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Theorem stoweidlem40 38933
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 476 . . . . . . 7 ((𝜑𝑡𝑇) → 𝑡𝑇)
4 1red 9934 . . . . . . . 8 ((𝜑𝑡𝑇) → 1 ∈ ℝ)
5 stoweidlem40.8 . . . . . . . . . 10 (𝜑𝑃:𝑇⟶ℝ)
65fnvinran 38196 . . . . . . . . 9 ((𝜑𝑡𝑇) → (𝑃𝑡) ∈ ℝ)
7 stoweidlem40.13 . . . . . . . . . . 11 (𝜑𝑁 ∈ ℕ)
87nnnn0d 11228 . . . . . . . . . 10 (𝜑𝑁 ∈ ℕ0)
98adantr 480 . . . . . . . . 9 ((𝜑𝑡𝑇) → 𝑁 ∈ ℕ0)
106, 9reexpcld 12887 . . . . . . . 8 ((𝜑𝑡𝑇) → ((𝑃𝑡)↑𝑁) ∈ ℝ)
114, 10resubcld 10337 . . . . . . 7 ((𝜑𝑡𝑇) → (1 − ((𝑃𝑡)↑𝑁)) ∈ ℝ)
12 stoweidlem40.4 . . . . . . . 8 𝐹 = (𝑡𝑇 ↦ (1 − ((𝑃𝑡)↑𝑁)))
1312fvmpt2 6200 . . . . . . 7 ((𝑡𝑇 ∧ (1 − ((𝑃𝑡)↑𝑁)) ∈ ℝ) → (𝐹𝑡) = (1 − ((𝑃𝑡)↑𝑁)))
143, 11, 13syl2anc 691 . . . . . 6 ((𝜑𝑡𝑇) → (𝐹𝑡) = (1 − ((𝑃𝑡)↑𝑁)))
1514eqcomd 2616 . . . . 5 ((𝜑𝑡𝑇) → (1 − ((𝑃𝑡)↑𝑁)) = (𝐹𝑡))
1615oveq1d 6564 . . . 4 ((𝜑𝑡𝑇) → ((1 − ((𝑃𝑡)↑𝑁))↑𝑀) = ((𝐹𝑡)↑𝑀))
172, 16mpteq2da 4671 . . 3 (𝜑 → (𝑡𝑇 ↦ ((1 − ((𝑃𝑡)↑𝑁))↑𝑀)) = (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑀)))
181, 17syl5eq 2656 . 2 (𝜑𝑄 = (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑀)))
19 nfmpt1 4675 . . . 4 𝑡(𝑡𝑇 ↦ (1 − ((𝑃𝑡)↑𝑁)))
2012, 19nfcxfr 2749 . . 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 9918 . . . . . . . . . 10 1 ∈ ℝ
25 stoweidlem40.5 . . . . . . . . . . 11 𝐺 = (𝑡𝑇 ↦ 1)
2625fvmpt2 6200 . . . . . . . . . 10 ((𝑡𝑇 ∧ 1 ∈ ℝ) → (𝐺𝑡) = 1)
2724, 26mpan2 703 . . . . . . . . 9 (𝑡𝑇 → (𝐺𝑡) = 1)
2827eqcomd 2616 . . . . . . . 8 (𝑡𝑇 → 1 = (𝐺𝑡))
2928adantl 481 . . . . . . 7 ((𝜑𝑡𝑇) → 1 = (𝐺𝑡))
30 stoweidlem40.6 . . . . . . . . . 10 𝐻 = (𝑡𝑇 ↦ ((𝑃𝑡)↑𝑁))
3130fvmpt2 6200 . . . . . . . . 9 ((𝑡𝑇 ∧ ((𝑃𝑡)↑𝑁) ∈ ℝ) → (𝐻𝑡) = ((𝑃𝑡)↑𝑁))
323, 10, 31syl2anc 691 . . . . . . . 8 ((𝜑𝑡𝑇) → (𝐻𝑡) = ((𝑃𝑡)↑𝑁))
3332eqcomd 2616 . . . . . . 7 ((𝜑𝑡𝑇) → ((𝑃𝑡)↑𝑁) = (𝐻𝑡))
3429, 33oveq12d 6567 . . . . . 6 ((𝜑𝑡𝑇) → (1 − ((𝑃𝑡)↑𝑁)) = ((𝐺𝑡) − (𝐻𝑡)))
352, 34mpteq2da 4671 . . . . 5 (𝜑 → (𝑡𝑇 ↦ (1 − ((𝑃𝑡)↑𝑁))) = (𝑡𝑇 ↦ ((𝐺𝑡) − (𝐻𝑡))))
3612, 35syl5eq 2656 . . . 4 (𝜑𝐹 = (𝑡𝑇 ↦ ((𝐺𝑡) − (𝐻𝑡))))
3723stoweidlem4 38897 . . . . . . 7 ((𝜑 ∧ 1 ∈ ℝ) → (𝑡𝑇 ↦ 1) ∈ 𝐴)
3824, 37mpan2 703 . . . . . 6 (𝜑 → (𝑡𝑇 ↦ 1) ∈ 𝐴)
3925, 38syl5eqel 2692 . . . . 5 (𝜑𝐺𝐴)
40 stoweidlem40.1 . . . . . . 7 𝑡𝑃
41 stoweidlem40.7 . . . . . . 7 (𝜑𝑃𝐴)
4240, 2, 21, 22, 23, 41, 8stoweidlem19 38912 . . . . . 6 (𝜑 → (𝑡𝑇 ↦ ((𝑃𝑡)↑𝑁)) ∈ 𝐴)
4330, 42syl5eqel 2692 . . . . 5 (𝜑𝐻𝐴)
44 nfmpt1 4675 . . . . . . 7 𝑡(𝑡𝑇 ↦ 1)
4525, 44nfcxfr 2749 . . . . . 6 𝑡𝐺
46 nfmpt1 4675 . . . . . . 7 𝑡(𝑡𝑇 ↦ ((𝑃𝑡)↑𝑁))
4730, 46nfcxfr 2749 . . . . . 6 𝑡𝐻
48 stoweidlem40.10 . . . . . 6 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)
4945, 47, 2, 21, 48, 22, 23stoweidlem33 38926 . . . . 5 ((𝜑𝐺𝐴𝐻𝐴) → (𝑡𝑇 ↦ ((𝐺𝑡) − (𝐻𝑡))) ∈ 𝐴)
5039, 43, 49mpd3an23 1418 . . . 4 (𝜑 → (𝑡𝑇 ↦ ((𝐺𝑡) − (𝐻𝑡))) ∈ 𝐴)
5136, 50eqeltrd 2688 . . 3 (𝜑𝐹𝐴)
52 stoweidlem40.14 . . . 4 (𝜑𝑀 ∈ ℕ)
5352nnnn0d 11228 . . 3 (𝜑𝑀 ∈ ℕ0)
5420, 2, 21, 22, 23, 51, 53stoweidlem19 38912 . 2 (𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑀)) ∈ 𝐴)
5518, 54eqeltrd 2688 1 (𝜑𝑄𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1031   = wceq 1475  wnf 1699  wcel 1977  wnfc 2738  cmpt 4643  wf 5800  cfv 5804  (class class class)co 6549  cr 9814  1c1 9816   + caddc 9818   · cmul 9820  cmin 10145  cn 10897  0cn0 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
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