Theorem stoweidlem40 46055

Description: This lemma proves that q_n is in the subalgebra, as in the proof of Lemma 1 in [BrosowskiDeutsh] p. 90. Q is used to represent q_n in the paper, N is used to represent n in the paper, and M is used to represent k^n in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Hypotheses

Ref	Expression
stoweidlem40.1	⊢ Ⅎ𝑡𝑃
stoweidlem40.2	⊢ Ⅎ𝑡𝜑
stoweidlem40.3	⊢ 𝑄 = (𝑡 ∈ 𝑇 ↦ ((1 − ((𝑃‘𝑡)↑𝑁))↑𝑀))
stoweidlem40.4	⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ (1 − ((𝑃‘𝑡)↑𝑁)))
stoweidlem40.5	⊢ 𝐺 = (𝑡 ∈ 𝑇 ↦ 1)
stoweidlem40.6	⊢ 𝐻 = (𝑡 ∈ 𝑇 ↦ ((𝑃‘𝑡)↑𝑁))
stoweidlem40.7	⊢ (𝜑 → 𝑃 ∈ 𝐴)
stoweidlem40.8	⊢ (𝜑 → 𝑃:𝑇⟶ℝ)
stoweidlem40.9	⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ)
stoweidlem40.10	⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)
stoweidlem40.11	⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
stoweidlem40.12	⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴)
stoweidlem40.13	⊢ (𝜑 → 𝑁 ∈ ℕ)
stoweidlem40.14	⊢ (𝜑 → 𝑀 ∈ ℕ)

Assertion

Ref	Expression
stoweidlem40	⊢ (𝜑 → 𝑄 ∈ 𝐴)

Distinct variable groups:   𝑓,𝑔,𝑡,𝐴   𝑓,𝐹,𝑔   𝑓,𝐺,𝑔   𝑓,𝐻,𝑔   𝑃,𝑓,𝑔   𝑇,𝑓,𝑔,𝑡   𝜑,𝑓,𝑔   𝑥,𝑡,𝐴   𝑡,𝑀   𝑡,𝑁   𝑥,𝑇   𝜑,𝑥

Allowed substitution hints:   𝜑(𝑡)   𝑃(𝑥,𝑡)   𝑄(𝑥,𝑡,𝑓,𝑔)   𝐹(𝑥,𝑡)   𝐺(𝑥,𝑡)   𝐻(𝑥,𝑡)   𝑀(𝑥,𝑓,𝑔)   𝑁(𝑥,𝑓,𝑔)

Proof of Theorem stoweidlem40

Step	Hyp	Ref	Expression
1		stoweidlem40.3	. . 3 ⊢ 𝑄 = (𝑡 ∈ 𝑇 ↦ ((1 − ((𝑃‘𝑡)↑𝑁))↑𝑀))
2		stoweidlem40.2	. . . 4 ⊢ Ⅎ𝑡𝜑
3		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇)
4		1red 11262	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 1 ∈ ℝ)
5		stoweidlem40.8	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃:𝑇⟶ℝ)
6	5	ffvelcdmda 7104	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑃‘𝑡) ∈ ℝ)
7		stoweidlem40.13	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ)
8	7	nnnn0d 12587	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
9	8	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑁 ∈ ℕ0)
10	6, 9	reexpcld 14203	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝑃‘𝑡)↑𝑁) ∈ ℝ)
11	4, 10	resubcld 11691	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (1 − ((𝑃‘𝑡)↑𝑁)) ∈ ℝ)
12		stoweidlem40.4	. . . . . . . 8 ⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ (1 − ((𝑃‘𝑡)↑𝑁)))
13	12	fvmpt2 7027	. . . . . . 7 ⊢ ((𝑡 ∈ 𝑇 ∧ (1 − ((𝑃‘𝑡)↑𝑁)) ∈ ℝ) → (𝐹‘𝑡) = (1 − ((𝑃‘𝑡)↑𝑁)))
14	3, 11, 13	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) = (1 − ((𝑃‘𝑡)↑𝑁)))
15	14	eqcomd 2743	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (1 − ((𝑃‘𝑡)↑𝑁)) = (𝐹‘𝑡))
16	15	oveq1d 7446	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((1 − ((𝑃‘𝑡)↑𝑁))↑𝑀) = ((𝐹‘𝑡)↑𝑀))
17	2, 16	mpteq2da 5240	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((1 − ((𝑃‘𝑡)↑𝑁))↑𝑀)) = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑀)))
18	1, 17	eqtrid 2789	. 2 ⊢ (𝜑 → 𝑄 = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑀)))
19		nfmpt1 5250	. . . 4 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ (1 − ((𝑃‘𝑡)↑𝑁)))
20	12, 19	nfcxfr 2903	. . 3 ⊢ Ⅎ𝑡𝐹
21		stoweidlem40.9	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ)
22		stoweidlem40.11	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
23		stoweidlem40.12	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴)
24		1re 11261	. . . . . . . . . 10 ⊢ 1 ∈ ℝ
25		stoweidlem40.5	. . . . . . . . . . 11 ⊢ 𝐺 = (𝑡 ∈ 𝑇 ↦ 1)
26	25	fvmpt2 7027	. . . . . . . . . 10 ⊢ ((𝑡 ∈ 𝑇 ∧ 1 ∈ ℝ) → (𝐺‘𝑡) = 1)
27	24, 26	mpan2 691	. . . . . . . . 9 ⊢ (𝑡 ∈ 𝑇 → (𝐺‘𝑡) = 1)
28	27	eqcomd 2743	. . . . . . . 8 ⊢ (𝑡 ∈ 𝑇 → 1 = (𝐺‘𝑡))
29	28	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 1 = (𝐺‘𝑡))
30		stoweidlem40.6	. . . . . . . . . 10 ⊢ 𝐻 = (𝑡 ∈ 𝑇 ↦ ((𝑃‘𝑡)↑𝑁))
31	30	fvmpt2 7027	. . . . . . . . 9 ⊢ ((𝑡 ∈ 𝑇 ∧ ((𝑃‘𝑡)↑𝑁) ∈ ℝ) → (𝐻‘𝑡) = ((𝑃‘𝑡)↑𝑁))
32	3, 10, 31	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐻‘𝑡) = ((𝑃‘𝑡)↑𝑁))
33	32	eqcomd 2743	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝑃‘𝑡)↑𝑁) = (𝐻‘𝑡))
34	29, 33	oveq12d 7449	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (1 − ((𝑃‘𝑡)↑𝑁)) = ((𝐺‘𝑡) − (𝐻‘𝑡)))
35	2, 34	mpteq2da 5240	. . . . 5 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ (1 − ((𝑃‘𝑡)↑𝑁))) = (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) − (𝐻‘𝑡))))
36	12, 35	eqtrid 2789	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) − (𝐻‘𝑡))))
37	23	stoweidlem4 46019	. . . . . . 7 ⊢ ((𝜑 ∧ 1 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 1) ∈ 𝐴)
38	24, 37	mpan2 691	. . . . . 6 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ 1) ∈ 𝐴)
39	25, 38	eqeltrid 2845	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ 𝐴)
40		stoweidlem40.1	. . . . . . 7 ⊢ Ⅎ𝑡𝑃
41		stoweidlem40.7	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
42	40, 2, 21, 22, 23, 41, 8	stoweidlem19 46034	. . . . . 6 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝑃‘𝑡)↑𝑁)) ∈ 𝐴)
43	30, 42	eqeltrid 2845	. . . . 5 ⊢ (𝜑 → 𝐻 ∈ 𝐴)
44		nfmpt1 5250	. . . . . . 7 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ 1)
45	25, 44	nfcxfr 2903	. . . . . 6 ⊢ Ⅎ𝑡𝐺
46		nfmpt1 5250	. . . . . . 7 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ ((𝑃‘𝑡)↑𝑁))
47	30, 46	nfcxfr 2903	. . . . . 6 ⊢ Ⅎ𝑡𝐻
48		stoweidlem40.10	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)
49	45, 47, 2, 21, 48, 22, 23	stoweidlem33 46048	. . . . 5 ⊢ ((𝜑 ∧ 𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) − (𝐻‘𝑡))) ∈ 𝐴)
50	39, 43, 49	mpd3an23 1465	. . . 4 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) − (𝐻‘𝑡))) ∈ 𝐴)
51	36, 50	eqeltrd 2841	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐴)
52		stoweidlem40.14	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ)
53	52	nnnn0d 12587	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℕ0)
54	20, 2, 21, 22, 23, 51, 53	stoweidlem19 46034	. 2 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑀)) ∈ 𝐴)
55	18, 54	eqeltrd 2841	1 ⊢ (𝜑 → 𝑄 ∈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1540  Ⅎwnf 1783   ∈ wcel 2108  Ⅎwnfc 2890   ↦ cmpt 5225  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431  ℝcr 11154  1c1 11156   + caddc 11158   · cmul 11160   − cmin 11492  ℕcn 12266  ℕ₀cn0 12526  ↑cexp 14102

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-n0 12527  df-z 12614  df-uz 12879  df-seq 14043  df-exp 14103

This theorem is referenced by:  stoweidlem45  46060