Theorem stoweidlem40 45056

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 11220	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 1 ∈ ℝ)
5		stoweidlem40.8	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃:𝑇⟶ℝ)
6	5	ffvelcdmda 7087	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑃‘𝑡) ∈ ℝ)
7		stoweidlem40.13	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ)
8	7	nnnn0d 12537	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
9	8	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑁 ∈ ℕ0)
10	6, 9	reexpcld 14133	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝑃‘𝑡)↑𝑁) ∈ ℝ)
11	4, 10	resubcld 11647	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (1 − ((𝑃‘𝑡)↑𝑁)) ∈ ℝ)
12		stoweidlem40.4	. . . . . . . 8 ⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ (1 − ((𝑃‘𝑡)↑𝑁)))
13	12	fvmpt2 7010	. . . . . . 7 ⊢ ((𝑡 ∈ 𝑇 ∧ (1 − ((𝑃‘𝑡)↑𝑁)) ∈ ℝ) → (𝐹‘𝑡) = (1 − ((𝑃‘𝑡)↑𝑁)))
14	3, 11, 13	syl2anc 583	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) = (1 − ((𝑃‘𝑡)↑𝑁)))
15	14	eqcomd 2737	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (1 − ((𝑃‘𝑡)↑𝑁)) = (𝐹‘𝑡))
16	15	oveq1d 7427	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((1 − ((𝑃‘𝑡)↑𝑁))↑𝑀) = ((𝐹‘𝑡)↑𝑀))
17	2, 16	mpteq2da 5247	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((1 − ((𝑃‘𝑡)↑𝑁))↑𝑀)) = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑀)))
18	1, 17	eqtrid 2783	. 2 ⊢ (𝜑 → 𝑄 = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑀)))
19		nfmpt1 5257	. . . 4 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ (1 − ((𝑃‘𝑡)↑𝑁)))
20	12, 19	nfcxfr 2900	. . 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 11219	. . . . . . . . . 10 ⊢ 1 ∈ ℝ
25		stoweidlem40.5	. . . . . . . . . . 11 ⊢ 𝐺 = (𝑡 ∈ 𝑇 ↦ 1)
26	25	fvmpt2 7010	. . . . . . . . . 10 ⊢ ((𝑡 ∈ 𝑇 ∧ 1 ∈ ℝ) → (𝐺‘𝑡) = 1)
27	24, 26	mpan2 688	. . . . . . . . 9 ⊢ (𝑡 ∈ 𝑇 → (𝐺‘𝑡) = 1)
28	27	eqcomd 2737	. . . . . . . 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 7010	. . . . . . . . 9 ⊢ ((𝑡 ∈ 𝑇 ∧ ((𝑃‘𝑡)↑𝑁) ∈ ℝ) → (𝐻‘𝑡) = ((𝑃‘𝑡)↑𝑁))
32	3, 10, 31	syl2anc 583	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐻‘𝑡) = ((𝑃‘𝑡)↑𝑁))
33	32	eqcomd 2737	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝑃‘𝑡)↑𝑁) = (𝐻‘𝑡))
34	29, 33	oveq12d 7430	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (1 − ((𝑃‘𝑡)↑𝑁)) = ((𝐺‘𝑡) − (𝐻‘𝑡)))
35	2, 34	mpteq2da 5247	. . . . 5 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ (1 − ((𝑃‘𝑡)↑𝑁))) = (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) − (𝐻‘𝑡))))
36	12, 35	eqtrid 2783	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) − (𝐻‘𝑡))))
37	23	stoweidlem4 45020	. . . . . . 7 ⊢ ((𝜑 ∧ 1 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 1) ∈ 𝐴)
38	24, 37	mpan2 688	. . . . . 6 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ 1) ∈ 𝐴)
39	25, 38	eqeltrid 2836	. . . . 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 45035	. . . . . 6 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝑃‘𝑡)↑𝑁)) ∈ 𝐴)
43	30, 42	eqeltrid 2836	. . . . 5 ⊢ (𝜑 → 𝐻 ∈ 𝐴)
44		nfmpt1 5257	. . . . . . 7 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ 1)
45	25, 44	nfcxfr 2900	. . . . . 6 ⊢ Ⅎ𝑡𝐺
46		nfmpt1 5257	. . . . . . 7 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ ((𝑃‘𝑡)↑𝑁))
47	30, 46	nfcxfr 2900	. . . . . 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 45049	. . . . 5 ⊢ ((𝜑 ∧ 𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) − (𝐻‘𝑡))) ∈ 𝐴)
50	39, 43, 49	mpd3an23 1462	. . . 4 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) − (𝐻‘𝑡))) ∈ 𝐴)
51	36, 50	eqeltrd 2832	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐴)
52		stoweidlem40.14	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ)
53	52	nnnn0d 12537	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℕ0)
54	20, 2, 21, 22, 23, 51, 53	stoweidlem19 45035	. 2 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑀)) ∈ 𝐴)
55	18, 54	eqeltrd 2832	1 ⊢ (𝜑 → 𝑄 ∈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540  Ⅎwnf 1784   ∈ wcel 2105  Ⅎwnfc 2882   ↦ cmpt 5232  ⟶wf 6540  ‘cfv 6544  (class class class)co 7412  ℝcr 11112  1c1 11114   + caddc 11116   · cmul 11118   − cmin 11449  ℕcn 12217  ℕ₀cn0 12477  ↑cexp 14032

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728  ax-cnex 11169  ax-resscn 11170  ax-1cn 11171  ax-icn 11172  ax-addcl 11173  ax-addrcl 11174  ax-mulcl 11175  ax-mulrcl 11176  ax-mulcom 11177  ax-addass 11178  ax-mulass 11179  ax-distr 11180  ax-i2m1 11181  ax-1ne0 11182  ax-1rid 11183  ax-rnegex 11184  ax-rrecex 11185  ax-cnre 11186  ax-pre-lttri 11187  ax-pre-lttrn 11188  ax-pre-ltadd 11189  ax-pre-mulgt0 11190

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7859  df-2nd 7979  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-rdg 8413  df-er 8706  df-en 8943  df-dom 8944  df-sdom 8945  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-nn 12218  df-n0 12478  df-z 12564  df-uz 12828  df-seq 13972  df-exp 14033

This theorem is referenced by:  stoweidlem45  45061