Theorem stoweidlem40 38933

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 476	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇)
4		1red 9934	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 1 ∈ ℝ)
5		stoweidlem40.8	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃:𝑇⟶ℝ)
6	5	fnvinran 38196	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑃‘𝑡) ∈ ℝ)
7		stoweidlem40.13	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ)
8	7	nnnn0d 11228	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
9	8	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑁 ∈ ℕ0)
10	6, 9	reexpcld 12887	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝑃‘𝑡)↑𝑁) ∈ ℝ)
11	4, 10	resubcld 10337	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (1 − ((𝑃‘𝑡)↑𝑁)) ∈ ℝ)
12		stoweidlem40.4	. . . . . . . 8 ⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ (1 − ((𝑃‘𝑡)↑𝑁)))
13	12	fvmpt2 6200	. . . . . . 7 ⊢ ((𝑡 ∈ 𝑇 ∧ (1 − ((𝑃‘𝑡)↑𝑁)) ∈ ℝ) → (𝐹‘𝑡) = (1 − ((𝑃‘𝑡)↑𝑁)))
14	3, 11, 13	syl2anc 691	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) = (1 − ((𝑃‘𝑡)↑𝑁)))
15	14	eqcomd 2616	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (1 − ((𝑃‘𝑡)↑𝑁)) = (𝐹‘𝑡))
16	15	oveq1d 6564	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((1 − ((𝑃‘𝑡)↑𝑁))↑𝑀) = ((𝐹‘𝑡)↑𝑀))
17	2, 16	mpteq2da 4671	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((1 − ((𝑃‘𝑡)↑𝑁))↑𝑀)) = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑀)))
18	1, 17	syl5eq 2656	. 2 ⊢ (𝜑 → 𝑄 = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑀)))
19		nfmpt1 4675	. . . 4 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ (1 − ((𝑃‘𝑡)↑𝑁)))
20	12, 19	nfcxfr 2749	. . 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 9918	. . . . . . . . . 10 ⊢ 1 ∈ ℝ
25		stoweidlem40.5	. . . . . . . . . . 11 ⊢ 𝐺 = (𝑡 ∈ 𝑇 ↦ 1)
26	25	fvmpt2 6200	. . . . . . . . . 10 ⊢ ((𝑡 ∈ 𝑇 ∧ 1 ∈ ℝ) → (𝐺‘𝑡) = 1)
27	24, 26	mpan2 703	. . . . . . . . 9 ⊢ (𝑡 ∈ 𝑇 → (𝐺‘𝑡) = 1)
28	27	eqcomd 2616	. . . . . . . 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 6200	. . . . . . . . 9 ⊢ ((𝑡 ∈ 𝑇 ∧ ((𝑃‘𝑡)↑𝑁) ∈ ℝ) → (𝐻‘𝑡) = ((𝑃‘𝑡)↑𝑁))
32	3, 10, 31	syl2anc 691	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐻‘𝑡) = ((𝑃‘𝑡)↑𝑁))
33	32	eqcomd 2616	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝑃‘𝑡)↑𝑁) = (𝐻‘𝑡))
34	29, 33	oveq12d 6567	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (1 − ((𝑃‘𝑡)↑𝑁)) = ((𝐺‘𝑡) − (𝐻‘𝑡)))
35	2, 34	mpteq2da 4671	. . . . 5 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ (1 − ((𝑃‘𝑡)↑𝑁))) = (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) − (𝐻‘𝑡))))
36	12, 35	syl5eq 2656	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) − (𝐻‘𝑡))))
37	23	stoweidlem4 38897	. . . . . . 7 ⊢ ((𝜑 ∧ 1 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 1) ∈ 𝐴)
38	24, 37	mpan2 703	. . . . . 6 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ 1) ∈ 𝐴)
39	25, 38	syl5eqel 2692	. . . . 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 38912	. . . . . 6 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝑃‘𝑡)↑𝑁)) ∈ 𝐴)
43	30, 42	syl5eqel 2692	. . . . 5 ⊢ (𝜑 → 𝐻 ∈ 𝐴)
44		nfmpt1 4675	. . . . . . 7 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ 1)
45	25, 44	nfcxfr 2749	. . . . . 6 ⊢ Ⅎ𝑡𝐺
46		nfmpt1 4675	. . . . . . 7 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ ((𝑃‘𝑡)↑𝑁))
47	30, 46	nfcxfr 2749	. . . . . 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 38926	. . . . 5 ⊢ ((𝜑 ∧ 𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) − (𝐻‘𝑡))) ∈ 𝐴)
50	39, 43, 49	mpd3an23 1418	. . . 4 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) − (𝐻‘𝑡))) ∈ 𝐴)
51	36, 50	eqeltrd 2688	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐴)
52		stoweidlem40.14	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ)
53	52	nnnn0d 11228	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℕ0)
54	20, 2, 21, 22, 23, 51, 53	stoweidlem19 38912	. 2 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑀)) ∈ 𝐴)
55	18, 54	eqeltrd 2688	1 ⊢ (𝜑 → 𝑄 ∈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475  Ⅎwnf 1699   ∈ wcel 1977  Ⅎwnfc 2738   ↦ cmpt 4643  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  ℝcr 9814  1c1 9816   + caddc 9818   · cmul 9820   − cmin 10145  ℕcn 10897  ℕ₀cn0 11169  ↑cexp 12722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-seq 12664  df-exp 12723

This theorem is referenced by:  stoweidlem45  38938