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Theorem stoweidlem19 40554
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 6698 . . . . . 6 (𝑛 = 0 → ((𝐹𝑡)↑𝑛) = ((𝐹𝑡)↑0))
32mpteq2dv 4778 . . . . 5 (𝑛 = 0 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑛)) = (𝑡𝑇 ↦ ((𝐹𝑡)↑0)))
43eleq1d 2715 . . . 4 (𝑛 = 0 → ((𝑡𝑇 ↦ ((𝐹𝑡)↑𝑛)) ∈ 𝐴 ↔ (𝑡𝑇 ↦ ((𝐹𝑡)↑0)) ∈ 𝐴))
54imbi2d 329 . . 3 (𝑛 = 0 → ((𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑛)) ∈ 𝐴) ↔ (𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑0)) ∈ 𝐴)))
6 oveq2 6698 . . . . . 6 (𝑛 = 𝑚 → ((𝐹𝑡)↑𝑛) = ((𝐹𝑡)↑𝑚))
76mpteq2dv 4778 . . . . 5 (𝑛 = 𝑚 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑛)) = (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)))
87eleq1d 2715 . . . 4 (𝑛 = 𝑚 → ((𝑡𝑇 ↦ ((𝐹𝑡)↑𝑛)) ∈ 𝐴 ↔ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴))
98imbi2d 329 . . 3 (𝑛 = 𝑚 → ((𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑛)) ∈ 𝐴) ↔ (𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴)))
10 oveq2 6698 . . . . . 6 (𝑛 = (𝑚 + 1) → ((𝐹𝑡)↑𝑛) = ((𝐹𝑡)↑(𝑚 + 1)))
1110mpteq2dv 4778 . . . . 5 (𝑛 = (𝑚 + 1) → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑛)) = (𝑡𝑇 ↦ ((𝐹𝑡)↑(𝑚 + 1))))
1211eleq1d 2715 . . . 4 (𝑛 = (𝑚 + 1) → ((𝑡𝑇 ↦ ((𝐹𝑡)↑𝑛)) ∈ 𝐴 ↔ (𝑡𝑇 ↦ ((𝐹𝑡)↑(𝑚 + 1))) ∈ 𝐴))
1312imbi2d 329 . . 3 (𝑛 = (𝑚 + 1) → ((𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑛)) ∈ 𝐴) ↔ (𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑(𝑚 + 1))) ∈ 𝐴)))
14 oveq2 6698 . . . . . 6 (𝑛 = 𝑁 → ((𝐹𝑡)↑𝑛) = ((𝐹𝑡)↑𝑁))
1514mpteq2dv 4778 . . . . 5 (𝑛 = 𝑁 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑛)) = (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑁)))
1615eleq1d 2715 . . . 4 (𝑛 = 𝑁 → ((𝑡𝑇 ↦ ((𝐹𝑡)↑𝑛)) ∈ 𝐴 ↔ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑁)) ∈ 𝐴))
1716imbi2d 329 . . 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 2718 . . . . . . . . . . . 12 (𝑓 = 𝐹 → (𝑓𝐴𝐹𝐴))
2221anbi2d 740 . . . . . . . . . . 11 (𝑓 = 𝐹 → ((𝜑𝑓𝐴) ↔ (𝜑𝐹𝐴)))
23 feq1 6064 . . . . . . . . . . 11 (𝑓 = 𝐹 → (𝑓:𝑇⟶ℝ ↔ 𝐹:𝑇⟶ℝ))
2422, 23imbi12d 333 . . . . . . . . . 10 (𝑓 = 𝐹 → (((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑𝐹𝐴) → 𝐹:𝑇⟶ℝ)))
25 stoweidlem19.3 . . . . . . . . . 10 ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)
2624, 25vtoclg 3297 . . . . . . . . 9 (𝐹𝐴 → ((𝜑𝐹𝐴) → 𝐹:𝑇⟶ℝ))
2719, 20, 26sylc 65 . . . . . . . 8 (𝜑𝐹:𝑇⟶ℝ)
2827ffvelrnda 6399 . . . . . . 7 ((𝜑𝑡𝑇) → (𝐹𝑡) ∈ ℝ)
29 recn 10064 . . . . . . 7 ((𝐹𝑡) ∈ ℝ → (𝐹𝑡) ∈ ℂ)
30 exp0 12904 . . . . . . 7 ((𝐹𝑡) ∈ ℂ → ((𝐹𝑡)↑0) = 1)
3128, 29, 303syl 18 . . . . . 6 ((𝜑𝑡𝑇) → ((𝐹𝑡)↑0) = 1)
3231eqcomd 2657 . . . . 5 ((𝜑𝑡𝑇) → 1 = ((𝐹𝑡)↑0))
3318, 32mpteq2da 4776 . . . 4 (𝜑 → (𝑡𝑇 ↦ 1) = (𝑡𝑇 ↦ ((𝐹𝑡)↑0)))
34 1re 10077 . . . . 5 1 ∈ ℝ
35 stoweidlem19.5 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴)
3635stoweidlem4 40539 . . . . 5 ((𝜑 ∧ 1 ∈ ℝ) → (𝑡𝑇 ↦ 1) ∈ 𝐴)
3734, 36mpan2 707 . . . 4 (𝜑 → (𝑡𝑇 ↦ 1) ∈ 𝐴)
3833, 37eqeltrrd 2731 . . 3 (𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑0)) ∈ 𝐴)
39 simpr 476 . . . . 5 (((𝑚 ∈ ℕ0 ∧ (𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴)) ∧ 𝜑) → 𝜑)
40 simpll 805 . . . . 5 (((𝑚 ∈ ℕ0 ∧ (𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴)) ∧ 𝜑) → 𝑚 ∈ ℕ0)
41 simplr 807 . . . . . 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 1883 . . . . . . . 8 𝑡 𝑚 ∈ ℕ0
44 nfmpt1 4780 . . . . . . . . 9 𝑡(𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚))
4544nfel1 2808 . . . . . . . 8 𝑡(𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴
4618, 43, 45nf3an 1871 . . . . . . 7 𝑡(𝜑𝑚 ∈ ℕ0 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴)
47 simpl1 1084 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ0 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡𝑇) → 𝜑)
48 simpr 476 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ0 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡𝑇) → 𝑡𝑇)
4928recnd 10106 . . . . . . . . 9 ((𝜑𝑡𝑇) → (𝐹𝑡) ∈ ℂ)
5047, 48, 49syl2anc 694 . . . . . . . 8 (((𝜑𝑚 ∈ ℕ0 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡𝑇) → (𝐹𝑡) ∈ ℂ)
51 simpl2 1085 . . . . . . . 8 (((𝜑𝑚 ∈ ℕ0 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡𝑇) → 𝑚 ∈ ℕ0)
5250, 51expp1d 13049 . . . . . . 7 (((𝜑𝑚 ∈ ℕ0 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡𝑇) → ((𝐹𝑡)↑(𝑚 + 1)) = (((𝐹𝑡)↑𝑚) · (𝐹𝑡)))
5346, 52mpteq2da 4776 . . . . . 6 ((𝜑𝑚 ∈ ℕ0 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡)↑(𝑚 + 1))) = (𝑡𝑇 ↦ (((𝐹𝑡)↑𝑚) · (𝐹𝑡))))
54283adant2 1100 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ0𝑡𝑇) → (𝐹𝑡) ∈ ℝ)
55 simp2 1082 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ0𝑡𝑇) → 𝑚 ∈ ℕ0)
5654, 55reexpcld 13065 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ0𝑡𝑇) → ((𝐹𝑡)↑𝑚) ∈ ℝ)
5747, 51, 48, 56syl3anc 1366 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ0 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡𝑇) → ((𝐹𝑡)↑𝑚) ∈ ℝ)
58 eqid 2651 . . . . . . . . . . . 12 (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) = (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚))
5958fvmpt2 6330 . . . . . . . . . . 11 ((𝑡𝑇 ∧ ((𝐹𝑡)↑𝑚) ∈ ℝ) → ((𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚))‘𝑡) = ((𝐹𝑡)↑𝑚))
6059eqcomd 2657 . . . . . . . . . 10 ((𝑡𝑇 ∧ ((𝐹𝑡)↑𝑚) ∈ ℝ) → ((𝐹𝑡)↑𝑚) = ((𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚))‘𝑡))
6148, 57, 60syl2anc 694 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ0 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡𝑇) → ((𝐹𝑡)↑𝑚) = ((𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚))‘𝑡))
6261oveq1d 6705 . . . . . . . 8 (((𝜑𝑚 ∈ ℕ0 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡𝑇) → (((𝐹𝑡)↑𝑚) · (𝐹𝑡)) = (((𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚))‘𝑡) · (𝐹𝑡)))
6346, 62mpteq2da 4776 . . . . . . 7 ((𝜑𝑚 ∈ ℕ0 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) → (𝑡𝑇 ↦ (((𝐹𝑡)↑𝑚) · (𝐹𝑡))) = (𝑡𝑇 ↦ (((𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚))‘𝑡) · (𝐹𝑡))))
6419adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) → 𝐹𝐴)
6544nfeq2 2809 . . . . . . . . . 10 𝑡 𝑓 = (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚))
66 stoweidlem19.1 . . . . . . . . . . 11 𝑡𝐹
6766nfeq2 2809 . . . . . . . . . 10 𝑡 𝑔 = 𝐹
68 stoweidlem19.4 . . . . . . . . . 10 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
6965, 67, 68stoweidlem6 40541 . . . . . . . . 9 ((𝜑 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴𝐹𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚))‘𝑡) · (𝐹𝑡))) ∈ 𝐴)
7064, 69mpd3an3 1465 . . . . . . . 8 ((𝜑 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚))‘𝑡) · (𝐹𝑡))) ∈ 𝐴)
71703adant2 1100 . . . . . . 7 ((𝜑𝑚 ∈ ℕ0 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚))‘𝑡) · (𝐹𝑡))) ∈ 𝐴)
7263, 71eqeltrd 2730 . . . . . 6 ((𝜑𝑚 ∈ ℕ0 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) → (𝑡𝑇 ↦ (((𝐹𝑡)↑𝑚) · (𝐹𝑡))) ∈ 𝐴)
7353, 72eqeltrd 2730 . . . . 5 ((𝜑𝑚 ∈ ℕ0 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡)↑(𝑚 + 1))) ∈ 𝐴)
7439, 40, 42, 73syl3anc 1366 . . . 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 11510 . 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 383  w3a 1054   = wceq 1523  wnf 1748  wcel 2030  wnfc 2780  cmpt 4762  wf 5922  cfv 5926  (class class class)co 6690  cc 9972  cr 9973  0cc0 9974  1c1 9975   + caddc 9977   · cmul 9979  0cn0 11330  cexp 12900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-n0 11331  df-z 11416  df-uz 11726  df-seq 12842  df-exp 12901
This theorem is referenced by:  stoweidlem40  40575
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