐴) ↔ (𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑0)) ∈ 𝐴)))
6 oveq2 6623 . . . . . 6 (𝑛 = 𝑚 → ((𝐹𝑡)↑𝑛) = ((𝐹𝑡)↑𝑚))
76mpteq2dv 4715 . . . . 5 (𝑛 = 𝑚 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑛)) = (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)))
87eleq1d 2683 . . . 4 (𝑛 = 𝑚 → ((𝑡𝑇 ↦ ((𝐹𝑡)↑𝑛)) ∈ 𝐴 ↔ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴))
98imbi2d 330 . . 3 (𝑛 = 𝑚 → ((𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑛)) ∈ 𝐴) ↔ (𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴)))
10 oveq2 6623 . . . . . 6 (𝑛 = (𝑚 + 1) → ((𝐹𝑡)↑𝑛) = ((𝐹𝑡)↑(𝑚 + 1)))
1110mpteq2dv 4715 . . . . 5 (𝑛 = (𝑚 + 1) → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑛)) = (𝑡𝑇 ↦ ((𝐹𝑡)↑(𝑚 + 1))))
1211eleq1d 2683 . . . 4 (𝑛 = (𝑚 + 1) → ((𝑡𝑇 ↦ ((𝐹𝑡)↑𝑛)) ∈ 𝐴 ↔ (𝑡𝑇 ↦ ((𝐹𝑡)↑(𝑚 + 1))) ∈ 𝐴))
1312imbi2d 330 . . 3 (𝑛 = (𝑚 + 1) → ((𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑛)) ∈ 𝐴) ↔ (𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑(𝑚 + 1))) ∈ 𝐴)))
14 oveq2 6623 . . . . . 6 (𝑛 = 𝑁 → ((𝐹𝑡)↑𝑛) = ((𝐹𝑡)↑𝑁))
1514mpteq2dv 4715 . . . . 5 (𝑛 = 𝑁 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑛)) = (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑁)))
1615eleq1d 2683 . . . 4 (𝑛 = 𝑁 → ((𝑡𝑇 ↦ ((𝐹𝑡)↑𝑛)) ∈ 𝐴 ↔ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑁)) ∈ 𝐴))
1716imbi2d 330 . . 3 (𝑛 = 𝑁 → ((𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑛)) ∈ 𝐴) ↔ (𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑁)) ∈ 𝐴)))
18 stoweidlem19.2 . . . . 5 𝑡𝜑
19 stoweidlem19.6 . . . . . . . . 9 (𝜑𝐹𝐴)
2019ancli 573 . . . . . . . . 9 (𝜑 → (𝜑𝐹𝐴))
21 eleq1 2686 . . . . . . . . . . . 12 (𝑓 = 𝐹 → (𝑓𝐴𝐹𝐴))
2221anbi2d 739 . . . . . . . . . . 11 (𝑓 = 𝐹 → ((𝜑𝑓𝐴) ↔ (𝜑𝐹𝐴)))
23 feq1 5993 . . . . . . . . . . 11 (𝑓 = 𝐹 → (𝑓:𝑇⟶ℝ ↔ 𝐹:𝑇⟶ℝ))
2422, 23imbi12d 334 . . . . . . . . . 10 (𝑓 = 𝐹 → (((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑𝐹𝐴) → 𝐹:𝑇⟶ℝ)))
25 stoweidlem19.3 . . . . . . . . . 10 ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)
2624, 25vtoclg 3256 . . . . . . . . 9 (𝐹𝐴 → ((𝜑𝐹𝐴) → 𝐹:𝑇⟶ℝ))
2719, 20, 26sylc 65 . . . . . . . 8 (𝜑𝐹:𝑇⟶ℝ)
2827ffvelrnda 6325 . . . . . . 7 ((𝜑𝑡𝑇) → (𝐹𝑡) ∈ ℝ)
29 recn 9986 . . . . . . 7 ((𝐹𝑡) ∈ ℝ → (𝐹𝑡) ∈ ℂ)
30 exp0 12820 . . . . . . 7 ((𝐹𝑡) ∈ ℂ → ((𝐹𝑡)↑0) = 1)
3128, 29, 303syl 18 . . . . . 6 ((𝜑𝑡𝑇) → ((𝐹𝑡)↑0) = 1)
3231eqcomd 2627 . . . . 5 ((𝜑𝑡𝑇) → 1 = ((𝐹𝑡)↑0))
3318, 32mpteq2da 4713 . . . 4 (𝜑 → (𝑡𝑇 ↦ 1) = (𝑡𝑇 ↦ ((𝐹𝑡)↑0)))
34 1re 9999 . . . . 5 1 ∈ ℝ
35 stoweidlem19.5 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴)
3635stoweidlem4 39558 . . . . 5 ((𝜑 ∧ 1 ∈ ℝ) → (𝑡𝑇 ↦ 1) ∈ 𝐴)
3734, 36mpan2 706 . . . 4 (𝜑 → (𝑡𝑇 ↦ 1) ∈ 𝐴)
3833, 37eqeltrrd 2699 . . 3 (𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑0)) ∈ 𝐴)
39 simpr 477 . . . . 5 (((𝑚 ∈ ℕ0 ∧ (𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴)) ∧ 𝜑) → 𝜑)
40 simpll 789 . . . . 5 (((𝑚 ∈ ℕ0 ∧ (𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴)) ∧ 𝜑) → 𝑚 ∈ ℕ0)
41 simplr 791 . . . . . 6 (((𝑚 ∈ ℕ0 ∧ (𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴)) ∧ 𝜑) → (𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴))
4239, 41mpd 15 . . . . 5 (((𝑚 ∈ ℕ0 ∧ (𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴)) ∧ 𝜑) → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴)
43 nfv 1840 . . . . . . . 8 𝑡 𝑚 ∈ ℕ0
44 nfmpt1 4717 . . . . . . . . 9 𝑡(𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚))
4544nfel1 2775 . . . . . . . 8 𝑡(𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴
4618, 43, 45nf3an 1828 . . . . . . 7 𝑡(𝜑𝑚 ∈ ℕ0 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴)
47 simpl1 1062 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ0 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡𝑇) → 𝜑)
48 simpr 477 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ0 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡𝑇) → 𝑡𝑇)
4928recnd 10028 . . . . . . . . 9 ((𝜑𝑡𝑇) → (𝐹𝑡) ∈ ℂ)
5047, 48, 49syl2anc 692 . . . . . . . 8 (((𝜑𝑚 ∈ ℕ0 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡𝑇) → (𝐹𝑡) ∈ ℂ)
51 simpl2 1063 . . . . . . . 8 (((𝜑𝑚 ∈ ℕ0 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡𝑇) → 𝑚 ∈ ℕ0)
5250, 51expp1d 12965 . . . . . . 7 (((𝜑𝑚 ∈ ℕ0 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡𝑇) → ((𝐹𝑡)↑(𝑚 + 1)) = (((𝐹𝑡)↑𝑚) · (𝐹𝑡)))
5346, 52mpteq2da 4713 . . . . . 6 ((𝜑𝑚 ∈ ℕ0 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡)↑(𝑚 + 1))) = (𝑡𝑇 ↦ (((𝐹𝑡)↑𝑚) · (𝐹𝑡))))
54283adant2 1078 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ0𝑡𝑇) → (𝐹𝑡) ∈ ℝ)
55 simp2 1060 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ0𝑡𝑇) → 𝑚 ∈ ℕ0)
5654, 55reexpcld 12981 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ0𝑡𝑇) → ((𝐹𝑡)↑𝑚) ∈ ℝ)
5747, 51, 48, 56syl3anc 1323 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ0 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡𝑇) → ((𝐹𝑡)↑𝑚) ∈ ℝ)
58 eqid 2621 . . . . . . . . . . . 12 (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) = (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚))
5958fvmpt2 6258 . . . . . . . . . . 11 ((𝑡𝑇 ∧ ((𝐹𝑡)↑𝑚) ∈ ℝ) → ((𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚))‘𝑡) = ((𝐹𝑡)↑𝑚))
6059eqcomd 2627 . . . . . . . . . 10 ((𝑡𝑇 ∧ ((𝐹𝑡)↑𝑚) ∈ ℝ) → ((𝐹𝑡)↑𝑚) = ((𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚))‘𝑡))
6148, 57, 60syl2anc 692 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ0 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡𝑇) → ((𝐹𝑡)↑𝑚) = ((𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚))‘𝑡))
6261oveq1d 6630 . . . . . . . 8 (((𝜑𝑚 ∈ ℕ0 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡𝑇) → (((𝐹𝑡)↑𝑚) · (𝐹𝑡)) = (((𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚))‘𝑡) · (𝐹𝑡)))
6346, 62mpteq2da 4713 . . . . . . 7 ((𝜑𝑚 ∈ ℕ0 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) → (𝑡𝑇 ↦ (((𝐹𝑡)↑𝑚) · (𝐹𝑡))) = (𝑡𝑇 ↦ (((𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚))‘𝑡) · (𝐹𝑡))))
6419adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) → 𝐹𝐴)
6544nfeq2 2776 . . . . . . . . . 10 𝑡 𝑓 = (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚))
66 stoweidlem19.1 . . . . . . . . . . 11 𝑡𝐹
6766nfeq2 2776 . . . . . . . . . 10 𝑡 𝑔 = 𝐹
68 stoweidlem19.4 . . . . . . . . . 10 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
6965, 67, 68stoweidlem6 39560 . . . . . . . . 9 ((𝜑 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴𝐹𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚))‘𝑡) · (𝐹𝑡))) ∈ 𝐴)
7064, 69mpd3an3 1422 . . . . . . . 8 ((𝜑 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚))‘𝑡) · (𝐹𝑡))) ∈ 𝐴)
71703adant2 1078 . . . . . . 7 ((𝜑𝑚 ∈ ℕ0 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚))‘𝑡) · (𝐹𝑡))) ∈ 𝐴)
7263, 71eqeltrd 2698 . . . . . 6 ((𝜑𝑚 ∈ ℕ0 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) → (𝑡𝑇 ↦ (((𝐹𝑡)↑𝑚) · (𝐹𝑡))) ∈ 𝐴)
7353, 72eqeltrd 2698 . . . . 5 ((𝜑𝑚 ∈ ℕ0 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡)↑(𝑚 + 1))) ∈ 𝐴)
7439, 40, 42, 73syl3anc 1323 . . . 4 (((𝑚 ∈ ℕ0 ∧ (𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴)) ∧ 𝜑) → (𝑡𝑇 ↦ ((𝐹𝑡)↑(𝑚 + 1))) ∈ 𝐴)
7574exp31 629 . . 3 (𝑚 ∈ ℕ0 → ((𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) → (𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑(𝑚 + 1))) ∈ 𝐴)))
765, 9, 13, 17, 38, 75nn0ind 11432 . 2 (𝑁 ∈ ℕ0 → (𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑁)) ∈ 𝐴))
771, 76mpcom 38 1 (𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑁)) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wnf 1705  wcel 1987  wnfc 2748  cmpt 4683  wf 5853  cfv 5857  (class class class)co 6615  cc 9894  cr 9895  0cc0 9896  1c1 9897   + caddc 9899   · cmul 9901  0cn0 11252  cexp 12816
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-er 7702  df-en 7916  df-dom 7917  df-sdom 7918  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-n0 11253  df-z 11338  df-uz 11648  df-seq 12758  df-exp 12817
This theorem is referenced by:  stoweidlem40  39594
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