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Theorem stoweidlem19 45975
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 7439 . . . . . 6 (𝑛 = 0 → ((𝐹𝑡)↑𝑛) = ((𝐹𝑡)↑0))
32mpteq2dv 5250 . . . . 5 (𝑛 = 0 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑛)) = (𝑡𝑇 ↦ ((𝐹𝑡)↑0)))
43eleq1d 2824 . . . 4 (𝑛 = 0 → ((𝑡𝑇 ↦ ((𝐹𝑡)↑𝑛)) ∈ 𝐴 ↔ (𝑡𝑇 ↦ ((𝐹𝑡)↑0)) ∈ 𝐴))
54imbi2d 340 . . 3 (𝑛 = 0 → ((𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑛)) ∈ 𝐴) ↔ (𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑0)) ∈ 𝐴)))
6 oveq2 7439 . . . . . 6 (𝑛 = 𝑚 → ((𝐹𝑡)↑𝑛) = ((𝐹𝑡)↑𝑚))
76mpteq2dv 5250 . . . . 5 (𝑛 = 𝑚 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑛)) = (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)))
87eleq1d 2824 . . . 4 (𝑛 = 𝑚 → ((𝑡𝑇 ↦ ((𝐹𝑡)↑𝑛)) ∈ 𝐴 ↔ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴))
98imbi2d 340 . . 3 (𝑛 = 𝑚 → ((𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑛)) ∈ 𝐴) ↔ (𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴)))
10 oveq2 7439 . . . . . 6 (𝑛 = (𝑚 + 1) → ((𝐹𝑡)↑𝑛) = ((𝐹𝑡)↑(𝑚 + 1)))
1110mpteq2dv 5250 . . . . 5 (𝑛 = (𝑚 + 1) → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑛)) = (𝑡𝑇 ↦ ((𝐹𝑡)↑(𝑚 + 1))))
1211eleq1d 2824 . . . 4 (𝑛 = (𝑚 + 1) → ((𝑡𝑇 ↦ ((𝐹𝑡)↑𝑛)) ∈ 𝐴 ↔ (𝑡𝑇 ↦ ((𝐹𝑡)↑(𝑚 + 1))) ∈ 𝐴))
1312imbi2d 340 . . 3 (𝑛 = (𝑚 + 1) → ((𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑛)) ∈ 𝐴) ↔ (𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑(𝑚 + 1))) ∈ 𝐴)))
14 oveq2 7439 . . . . . 6 (𝑛 = 𝑁 → ((𝐹𝑡)↑𝑛) = ((𝐹𝑡)↑𝑁))
1514mpteq2dv 5250 . . . . 5 (𝑛 = 𝑁 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑛)) = (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑁)))
1615eleq1d 2824 . . . 4 (𝑛 = 𝑁 → ((𝑡𝑇 ↦ ((𝐹𝑡)↑𝑛)) ∈ 𝐴 ↔ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑁)) ∈ 𝐴))
1716imbi2d 340 . . 3 (𝑛 = 𝑁 → ((𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑛)) ∈ 𝐴) ↔ (𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑁)) ∈ 𝐴)))
18 stoweidlem19.2 . . . . 5 𝑡𝜑
19 stoweidlem19.6 . . . . . . . . 9 (𝜑𝐹𝐴)
2019ancli 548 . . . . . . . . 9 (𝜑 → (𝜑𝐹𝐴))
21 eleq1 2827 . . . . . . . . . . . 12 (𝑓 = 𝐹 → (𝑓𝐴𝐹𝐴))
2221anbi2d 630 . . . . . . . . . . 11 (𝑓 = 𝐹 → ((𝜑𝑓𝐴) ↔ (𝜑𝐹𝐴)))
23 feq1 6717 . . . . . . . . . . 11 (𝑓 = 𝐹 → (𝑓:𝑇⟶ℝ ↔ 𝐹:𝑇⟶ℝ))
2422, 23imbi12d 344 . . . . . . . . . 10 (𝑓 = 𝐹 → (((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑𝐹𝐴) → 𝐹:𝑇⟶ℝ)))
25 stoweidlem19.3 . . . . . . . . . 10 ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)
2624, 25vtoclg 3554 . . . . . . . . 9 (𝐹𝐴 → ((𝜑𝐹𝐴) → 𝐹:𝑇⟶ℝ))
2719, 20, 26sylc 65 . . . . . . . 8 (𝜑𝐹:𝑇⟶ℝ)
2827ffvelcdmda 7104 . . . . . . 7 ((𝜑𝑡𝑇) → (𝐹𝑡) ∈ ℝ)
29 recn 11243 . . . . . . 7 ((𝐹𝑡) ∈ ℝ → (𝐹𝑡) ∈ ℂ)
30 exp0 14103 . . . . . . 7 ((𝐹𝑡) ∈ ℂ → ((𝐹𝑡)↑0) = 1)
3128, 29, 303syl 18 . . . . . 6 ((𝜑𝑡𝑇) → ((𝐹𝑡)↑0) = 1)
3231eqcomd 2741 . . . . 5 ((𝜑𝑡𝑇) → 1 = ((𝐹𝑡)↑0))
3318, 32mpteq2da 5246 . . . 4 (𝜑 → (𝑡𝑇 ↦ 1) = (𝑡𝑇 ↦ ((𝐹𝑡)↑0)))
34 1re 11259 . . . . 5 1 ∈ ℝ
35 stoweidlem19.5 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴)
3635stoweidlem4 45960 . . . . 5 ((𝜑 ∧ 1 ∈ ℝ) → (𝑡𝑇 ↦ 1) ∈ 𝐴)
3734, 36mpan2 691 . . . 4 (𝜑 → (𝑡𝑇 ↦ 1) ∈ 𝐴)
3833, 37eqeltrrd 2840 . . 3 (𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑0)) ∈ 𝐴)
39 simpr 484 . . . . 5 (((𝑚 ∈ ℕ0 ∧ (𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴)) ∧ 𝜑) → 𝜑)
40 simpll 767 . . . . 5 (((𝑚 ∈ ℕ0 ∧ (𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴)) ∧ 𝜑) → 𝑚 ∈ ℕ0)
41 simplr 769 . . . . . 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 1912 . . . . . . . 8 𝑡 𝑚 ∈ ℕ0
44 nfmpt1 5256 . . . . . . . . 9 𝑡(𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚))
4544nfel1 2920 . . . . . . . 8 𝑡(𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴
4618, 43, 45nf3an 1899 . . . . . . 7 𝑡(𝜑𝑚 ∈ ℕ0 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴)
47 simpl1 1190 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ0 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡𝑇) → 𝜑)
48 simpr 484 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ0 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡𝑇) → 𝑡𝑇)
4928recnd 11287 . . . . . . . . 9 ((𝜑𝑡𝑇) → (𝐹𝑡) ∈ ℂ)
5047, 48, 49syl2anc 584 . . . . . . . 8 (((𝜑𝑚 ∈ ℕ0 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡𝑇) → (𝐹𝑡) ∈ ℂ)
51 simpl2 1191 . . . . . . . 8 (((𝜑𝑚 ∈ ℕ0 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡𝑇) → 𝑚 ∈ ℕ0)
5250, 51expp1d 14184 . . . . . . 7 (((𝜑𝑚 ∈ ℕ0 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡𝑇) → ((𝐹𝑡)↑(𝑚 + 1)) = (((𝐹𝑡)↑𝑚) · (𝐹𝑡)))
5346, 52mpteq2da 5246 . . . . . 6 ((𝜑𝑚 ∈ ℕ0 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡)↑(𝑚 + 1))) = (𝑡𝑇 ↦ (((𝐹𝑡)↑𝑚) · (𝐹𝑡))))
54283adant2 1130 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ0𝑡𝑇) → (𝐹𝑡) ∈ ℝ)
55 simp2 1136 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ0𝑡𝑇) → 𝑚 ∈ ℕ0)
5654, 55reexpcld 14200 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ0𝑡𝑇) → ((𝐹𝑡)↑𝑚) ∈ ℝ)
5747, 51, 48, 56syl3anc 1370 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ0 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡𝑇) → ((𝐹𝑡)↑𝑚) ∈ ℝ)
58 eqid 2735 . . . . . . . . . . . 12 (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) = (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚))
5958fvmpt2 7027 . . . . . . . . . . 11 ((𝑡𝑇 ∧ ((𝐹𝑡)↑𝑚) ∈ ℝ) → ((𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚))‘𝑡) = ((𝐹𝑡)↑𝑚))
6059eqcomd 2741 . . . . . . . . . 10 ((𝑡𝑇 ∧ ((𝐹𝑡)↑𝑚) ∈ ℝ) → ((𝐹𝑡)↑𝑚) = ((𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚))‘𝑡))
6148, 57, 60syl2anc 584 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ0 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡𝑇) → ((𝐹𝑡)↑𝑚) = ((𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚))‘𝑡))
6261oveq1d 7446 . . . . . . . 8 (((𝜑𝑚 ∈ ℕ0 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡𝑇) → (((𝐹𝑡)↑𝑚) · (𝐹𝑡)) = (((𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚))‘𝑡) · (𝐹𝑡)))
6346, 62mpteq2da 5246 . . . . . . 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 2921 . . . . . . . . . 10 𝑡 𝑓 = (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚))
66 stoweidlem19.1 . . . . . . . . . . 11 𝑡𝐹
6766nfeq2 2921 . . . . . . . . . 10 𝑡 𝑔 = 𝐹
68 stoweidlem19.4 . . . . . . . . . 10 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
6965, 67, 68stoweidlem6 45962 . . . . . . . . 9 ((𝜑 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴𝐹𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚))‘𝑡) · (𝐹𝑡))) ∈ 𝐴)
7064, 69mpd3an3 1461 . . . . . . . 8 ((𝜑 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚))‘𝑡) · (𝐹𝑡))) ∈ 𝐴)
71703adant2 1130 . . . . . . 7 ((𝜑𝑚 ∈ ℕ0 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚))‘𝑡) · (𝐹𝑡))) ∈ 𝐴)
7263, 71eqeltrd 2839 . . . . . 6 ((𝜑𝑚 ∈ ℕ0 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) → (𝑡𝑇 ↦ (((𝐹𝑡)↑𝑚) · (𝐹𝑡))) ∈ 𝐴)
7353, 72eqeltrd 2839 . . . . 5 ((𝜑𝑚 ∈ ℕ0 ∧ (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡)↑(𝑚 + 1))) ∈ 𝐴)
7439, 40, 42, 73syl3anc 1370 . . . 4 (((𝑚 ∈ ℕ0 ∧ (𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴)) ∧ 𝜑) → (𝑡𝑇 ↦ ((𝐹𝑡)↑(𝑚 + 1))) ∈ 𝐴)
7574exp31 419 . . 3 (𝑚 ∈ ℕ0 → ((𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑚)) ∈ 𝐴) → (𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑(𝑚 + 1))) ∈ 𝐴)))
765, 9, 13, 17, 38, 75nn0ind 12711 . 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 395  w3a 1086   = wceq 1537  wnf 1780  wcel 2106  wnfc 2888  cmpt 5231  wf 6559  cfv 6563  (class class class)co 7431  cc 11151  cr 11152  0cc0 11153  1c1 11154   + caddc 11156   · cmul 11158  0cn0 12524  cexp 14099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-n0 12525  df-z 12612  df-uz 12877  df-seq 14040  df-exp 14100
This theorem is referenced by:  stoweidlem40  45996
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