Theorem stoweidlem40 46305

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 11135	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 1 ∈ ℝ)
5		stoweidlem40.8	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃:𝑇⟶ℝ)
6	5	ffvelcdmda 7029	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑃‘𝑡) ∈ ℝ)
7		stoweidlem40.13	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ)
8	7	nnnn0d 12464	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
9	8	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑁 ∈ ℕ0)
10	6, 9	reexpcld 14088	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝑃‘𝑡)↑𝑁) ∈ ℝ)
11	4, 10	resubcld 11567	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (1 − ((𝑃‘𝑡)↑𝑁)) ∈ ℝ)
12		stoweidlem40.4	. . . . . . . 8 ⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ (1 − ((𝑃‘𝑡)↑𝑁)))
13	12	fvmpt2 6952	. . . . . . 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 2742	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (1 − ((𝑃‘𝑡)↑𝑁)) = (𝐹‘𝑡))
16	15	oveq1d 7373	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((1 − ((𝑃‘𝑡)↑𝑁))↑𝑀) = ((𝐹‘𝑡)↑𝑀))
17	2, 16	mpteq2da 5190	. . 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 5197	. . . 4 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ (1 − ((𝑃‘𝑡)↑𝑁)))
20	12, 19	nfcxfr 2896	. . 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 11134	. . . . . . . . . 10 ⊢ 1 ∈ ℝ
25		stoweidlem40.5	. . . . . . . . . . 11 ⊢ 𝐺 = (𝑡 ∈ 𝑇 ↦ 1)
26	25	fvmpt2 6952	. . . . . . . . . 10 ⊢ ((𝑡 ∈ 𝑇 ∧ 1 ∈ ℝ) → (𝐺‘𝑡) = 1)
27	24, 26	mpan2 691	. . . . . . . . 9 ⊢ (𝑡 ∈ 𝑇 → (𝐺‘𝑡) = 1)
28	27	eqcomd 2742	. . . . . . . 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 6952	. . . . . . . . 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 2742	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝑃‘𝑡)↑𝑁) = (𝐻‘𝑡))
34	29, 33	oveq12d 7376	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (1 − ((𝑃‘𝑡)↑𝑁)) = ((𝐺‘𝑡) − (𝐻‘𝑡)))
35	2, 34	mpteq2da 5190	. . . . 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 46269	. . . . . . 7 ⊢ ((𝜑 ∧ 1 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 1) ∈ 𝐴)
38	24, 37	mpan2 691	. . . . . 6 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ 1) ∈ 𝐴)
39	25, 38	eqeltrid 2840	. . . . 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 46284	. . . . . 6 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝑃‘𝑡)↑𝑁)) ∈ 𝐴)
43	30, 42	eqeltrid 2840	. . . . 5 ⊢ (𝜑 → 𝐻 ∈ 𝐴)
44		nfmpt1 5197	. . . . . . 7 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ 1)
45	25, 44	nfcxfr 2896	. . . . . 6 ⊢ Ⅎ𝑡𝐺
46		nfmpt1 5197	. . . . . . 7 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ ((𝑃‘𝑡)↑𝑁))
47	30, 46	nfcxfr 2896	. . . . . 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 46298	. . . . 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 2836	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐴)
52		stoweidlem40.14	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ)
53	52	nnnn0d 12464	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℕ0)
54	20, 2, 21, 22, 23, 51, 53	stoweidlem19 46284	. 2 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑀)) ∈ 𝐴)
55	18, 54	eqeltrd 2836	1 ⊢ (𝜑 → 𝑄 ∈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541  Ⅎwnf 1784   ∈ wcel 2113  Ⅎwnfc 2883   ↦ cmpt 5179  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358  ℝcr 11027  1c1 11029   + caddc 11031   · cmul 11033   − cmin 11366  ℕcn 12147  ℕ₀cn0 12403  ↑cexp 13986

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-er 8635  df-en 8886  df-dom 8887  df-sdom 8888  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-nn 12148  df-n0 12404  df-z 12491  df-uz 12754  df-seq 13927  df-exp 13987

This theorem is referenced by:  stoweidlem45  46310