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Theorem stoweidlem40 38737
Description: This lemma proves that qn is in the subalgebra, as in the proof of Lemma 1 in [BrosowskiDeutsh] p. 90. Q is used to represent qn 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 𝑡𝑃
stoweidlem40.2 𝑡𝜑
stoweidlem40.3 𝑄 = (𝑡𝑇 ↦ ((1 − ((𝑃𝑡)↑𝑁))↑𝑀))
stoweidlem40.4 𝐹 = (𝑡𝑇 ↦ (1 − ((𝑃𝑡)↑𝑁)))
stoweidlem40.5 𝐺 = (𝑡𝑇 ↦ 1)
stoweidlem40.6 𝐻 = (𝑡𝑇 ↦ ((𝑃𝑡)↑𝑁))
stoweidlem40.7 (𝜑𝑃𝐴)
stoweidlem40.8 (𝜑𝑃:𝑇⟶ℝ)
stoweidlem40.9 ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)
stoweidlem40.10 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)
stoweidlem40.11 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
stoweidlem40.12 ((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴)
stoweidlem40.13 (𝜑𝑁 ∈ ℕ)
stoweidlem40.14 (𝜑𝑀 ∈ ℕ)
Assertion
Ref Expression
stoweidlem40 (𝜑𝑄𝐴)
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
StepHypRef Expression
1 stoweidlem40.3 . . 3 𝑄 = (𝑡𝑇 ↦ ((1 − ((𝑃𝑡)↑𝑁))↑𝑀))
2 stoweidlem40.2 . . . 4 𝑡𝜑
3 simpr 475 . . . . . . 7 ((𝜑𝑡𝑇) → 𝑡𝑇)
4 1red 9911 . . . . . . . 8 ((𝜑𝑡𝑇) → 1 ∈ ℝ)
5 stoweidlem40.8 . . . . . . . . . 10 (𝜑𝑃:𝑇⟶ℝ)
65fnvinran 37999 . . . . . . . . 9 ((𝜑𝑡𝑇) → (𝑃𝑡) ∈ ℝ)
7 stoweidlem40.13 . . . . . . . . . . 11 (𝜑𝑁 ∈ ℕ)
87nnnn0d 11198 . . . . . . . . . 10 (𝜑𝑁 ∈ ℕ0)
98adantr 479 . . . . . . . . 9 ((𝜑𝑡𝑇) → 𝑁 ∈ ℕ0)
106, 9reexpcld 12842 . . . . . . . 8 ((𝜑𝑡𝑇) → ((𝑃𝑡)↑𝑁) ∈ ℝ)
114, 10resubcld 10309 . . . . . . 7 ((𝜑𝑡𝑇) → (1 − ((𝑃𝑡)↑𝑁)) ∈ ℝ)
12 stoweidlem40.4 . . . . . . . 8 𝐹 = (𝑡𝑇 ↦ (1 − ((𝑃𝑡)↑𝑁)))
1312fvmpt2 6185 . . . . . . 7 ((𝑡𝑇 ∧ (1 − ((𝑃𝑡)↑𝑁)) ∈ ℝ) → (𝐹𝑡) = (1 − ((𝑃𝑡)↑𝑁)))
143, 11, 13syl2anc 690 . . . . . 6 ((𝜑𝑡𝑇) → (𝐹𝑡) = (1 − ((𝑃𝑡)↑𝑁)))
1514eqcomd 2615 . . . . 5 ((𝜑𝑡𝑇) → (1 − ((𝑃𝑡)↑𝑁)) = (𝐹𝑡))
1615oveq1d 6542 . . . 4 ((𝜑𝑡𝑇) → ((1 − ((𝑃𝑡)↑𝑁))↑𝑀) = ((𝐹𝑡)↑𝑀))
172, 16mpteq2da 4665 . . 3 (𝜑 → (𝑡𝑇 ↦ ((1 − ((𝑃𝑡)↑𝑁))↑𝑀)) = (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑀)))
181, 17syl5eq 2655 . 2 (𝜑𝑄 = (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑀)))
19 nfmpt1 4669 . . . 4 𝑡(𝑡𝑇 ↦ (1 − ((𝑃𝑡)↑𝑁)))
2012, 19nfcxfr 2748 . . 3 𝑡𝐹
21 stoweidlem40.9 . . 3 ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)
22 stoweidlem40.11 . . 3 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
23 stoweidlem40.12 . . 3 ((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴)
24 1re 9895 . . . . . . . . . 10 1 ∈ ℝ
25 stoweidlem40.5 . . . . . . . . . . 11 𝐺 = (𝑡𝑇 ↦ 1)
2625fvmpt2 6185 . . . . . . . . . 10 ((𝑡𝑇 ∧ 1 ∈ ℝ) → (𝐺𝑡) = 1)
2724, 26mpan2 702 . . . . . . . . 9 (𝑡𝑇 → (𝐺𝑡) = 1)
2827eqcomd 2615 . . . . . . . 8 (𝑡𝑇 → 1 = (𝐺𝑡))
2928adantl 480 . . . . . . 7 ((𝜑𝑡𝑇) → 1 = (𝐺𝑡))
30 stoweidlem40.6 . . . . . . . . . 10 𝐻 = (𝑡𝑇 ↦ ((𝑃𝑡)↑𝑁))
3130fvmpt2 6185 . . . . . . . . 9 ((𝑡𝑇 ∧ ((𝑃𝑡)↑𝑁) ∈ ℝ) → (𝐻𝑡) = ((𝑃𝑡)↑𝑁))
323, 10, 31syl2anc 690 . . . . . . . 8 ((𝜑𝑡𝑇) → (𝐻𝑡) = ((𝑃𝑡)↑𝑁))
3332eqcomd 2615 . . . . . . 7 ((𝜑𝑡𝑇) → ((𝑃𝑡)↑𝑁) = (𝐻𝑡))
3429, 33oveq12d 6545 . . . . . 6 ((𝜑𝑡𝑇) → (1 − ((𝑃𝑡)↑𝑁)) = ((𝐺𝑡) − (𝐻𝑡)))
352, 34mpteq2da 4665 . . . . 5 (𝜑 → (𝑡𝑇 ↦ (1 − ((𝑃𝑡)↑𝑁))) = (𝑡𝑇 ↦ ((𝐺𝑡) − (𝐻𝑡))))
3612, 35syl5eq 2655 . . . 4 (𝜑𝐹 = (𝑡𝑇 ↦ ((𝐺𝑡) − (𝐻𝑡))))
3723stoweidlem4 38701 . . . . . . 7 ((𝜑 ∧ 1 ∈ ℝ) → (𝑡𝑇 ↦ 1) ∈ 𝐴)
3824, 37mpan2 702 . . . . . 6 (𝜑 → (𝑡𝑇 ↦ 1) ∈ 𝐴)
3925, 38syl5eqel 2691 . . . . 5 (𝜑𝐺𝐴)
40 stoweidlem40.1 . . . . . . 7 𝑡𝑃
41 stoweidlem40.7 . . . . . . 7 (𝜑𝑃𝐴)
4240, 2, 21, 22, 23, 41, 8stoweidlem19 38716 . . . . . 6 (𝜑 → (𝑡𝑇 ↦ ((𝑃𝑡)↑𝑁)) ∈ 𝐴)
4330, 42syl5eqel 2691 . . . . 5 (𝜑𝐻𝐴)
44 nfmpt1 4669 . . . . . . 7 𝑡(𝑡𝑇 ↦ 1)
4525, 44nfcxfr 2748 . . . . . 6 𝑡𝐺
46 nfmpt1 4669 . . . . . . 7 𝑡(𝑡𝑇 ↦ ((𝑃𝑡)↑𝑁))
4730, 46nfcxfr 2748 . . . . . 6 𝑡𝐻
48 stoweidlem40.10 . . . . . 6 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)
4945, 47, 2, 21, 48, 22, 23stoweidlem33 38730 . . . . 5 ((𝜑𝐺𝐴𝐻𝐴) → (𝑡𝑇 ↦ ((𝐺𝑡) − (𝐻𝑡))) ∈ 𝐴)
5039, 43, 49mpd3an23 1417 . . . 4 (𝜑 → (𝑡𝑇 ↦ ((𝐺𝑡) − (𝐻𝑡))) ∈ 𝐴)
5136, 50eqeltrd 2687 . . 3 (𝜑𝐹𝐴)
52 stoweidlem40.14 . . . 4 (𝜑𝑀 ∈ ℕ)
5352nnnn0d 11198 . . 3 (𝜑𝑀 ∈ ℕ0)
5420, 2, 21, 22, 23, 51, 53stoweidlem19 38716 . 2 (𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑀)) ∈ 𝐴)
5518, 54eqeltrd 2687 1 (𝜑𝑄𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030   = wceq 1474  wnf 1698  wcel 1976  wnfc 2737  cmpt 4637  wf 5786  cfv 5790  (class class class)co 6527  cr 9791  1c1 9793   + caddc 9795   · cmul 9797  cmin 10117  cn 10867  0cn0 11139  cexp 12677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10868  df-n0 11140  df-z 11211  df-uz 11520  df-seq 12619  df-exp 12678
This theorem is referenced by:  stoweidlem45  38742
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