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Theorem stoweidlem44 45432
Description: This lemma is used to prove the existence of a function p as in Lemma 1 of [BrosowskiDeutsh] p. 90: p is in the subalgebra, such that 0 <= p <= 1, p_(t0) = 0, and p > 0 on T - U. Z is used to represent t0 in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
stoweidlem44.1 𝑗𝜑
stoweidlem44.2 𝑡𝜑
stoweidlem44.3 𝐾 = (topGen‘ran (,))
stoweidlem44.4 𝑄 = {𝐴 ∣ ((𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1))}
stoweidlem44.5 𝑃 = (𝑡𝑇 ↦ ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡)))
stoweidlem44.6 (𝜑𝑀 ∈ ℕ)
stoweidlem44.7 (𝜑𝐺:(1...𝑀)⟶𝑄)
stoweidlem44.8 (𝜑 → ∀𝑡 ∈ (𝑇𝑈)∃𝑗 ∈ (1...𝑀)0 < ((𝐺𝑗)‘𝑡))
stoweidlem44.9 𝑇 = 𝐽
stoweidlem44.10 (𝜑𝐴 ⊆ (𝐽 Cn 𝐾))
stoweidlem44.11 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)
stoweidlem44.12 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
stoweidlem44.13 ((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴)
stoweidlem44.14 (𝜑𝑍𝑇)
Assertion
Ref Expression
stoweidlem44 (𝜑 → ∃𝑝𝐴 (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)))
Distinct variable groups:   𝑓,𝑔,𝑖,𝑡,𝐺   𝑓,𝑗,𝑖,𝑡,𝐺   𝐴,𝑓,𝑔   𝑓,𝑀,𝑔,𝑖,𝑡   𝑇,𝑓,𝑔,𝑖,𝑡   𝜑,𝑓,𝑔,𝑖   ,𝑖,𝑗,𝑡,𝐺   𝐴,   𝑇,,𝑗   ,𝑍,𝑖,𝑡   𝑥,𝑗,𝑀,𝑡   𝑈,𝑗   𝑡,𝑝,𝑇   𝐴,𝑝   𝑃,𝑝   𝑈,𝑝   𝑍,𝑝   𝑥,𝐴   𝑥,𝑇   𝜑,𝑥
Allowed substitution hints:   𝜑(𝑡,,𝑗,𝑝)   𝐴(𝑡,𝑖,𝑗)   𝑃(𝑥,𝑡,𝑓,𝑔,,𝑖,𝑗)   𝑄(𝑥,𝑡,𝑓,𝑔,,𝑖,𝑗,𝑝)   𝑈(𝑥,𝑡,𝑓,𝑔,,𝑖)   𝐺(𝑥,𝑝)   𝐽(𝑥,𝑡,𝑓,𝑔,,𝑖,𝑗,𝑝)   𝐾(𝑥,𝑡,𝑓,𝑔,,𝑖,𝑗,𝑝)   𝑀(,𝑝)   𝑍(𝑥,𝑓,𝑔,𝑗)

Proof of Theorem stoweidlem44
StepHypRef Expression
1 stoweidlem44.2 . . . 4 𝑡𝜑
2 stoweidlem44.5 . . . 4 𝑃 = (𝑡𝑇 ↦ ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡)))
3 eqid 2728 . . . 4 (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡)) = (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡))
4 eqid 2728 . . . 4 (𝑡𝑇 ↦ (1 / 𝑀)) = (𝑡𝑇 ↦ (1 / 𝑀))
5 stoweidlem44.6 . . . 4 (𝜑𝑀 ∈ ℕ)
65nnrecred 12294 . . . 4 (𝜑 → (1 / 𝑀) ∈ ℝ)
7 stoweidlem44.7 . . . . 5 (𝜑𝐺:(1...𝑀)⟶𝑄)
8 stoweidlem44.4 . . . . . 6 𝑄 = {𝐴 ∣ ((𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1))}
9 ssrab2 4075 . . . . . 6 {𝐴 ∣ ((𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1))} ⊆ 𝐴
108, 9eqsstri 4014 . . . . 5 𝑄𝐴
11 fss 6739 . . . . 5 ((𝐺:(1...𝑀)⟶𝑄𝑄𝐴) → 𝐺:(1...𝑀)⟶𝐴)
127, 10, 11sylancl 585 . . . 4 (𝜑𝐺:(1...𝑀)⟶𝐴)
13 stoweidlem44.11 . . . 4 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)
14 stoweidlem44.12 . . . 4 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
15 stoweidlem44.13 . . . 4 ((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴)
16 stoweidlem44.3 . . . . 5 𝐾 = (topGen‘ran (,))
17 stoweidlem44.9 . . . . 5 𝑇 = 𝐽
18 eqid 2728 . . . . 5 (𝐽 Cn 𝐾) = (𝐽 Cn 𝐾)
19 stoweidlem44.10 . . . . . 6 (𝜑𝐴 ⊆ (𝐽 Cn 𝐾))
2019sselda 3980 . . . . 5 ((𝜑𝑓𝐴) → 𝑓 ∈ (𝐽 Cn 𝐾))
2116, 17, 18, 20fcnre 44387 . . . 4 ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)
221, 2, 3, 4, 5, 6, 12, 13, 14, 15, 21stoweidlem32 45420 . . 3 (𝜑𝑃𝐴)
238, 2, 5, 7, 21stoweidlem38 45426 . . . . . 6 ((𝜑𝑡𝑇) → (0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1))
2423ex 412 . . . . 5 (𝜑 → (𝑡𝑇 → (0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1)))
251, 24ralrimi 3251 . . . 4 (𝜑 → ∀𝑡𝑇 (0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1))
26 stoweidlem44.14 . . . . 5 (𝜑𝑍𝑇)
278, 2, 5, 7, 21, 26stoweidlem37 45425 . . . 4 (𝜑 → (𝑃𝑍) = 0)
28 stoweidlem44.1 . . . . . . . . 9 𝑗𝜑
29 nfv 1910 . . . . . . . . 9 𝑗 𝑡 ∈ (𝑇𝑈)
3028, 29nfan 1895 . . . . . . . 8 𝑗(𝜑𝑡 ∈ (𝑇𝑈))
31 nfv 1910 . . . . . . . 8 𝑗0 < ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡))
32 stoweidlem44.8 . . . . . . . . . 10 (𝜑 → ∀𝑡 ∈ (𝑇𝑈)∃𝑗 ∈ (1...𝑀)0 < ((𝐺𝑗)‘𝑡))
3332r19.21bi 3245 . . . . . . . . 9 ((𝜑𝑡 ∈ (𝑇𝑈)) → ∃𝑗 ∈ (1...𝑀)0 < ((𝐺𝑗)‘𝑡))
34 df-rex 3068 . . . . . . . . 9 (∃𝑗 ∈ (1...𝑀)0 < ((𝐺𝑗)‘𝑡) ↔ ∃𝑗(𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡)))
3533, 34sylib 217 . . . . . . . 8 ((𝜑𝑡 ∈ (𝑇𝑈)) → ∃𝑗(𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡)))
366ad2antrr 725 . . . . . . . . 9 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → (1 / 𝑀) ∈ ℝ)
37 simpll 766 . . . . . . . . . 10 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → 𝜑)
38 eldifi 4125 . . . . . . . . . . 11 (𝑡 ∈ (𝑇𝑈) → 𝑡𝑇)
3938ad2antlr 726 . . . . . . . . . 10 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → 𝑡𝑇)
40 fzfid 13971 . . . . . . . . . . 11 ((𝜑𝑡𝑇) → (1...𝑀) ∈ Fin)
418, 7, 21stoweidlem15 45403 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡𝑇) → (((𝐺𝑖)‘𝑡) ∈ ℝ ∧ 0 ≤ ((𝐺𝑖)‘𝑡) ∧ ((𝐺𝑖)‘𝑡) ≤ 1))
4241an32s 651 . . . . . . . . . . . 12 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → (((𝐺𝑖)‘𝑡) ∈ ℝ ∧ 0 ≤ ((𝐺𝑖)‘𝑡) ∧ ((𝐺𝑖)‘𝑡) ≤ 1))
4342simp1d 1140 . . . . . . . . . . 11 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐺𝑖)‘𝑡) ∈ ℝ)
4440, 43fsumrecl 15713 . . . . . . . . . 10 ((𝜑𝑡𝑇) → Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡) ∈ ℝ)
4537, 39, 44syl2anc 583 . . . . . . . . 9 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡) ∈ ℝ)
465nnred 12258 . . . . . . . . . . 11 (𝜑𝑀 ∈ ℝ)
475nngt0d 12292 . . . . . . . . . . 11 (𝜑 → 0 < 𝑀)
4846, 47recgt0d 12179 . . . . . . . . . 10 (𝜑 → 0 < (1 / 𝑀))
4948ad2antrr 725 . . . . . . . . 9 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → 0 < (1 / 𝑀))
50 0red 11248 . . . . . . . . . . 11 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → 0 ∈ ℝ)
51 simprl 770 . . . . . . . . . . . . . 14 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → 𝑗 ∈ (1...𝑀))
5237, 51, 393jca 1126 . . . . . . . . . . . . 13 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → (𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇))
53 snfi 9069 . . . . . . . . . . . . . . 15 {𝑗} ∈ Fin
5453a1i 11 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) → {𝑗} ∈ Fin)
55 simpl1 1189 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ {𝑗}) → 𝜑)
56 simpl3 1191 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ {𝑗}) → 𝑡𝑇)
57 elsni 4646 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ {𝑗} → 𝑖 = 𝑗)
5857adantl 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ {𝑗}) → 𝑖 = 𝑗)
59 simpl2 1190 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ {𝑗}) → 𝑗 ∈ (1...𝑀))
6058, 59eqeltrd 2829 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ {𝑗}) → 𝑖 ∈ (1...𝑀))
6155, 56, 60, 43syl21anc 837 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ {𝑗}) → ((𝐺𝑖)‘𝑡) ∈ ℝ)
6254, 61fsumrecl 15713 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) → Σ𝑖 ∈ {𝑗} ((𝐺𝑖)‘𝑡) ∈ ℝ)
6352, 62syl 17 . . . . . . . . . . . 12 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → Σ𝑖 ∈ {𝑗} ((𝐺𝑖)‘𝑡) ∈ ℝ)
6450, 63readdcld 11274 . . . . . . . . . . 11 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → (0 + Σ𝑖 ∈ {𝑗} ((𝐺𝑖)‘𝑡)) ∈ ℝ)
65 fzfi 13970 . . . . . . . . . . . . . . 15 (1...𝑀) ∈ Fin
66 diffi 9204 . . . . . . . . . . . . . . 15 ((1...𝑀) ∈ Fin → ((1...𝑀) ∖ {𝑗}) ∈ Fin)
6765, 66mp1i 13 . . . . . . . . . . . . . 14 ((𝜑𝑡𝑇) → ((1...𝑀) ∖ {𝑗}) ∈ Fin)
68 eldifi 4125 . . . . . . . . . . . . . . 15 (𝑖 ∈ ((1...𝑀) ∖ {𝑗}) → 𝑖 ∈ (1...𝑀))
6968, 43sylan2 592 . . . . . . . . . . . . . 14 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ ((1...𝑀) ∖ {𝑗})) → ((𝐺𝑖)‘𝑡) ∈ ℝ)
7067, 69fsumrecl 15713 . . . . . . . . . . . . 13 ((𝜑𝑡𝑇) → Σ𝑖 ∈ ((1...𝑀) ∖ {𝑗})((𝐺𝑖)‘𝑡) ∈ ℝ)
7137, 39, 70syl2anc 583 . . . . . . . . . . . 12 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → Σ𝑖 ∈ ((1...𝑀) ∖ {𝑗})((𝐺𝑖)‘𝑡) ∈ ℝ)
7271, 63readdcld 11274 . . . . . . . . . . 11 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → (Σ𝑖 ∈ ((1...𝑀) ∖ {𝑗})((𝐺𝑖)‘𝑡) + Σ𝑖 ∈ {𝑗} ((𝐺𝑖)‘𝑡)) ∈ ℝ)
73 00id 11420 . . . . . . . . . . . 12 (0 + 0) = 0
74 simprr 772 . . . . . . . . . . . . . 14 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → 0 < ((𝐺𝑗)‘𝑡))
758, 7, 21stoweidlem15 45403 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ (1...𝑀)) ∧ 𝑡𝑇) → (((𝐺𝑗)‘𝑡) ∈ ℝ ∧ 0 ≤ ((𝐺𝑗)‘𝑡) ∧ ((𝐺𝑗)‘𝑡) ≤ 1))
7675simp1d 1140 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ (1...𝑀)) ∧ 𝑡𝑇) → ((𝐺𝑗)‘𝑡) ∈ ℝ)
7737, 51, 39, 76syl21anc 837 . . . . . . . . . . . . . . . 16 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → ((𝐺𝑗)‘𝑡) ∈ ℝ)
7877recnd 11273 . . . . . . . . . . . . . . 15 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → ((𝐺𝑗)‘𝑡) ∈ ℂ)
79 fveq2 6897 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝑗 → (𝐺𝑖) = (𝐺𝑗))
8079fveq1d 6899 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑗 → ((𝐺𝑖)‘𝑡) = ((𝐺𝑗)‘𝑡))
8180sumsn 15725 . . . . . . . . . . . . . . 15 ((𝑗 ∈ (1...𝑀) ∧ ((𝐺𝑗)‘𝑡) ∈ ℂ) → Σ𝑖 ∈ {𝑗} ((𝐺𝑖)‘𝑡) = ((𝐺𝑗)‘𝑡))
8251, 78, 81syl2anc 583 . . . . . . . . . . . . . 14 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → Σ𝑖 ∈ {𝑗} ((𝐺𝑖)‘𝑡) = ((𝐺𝑗)‘𝑡))
8374, 82breqtrrd 5176 . . . . . . . . . . . . 13 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → 0 < Σ𝑖 ∈ {𝑗} ((𝐺𝑖)‘𝑡))
8450, 63, 50, 83ltadd2dd 11404 . . . . . . . . . . . 12 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → (0 + 0) < (0 + Σ𝑖 ∈ {𝑗} ((𝐺𝑖)‘𝑡)))
8573, 84eqbrtrrid 5184 . . . . . . . . . . 11 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → 0 < (0 + Σ𝑖 ∈ {𝑗} ((𝐺𝑖)‘𝑡)))
86 0red 11248 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) → 0 ∈ ℝ)
87703adant2 1129 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) → Σ𝑖 ∈ ((1...𝑀) ∖ {𝑗})((𝐺𝑖)‘𝑡) ∈ ℝ)
88 simpll 766 . . . . . . . . . . . . . . . . 17 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ ((1...𝑀) ∖ {𝑗})) → 𝜑)
8968adantl 481 . . . . . . . . . . . . . . . . 17 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ ((1...𝑀) ∖ {𝑗})) → 𝑖 ∈ (1...𝑀))
90 simplr 768 . . . . . . . . . . . . . . . . 17 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ ((1...𝑀) ∖ {𝑗})) → 𝑡𝑇)
9188, 89, 90, 41syl21anc 837 . . . . . . . . . . . . . . . 16 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ ((1...𝑀) ∖ {𝑗})) → (((𝐺𝑖)‘𝑡) ∈ ℝ ∧ 0 ≤ ((𝐺𝑖)‘𝑡) ∧ ((𝐺𝑖)‘𝑡) ≤ 1))
9291simp2d 1141 . . . . . . . . . . . . . . 15 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ ((1...𝑀) ∖ {𝑗})) → 0 ≤ ((𝐺𝑖)‘𝑡))
9367, 69, 92fsumge0 15774 . . . . . . . . . . . . . 14 ((𝜑𝑡𝑇) → 0 ≤ Σ𝑖 ∈ ((1...𝑀) ∖ {𝑗})((𝐺𝑖)‘𝑡))
94933adant2 1129 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) → 0 ≤ Σ𝑖 ∈ ((1...𝑀) ∖ {𝑗})((𝐺𝑖)‘𝑡))
9586, 87, 62, 94leadd1dd 11859 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) → (0 + Σ𝑖 ∈ {𝑗} ((𝐺𝑖)‘𝑡)) ≤ (Σ𝑖 ∈ ((1...𝑀) ∖ {𝑗})((𝐺𝑖)‘𝑡) + Σ𝑖 ∈ {𝑗} ((𝐺𝑖)‘𝑡)))
9652, 95syl 17 . . . . . . . . . . 11 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → (0 + Σ𝑖 ∈ {𝑗} ((𝐺𝑖)‘𝑡)) ≤ (Σ𝑖 ∈ ((1...𝑀) ∖ {𝑗})((𝐺𝑖)‘𝑡) + Σ𝑖 ∈ {𝑗} ((𝐺𝑖)‘𝑡)))
9750, 64, 72, 85, 96ltletrd 11405 . . . . . . . . . 10 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → 0 < (Σ𝑖 ∈ ((1...𝑀) ∖ {𝑗})((𝐺𝑖)‘𝑡) + Σ𝑖 ∈ {𝑗} ((𝐺𝑖)‘𝑡)))
98 eldifn 4126 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ((1...𝑀) ∖ {𝑗}) → ¬ 𝑥 ∈ {𝑗})
99 imnan 399 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ((1...𝑀) ∖ {𝑗}) → ¬ 𝑥 ∈ {𝑗}) ↔ ¬ (𝑥 ∈ ((1...𝑀) ∖ {𝑗}) ∧ 𝑥 ∈ {𝑗}))
10098, 99mpbi 229 . . . . . . . . . . . . . . 15 ¬ (𝑥 ∈ ((1...𝑀) ∖ {𝑗}) ∧ 𝑥 ∈ {𝑗})
101 elin 3963 . . . . . . . . . . . . . . 15 (𝑥 ∈ (((1...𝑀) ∖ {𝑗}) ∩ {𝑗}) ↔ (𝑥 ∈ ((1...𝑀) ∖ {𝑗}) ∧ 𝑥 ∈ {𝑗}))
102100, 101mtbir 323 . . . . . . . . . . . . . 14 ¬ 𝑥 ∈ (((1...𝑀) ∖ {𝑗}) ∩ {𝑗})
103102nel0 4351 . . . . . . . . . . . . 13 (((1...𝑀) ∖ {𝑗}) ∩ {𝑗}) = ∅
104103a1i 11 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) → (((1...𝑀) ∖ {𝑗}) ∩ {𝑗}) = ∅)
105 undif1 4476 . . . . . . . . . . . . 13 (((1...𝑀) ∖ {𝑗}) ∪ {𝑗}) = ((1...𝑀) ∪ {𝑗})
106 snssi 4812 . . . . . . . . . . . . . . 15 (𝑗 ∈ (1...𝑀) → {𝑗} ⊆ (1...𝑀))
1071063ad2ant2 1132 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) → {𝑗} ⊆ (1...𝑀))
108 ssequn2 4183 . . . . . . . . . . . . . 14 ({𝑗} ⊆ (1...𝑀) ↔ ((1...𝑀) ∪ {𝑗}) = (1...𝑀))
109107, 108sylib 217 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) → ((1...𝑀) ∪ {𝑗}) = (1...𝑀))
110105, 109eqtr2id 2781 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) → (1...𝑀) = (((1...𝑀) ∖ {𝑗}) ∪ {𝑗}))
111 fzfid 13971 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) → (1...𝑀) ∈ Fin)
112433adantl2 1165 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐺𝑖)‘𝑡) ∈ ℝ)
113112recnd 11273 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐺𝑖)‘𝑡) ∈ ℂ)
114104, 110, 111, 113fsumsplit 15720 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) → Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡) = (Σ𝑖 ∈ ((1...𝑀) ∖ {𝑗})((𝐺𝑖)‘𝑡) + Σ𝑖 ∈ {𝑗} ((𝐺𝑖)‘𝑡)))
11552, 114syl 17 . . . . . . . . . 10 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡) = (Σ𝑖 ∈ ((1...𝑀) ∖ {𝑗})((𝐺𝑖)‘𝑡) + Σ𝑖 ∈ {𝑗} ((𝐺𝑖)‘𝑡)))
11697, 115breqtrrd 5176 . . . . . . . . 9 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → 0 < Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡))
11736, 45, 49, 116mulgt0d 11400 . . . . . . . 8 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → 0 < ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡)))
11830, 31, 35, 117exlimdd 2209 . . . . . . 7 ((𝜑𝑡 ∈ (𝑇𝑈)) → 0 < ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡)))
1198, 2, 5, 7, 21stoweidlem30 45418 . . . . . . . 8 ((𝜑𝑡𝑇) → (𝑃𝑡) = ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡)))
12038, 119sylan2 592 . . . . . . 7 ((𝜑𝑡 ∈ (𝑇𝑈)) → (𝑃𝑡) = ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡)))
121118, 120breqtrrd 5176 . . . . . 6 ((𝜑𝑡 ∈ (𝑇𝑈)) → 0 < (𝑃𝑡))
122121ex 412 . . . . 5 (𝜑 → (𝑡 ∈ (𝑇𝑈) → 0 < (𝑃𝑡)))
1231, 122ralrimi 3251 . . . 4 (𝜑 → ∀𝑡 ∈ (𝑇𝑈)0 < (𝑃𝑡))
12425, 27, 1233jca 1126 . . 3 (𝜑 → (∀𝑡𝑇 (0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1) ∧ (𝑃𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑃𝑡)))
125 eleq1 2817 . . . . . 6 (𝑝 = 𝑃 → (𝑝𝐴𝑃𝐴))
126 nfmpt1 5256 . . . . . . . . . 10 𝑡(𝑡𝑇 ↦ ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡)))
1272, 126nfcxfr 2897 . . . . . . . . 9 𝑡𝑃
128127nfeq2 2917 . . . . . . . 8 𝑡 𝑝 = 𝑃
129 fveq1 6896 . . . . . . . . . 10 (𝑝 = 𝑃 → (𝑝𝑡) = (𝑃𝑡))
130129breq2d 5160 . . . . . . . . 9 (𝑝 = 𝑃 → (0 ≤ (𝑝𝑡) ↔ 0 ≤ (𝑃𝑡)))
131129breq1d 5158 . . . . . . . . 9 (𝑝 = 𝑃 → ((𝑝𝑡) ≤ 1 ↔ (𝑃𝑡) ≤ 1))
132130, 131anbi12d 631 . . . . . . . 8 (𝑝 = 𝑃 → ((0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ↔ (0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1)))
133128, 132ralbid 3267 . . . . . . 7 (𝑝 = 𝑃 → (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1)))
134 fveq1 6896 . . . . . . . 8 (𝑝 = 𝑃 → (𝑝𝑍) = (𝑃𝑍))
135134eqeq1d 2730 . . . . . . 7 (𝑝 = 𝑃 → ((𝑝𝑍) = 0 ↔ (𝑃𝑍) = 0))
136129breq2d 5160 . . . . . . . 8 (𝑝 = 𝑃 → (0 < (𝑝𝑡) ↔ 0 < (𝑃𝑡)))
137128, 136ralbid 3267 . . . . . . 7 (𝑝 = 𝑃 → (∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡) ↔ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑃𝑡)))
138133, 135, 1373anbi123d 1433 . . . . . 6 (𝑝 = 𝑃 → ((∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)) ↔ (∀𝑡𝑇 (0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1) ∧ (𝑃𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑃𝑡))))
139125, 138anbi12d 631 . . . . 5 (𝑝 = 𝑃 → ((𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡))) ↔ (𝑃𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1) ∧ (𝑃𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑃𝑡)))))
140139spcegv 3584 . . . 4 (𝑃𝐴 → ((𝑃𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1) ∧ (𝑃𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑃𝑡))) → ∃𝑝(𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)))))
14122, 140syl 17 . . 3 (𝜑 → ((𝑃𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1) ∧ (𝑃𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑃𝑡))) → ∃𝑝(𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)))))
14222, 124, 141mp2and 698 . 2 (𝜑 → ∃𝑝(𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡))))
143 df-rex 3068 . 2 (∃𝑝𝐴 (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)) ↔ ∃𝑝(𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡))))
144142, 143sylibr 233 1 (𝜑 → ∃𝑝𝐴 (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1085   = wceq 1534  wex 1774  wnf 1778  wcel 2099  wral 3058  wrex 3067  {crab 3429  cdif 3944  cun 3945  cin 3946  wss 3947  c0 4323  {csn 4629   cuni 4908   class class class wbr 5148  cmpt 5231  ran crn 5679  wf 6544  cfv 6548  (class class class)co 7420  Fincfn 8964  cc 11137  cr 11138  0cc0 11139  1c1 11140   + caddc 11142   · cmul 11144   < clt 11279  cle 11280   / cdiv 11902  cn 12243  (,)cioo 13357  ...cfz 13517  Σcsu 15665  topGenctg 17419   Cn ccn 23141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-inf2 9665  ax-cnex 11195  ax-resscn 11196  ax-1cn 11197  ax-icn 11198  ax-addcl 11199  ax-addrcl 11200  ax-mulcl 11201  ax-mulrcl 11202  ax-mulcom 11203  ax-addass 11204  ax-mulass 11205  ax-distr 11206  ax-i2m1 11207  ax-1ne0 11208  ax-1rid 11209  ax-rnegex 11210  ax-rrecex 11211  ax-cnre 11212  ax-pre-lttri 11213  ax-pre-lttrn 11214  ax-pre-ltadd 11215  ax-pre-mulgt0 11216  ax-pre-sup 11217
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-isom 6557  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-1st 7993  df-2nd 7994  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-er 8725  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-sup 9466  df-oi 9534  df-card 9963  df-pnf 11281  df-mnf 11282  df-xr 11283  df-ltxr 11284  df-le 11285  df-sub 11477  df-neg 11478  df-div 11903  df-nn 12244  df-2 12306  df-3 12307  df-n0 12504  df-z 12590  df-uz 12854  df-rp 13008  df-ioo 13361  df-ico 13363  df-fz 13518  df-fzo 13661  df-seq 14000  df-exp 14060  df-hash 14323  df-cj 15079  df-re 15080  df-im 15081  df-sqrt 15215  df-abs 15216  df-clim 15465  df-sum 15666  df-topgen 17425  df-top 22809  df-topon 22826  df-bases 22862  df-cn 23144
This theorem is referenced by:  stoweidlem53  45441
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