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Theorem stoweidlem19 42452
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 7138 . . . . . 6 (𝑛 = 0 → ((𝐹𝑡)↑𝑛) = ((𝐹𝑡)↑0))
32mpteq2dv 5135 . . . . 5 (𝑛 = 0 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑛)) = (𝑡𝑇 ↦ ((𝐹𝑡)↑0)))
43eleq1d 2896 . . . 4 (𝑛 = 0 → ((𝑡𝑇 ↦ ((𝐹𝑡)↑𝑛)) ∈ 𝐴 ↔ (𝑡𝑇 ↦ ((𝐹𝑡)↑0)) ∈ 𝐴))
54imbi2d 344 . . 3 (𝑛 = 0 → ((𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑛)) ∈ 𝐴) ↔ (𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑0)) ∈ 𝐴)))
6 oveq2 7138 . . . . . 6 (𝑛 = 𝑚 → ((𝐹𝑡)↑𝑛) = ((𝐹𝑡)↑𝑚))
76mpteq2dv 5135 . . . . 5 (𝑛 = 𝑚 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑛)) = (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)))
87eleq1d 2896 . . . 4 (𝑛 = 𝑚 → ((𝑡𝑇 ↦ ((𝐹𝑡)↑𝑛)) ∈ 𝐴 ↔ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴))
98imbi2d 344 . . 3 (𝑛 = 𝑚 → ((𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑛)) ∈ 𝐴) ↔ (𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴)))
10 oveq2 7138 . . . . . 6 (𝑛 = (𝑚 + 1) → ((𝐹𝑡)↑𝑛) = ((𝐹𝑡)↑(𝑚 + 1)))
1110mpteq2dv 5135 . . . . 5 (𝑛 = (𝑚 + 1) → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑛)) = (𝑡𝑇 ↦ ((𝐹𝑡)↑(𝑚 + 1))))
1211eleq1d 2896 . . . 4 (𝑛 = (𝑚 + 1) → ((𝑡𝑇 ↦ ((𝐹𝑡)↑𝑛)) ∈ 𝐴 ↔ (𝑡𝑇 ↦ ((𝐹𝑡)↑(𝑚 + 1))) ∈ 𝐴))
1312imbi2d 344 . . 3 (𝑛 = (𝑚 + 1) → ((𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑛)) ∈ 𝐴) ↔ (𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑(𝑚 + 1))) ∈ 𝐴)))
14 oveq2 7138 . . . . . 6 (𝑛 = 𝑁 → ((𝐹𝑡)↑𝑛) = ((𝐹𝑡)↑𝑁))
1514mpteq2dv 5135 . . . . 5 (𝑛 = 𝑁 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑛)) = (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑁)))
1615eleq1d 2896 . . . 4 (𝑛 = 𝑁 → ((𝑡𝑇 ↦ ((𝐹𝑡)↑𝑛)) ∈ 𝐴 ↔ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑁)) ∈ 𝐴))
1716imbi2d 344 . . 3 (𝑛 = 𝑁 → ((𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑛)) ∈ 𝐴) ↔ (𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑁)) ∈ 𝐴)))
18 stoweidlem19.2 . . . . 5 𝑡𝜑
19 stoweidlem19.6 . . . . . . . . 9 (𝜑𝐹𝐴)
2019ancli 552 . . . . . . . . 9 (𝜑 → (𝜑𝐹𝐴))
21 eleq1 2899 . . . . . . . . . . . 12 (𝑓 = 𝐹 → (𝑓𝐴𝐹𝐴))
2221anbi2d 631 . . . . . . . . . . 11 (𝑓 = 𝐹 → ((𝜑𝑓𝐴) ↔ (𝜑𝐹𝐴)))
23 feq1 6468 . . . . . . . . . . 11 (𝑓 = 𝐹 → (𝑓:𝑇⟶ℝ ↔ 𝐹:𝑇⟶ℝ))
2422, 23imbi12d 348 . . . . . . . . . 10 (𝑓 = 𝐹 → (((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑𝐹𝐴) → 𝐹:𝑇⟶ℝ)))
25 stoweidlem19.3 . . . . . . . . . 10 ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)
2624, 25vtoclg 3544 . . . . . . . . 9 (𝐹𝐴 → ((𝜑𝐹𝐴) → 𝐹:𝑇⟶ℝ))
2719, 20, 26sylc 65 . . . . . . . 8 (𝜑𝐹:𝑇⟶ℝ)
2827ffvelrnda 6824 . . . . . . 7 ((𝜑𝑡𝑇) → (𝐹𝑡) ∈ ℝ)
29 recn 10604 . . . . . . 7 ((𝐹𝑡) ∈ ℝ → (𝐹𝑡) ∈ ℂ)
30 exp0 13417 . . . . . . 7 ((𝐹𝑡) ∈ ℂ → ((𝐹𝑡)↑0) = 1)
3128, 29, 303syl 18 . . . . . 6 ((𝜑𝑡𝑇) → ((𝐹𝑡)↑0) = 1)
3231eqcomd 2827 . . . . 5 ((𝜑𝑡𝑇) → 1 = ((𝐹𝑡)↑0))
3318, 32mpteq2da 5133 . . . 4 (𝜑 → (𝑡𝑇 ↦ 1) = (𝑡𝑇 ↦ ((𝐹𝑡)↑0)))
34 1re 10618 . . . . 5 1 ∈ ℝ
35 stoweidlem19.5 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴)
3635stoweidlem4 42437 . . . . 5 ((𝜑 ∧ 1 ∈ ℝ) → (𝑡𝑇 ↦ 1) ∈ 𝐴)
3734, 36mpan2 690 . . . 4 (𝜑 → (𝑡𝑇 ↦ 1) ∈ 𝐴)
3833, 37eqeltrrd 2913 . . 3 (𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑0)) ∈ 𝐴)
39 simpr 488 . . . . 5 (((𝑚 ∈ ℕ0 ∧ (𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴)) ∧ 𝜑) → 𝜑)
40 simpll 766 . . . . 5 (((𝑚 ∈ ℕ0 ∧ (𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴)) ∧ 𝜑) → 𝑚 ∈ ℕ0)
41 simplr 768 . . . . . 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 1916 . . . . . . . 8 𝑡 𝑚 ∈ ℕ0
44 nfmpt1 5137 . . . . . . . . 9 𝑡(𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚))
4544nfel1 2990 . . . . . . . 8 𝑡(𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴
4618, 43, 45nf3an 1903 . . . . . . 7 𝑡(𝜑𝑚 ∈ ℕ0 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴)
47 simpl1 1188 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ0 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡𝑇) → 𝜑)
48 simpr 488 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ0 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡𝑇) → 𝑡𝑇)
4928recnd 10646 . . . . . . . . 9 ((𝜑𝑡𝑇) → (𝐹𝑡) ∈ ℂ)
5047, 48, 49syl2anc 587 . . . . . . . 8 (((𝜑𝑚 ∈ ℕ0 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡𝑇) → (𝐹𝑡) ∈ ℂ)
51 simpl2 1189 . . . . . . . 8 (((𝜑𝑚 ∈ ℕ0 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡𝑇) → 𝑚 ∈ ℕ0)
5250, 51expp1d 13495 . . . . . . 7 (((𝜑𝑚 ∈ ℕ0 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡𝑇) → ((𝐹𝑡)↑(𝑚 + 1)) = (((𝐹𝑡)↑𝑚) · (𝐹𝑡)))
5346, 52mpteq2da 5133 . . . . . 6 ((𝜑𝑚 ∈ ℕ0 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡)↑(𝑚 + 1))) = (𝑡𝑇 ↦ (((𝐹𝑡)↑𝑚) · (𝐹𝑡))))
54283adant2 1128 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ0𝑡𝑇) → (𝐹𝑡) ∈ ℝ)
55 simp2 1134 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ0𝑡𝑇) → 𝑚 ∈ ℕ0)
5654, 55reexpcld 13511 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ0𝑡𝑇) → ((𝐹𝑡)↑𝑚) ∈ ℝ)
5747, 51, 48, 56syl3anc 1368 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ0 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡𝑇) → ((𝐹𝑡)↑𝑚) ∈ ℝ)
58 eqid 2821 . . . . . . . . . . . 12 (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) = (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚))
5958fvmpt2 6752 . . . . . . . . . . 11 ((𝑡𝑇 ∧ ((𝐹𝑡)↑𝑚) ∈ ℝ) → ((𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚))‘𝑡) = ((𝐹𝑡)↑𝑚))
6059eqcomd 2827 . . . . . . . . . 10 ((𝑡𝑇 ∧ ((𝐹𝑡)↑𝑚) ∈ ℝ) → ((𝐹𝑡)↑𝑚) = ((𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚))‘𝑡))
6148, 57, 60syl2anc 587 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ0 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡𝑇) → ((𝐹𝑡)↑𝑚) = ((𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚))‘𝑡))
6261oveq1d 7145 . . . . . . . 8 (((𝜑𝑚 ∈ ℕ0 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡𝑇) → (((𝐹𝑡)↑𝑚) · (𝐹𝑡)) = (((𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚))‘𝑡) · (𝐹𝑡)))
6346, 62mpteq2da 5133 . . . . . . 7 ((𝜑𝑚 ∈ ℕ0 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) → (𝑡𝑇 ↦ (((𝐹𝑡)↑𝑚) · (𝐹𝑡))) = (𝑡𝑇 ↦ (((𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚))‘𝑡) · (𝐹𝑡))))
6419adantr 484 . . . . . . . . 9 ((𝜑 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) → 𝐹𝐴)
6544nfeq2 2991 . . . . . . . . . 10 𝑡 𝑓 = (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚))
66 stoweidlem19.1 . . . . . . . . . . 11 𝑡𝐹
6766nfeq2 2991 . . . . . . . . . 10 𝑡 𝑔 = 𝐹
68 stoweidlem19.4 . . . . . . . . . 10 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
6965, 67, 68stoweidlem6 42439 . . . . . . . . 9 ((𝜑 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴𝐹𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚))‘𝑡) · (𝐹𝑡))) ∈ 𝐴)
7064, 69mpd3an3 1459 . . . . . . . 8 ((𝜑 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚))‘𝑡) · (𝐹𝑡))) ∈ 𝐴)
71703adant2 1128 . . . . . . 7 ((𝜑𝑚 ∈ ℕ0 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚))‘𝑡) · (𝐹𝑡))) ∈ 𝐴)
7263, 71eqeltrd 2912 . . . . . 6 ((𝜑𝑚 ∈ ℕ0 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) → (𝑡𝑇 ↦ (((𝐹𝑡)↑𝑚) · (𝐹𝑡))) ∈ 𝐴)
7353, 72eqeltrd 2912 . . . . 5 ((𝜑𝑚 ∈ ℕ0 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡)↑(𝑚 + 1))) ∈ 𝐴)
7439, 40, 42, 73syl3anc 1368 . . . 4 (((𝑚 ∈ ℕ0 ∧ (𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴)) ∧ 𝜑) → (𝑡𝑇 ↦ ((𝐹𝑡)↑(𝑚 + 1))) ∈ 𝐴)
7574exp31 423 . . 3 (𝑚 ∈ ℕ0 → ((𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) → (𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑(𝑚 + 1))) ∈ 𝐴)))
765, 9, 13, 17, 38, 75nn0ind 12055 . 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 399  w3a 1084   = wceq 1538  wnf 1785  wcel 2115  wnfc 2958  cmpt 5119  wf 6324  cfv 6328  (class class class)co 7130  cc 10512  cr 10513  0cc0 10514  1c1 10515   + caddc 10517   · cmul 10519  0cn0 11875  cexp 13413
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-cnex 10570  ax-resscn 10571  ax-1cn 10572  ax-icn 10573  ax-addcl 10574  ax-addrcl 10575  ax-mulcl 10576  ax-mulrcl 10577  ax-mulcom 10578  ax-addass 10579  ax-mulass 10580  ax-distr 10581  ax-i2m1 10582  ax-1ne0 10583  ax-1rid 10584  ax-rnegex 10585  ax-rrecex 10586  ax-cnre 10587  ax-pre-lttri 10588  ax-pre-lttrn 10589  ax-pre-ltadd 10590  ax-pre-mulgt0 10591
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-nel 3112  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7088  df-ov 7133  df-oprab 7134  df-mpo 7135  df-om 7556  df-2nd 7665  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-er 8264  df-en 8485  df-dom 8486  df-sdom 8487  df-pnf 10654  df-mnf 10655  df-xr 10656  df-ltxr 10657  df-le 10658  df-sub 10849  df-neg 10850  df-nn 11616  df-n0 11876  df-z 11960  df-uz 12222  df-seq 13353  df-exp 13414
This theorem is referenced by:  stoweidlem40  42473
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