Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  stoweidlem44 Structured version   Visualization version   GIF version

Theorem stoweidlem44 42206
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 2818 . . . 4 (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡)) = (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡))
4 eqid 2818 . . . 4 (𝑡𝑇 ↦ (1 / 𝑀)) = (𝑡𝑇 ↦ (1 / 𝑀))
5 stoweidlem44.6 . . . 4 (𝜑𝑀 ∈ ℕ)
65nnrecred 11676 . . . 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 4053 . . . . . 6 {𝐴 ∣ ((𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1))} ⊆ 𝐴
108, 9eqsstri 3998 . . . . 5 𝑄𝐴
11 fss 6520 . . . . 5 ((𝐺:(1...𝑀)⟶𝑄𝑄𝐴) → 𝐺:(1...𝑀)⟶𝐴)
127, 10, 11sylancl 586 . . . 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 2818 . . . . 5 (𝐽 Cn 𝐾) = (𝐽 Cn 𝐾)
19 stoweidlem44.10 . . . . . 6 (𝜑𝐴 ⊆ (𝐽 Cn 𝐾))
2019sselda 3964 . . . . 5 ((𝜑𝑓𝐴) → 𝑓 ∈ (𝐽 Cn 𝐾))
2116, 17, 18, 20fcnre 41159 . . . 4 ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)
221, 2, 3, 4, 5, 6, 12, 13, 14, 15, 21stoweidlem32 42194 . . 3 (𝜑𝑃𝐴)
238, 2, 5, 7, 21stoweidlem38 42200 . . . . . 6 ((𝜑𝑡𝑇) → (0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1))
2423ex 413 . . . . 5 (𝜑 → (𝑡𝑇 → (0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1)))
251, 24ralrimi 3213 . . . 4 (𝜑 → ∀𝑡𝑇 (0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1))
26 stoweidlem44.14 . . . . 5 (𝜑𝑍𝑇)
278, 2, 5, 7, 21, 26stoweidlem37 42199 . . . 4 (𝜑 → (𝑃𝑍) = 0)
28 stoweidlem44.1 . . . . . . . . 9 𝑗𝜑
29 nfv 1906 . . . . . . . . 9 𝑗 𝑡 ∈ (𝑇𝑈)
3028, 29nfan 1891 . . . . . . . 8 𝑗(𝜑𝑡 ∈ (𝑇𝑈))
31 nfv 1906 . . . . . . . 8 𝑗0 < ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡))
32 stoweidlem44.8 . . . . . . . . . 10 (𝜑 → ∀𝑡 ∈ (𝑇𝑈)∃𝑗 ∈ (1...𝑀)0 < ((𝐺𝑗)‘𝑡))
3332r19.21bi 3205 . . . . . . . . 9 ((𝜑𝑡 ∈ (𝑇𝑈)) → ∃𝑗 ∈ (1...𝑀)0 < ((𝐺𝑗)‘𝑡))
34 df-rex 3141 . . . . . . . . 9 (∃𝑗 ∈ (1...𝑀)0 < ((𝐺𝑗)‘𝑡) ↔ ∃𝑗(𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡)))
3533, 34sylib 219 . . . . . . . 8 ((𝜑𝑡 ∈ (𝑇𝑈)) → ∃𝑗(𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡)))
366ad2antrr 722 . . . . . . . . 9 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → (1 / 𝑀) ∈ ℝ)
37 simpll 763 . . . . . . . . . 10 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → 𝜑)
38 eldifi 4100 . . . . . . . . . . 11 (𝑡 ∈ (𝑇𝑈) → 𝑡𝑇)
3938ad2antlr 723 . . . . . . . . . 10 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → 𝑡𝑇)
40 fzfid 13329 . . . . . . . . . . 11 ((𝜑𝑡𝑇) → (1...𝑀) ∈ Fin)
418, 7, 21stoweidlem15 42177 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡𝑇) → (((𝐺𝑖)‘𝑡) ∈ ℝ ∧ 0 ≤ ((𝐺𝑖)‘𝑡) ∧ ((𝐺𝑖)‘𝑡) ≤ 1))
4241an32s 648 . . . . . . . . . . . 12 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → (((𝐺𝑖)‘𝑡) ∈ ℝ ∧ 0 ≤ ((𝐺𝑖)‘𝑡) ∧ ((𝐺𝑖)‘𝑡) ≤ 1))
4342simp1d 1134 . . . . . . . . . . 11 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐺𝑖)‘𝑡) ∈ ℝ)
4440, 43fsumrecl 15079 . . . . . . . . . 10 ((𝜑𝑡𝑇) → Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡) ∈ ℝ)
4537, 39, 44syl2anc 584 . . . . . . . . 9 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡) ∈ ℝ)
465nnred 11641 . . . . . . . . . . 11 (𝜑𝑀 ∈ ℝ)
475nngt0d 11674 . . . . . . . . . . 11 (𝜑 → 0 < 𝑀)
4846, 47recgt0d 11562 . . . . . . . . . 10 (𝜑 → 0 < (1 / 𝑀))
4948ad2antrr 722 . . . . . . . . 9 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → 0 < (1 / 𝑀))
50 0red 10632 . . . . . . . . . . 11 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → 0 ∈ ℝ)
51 simprl 767 . . . . . . . . . . . . . 14 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → 𝑗 ∈ (1...𝑀))
5237, 51, 393jca 1120 . . . . . . . . . . . . 13 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → (𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇))
53 snfi 8582 . . . . . . . . . . . . . . 15 {𝑗} ∈ Fin
5453a1i 11 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) → {𝑗} ∈ Fin)
55 simpl1 1183 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ {𝑗}) → 𝜑)
56 simpl3 1185 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ {𝑗}) → 𝑡𝑇)
57 elsni 4574 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ {𝑗} → 𝑖 = 𝑗)
5857adantl 482 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ {𝑗}) → 𝑖 = 𝑗)
59 simpl2 1184 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ {𝑗}) → 𝑗 ∈ (1...𝑀))
6058, 59eqeltrd 2910 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ {𝑗}) → 𝑖 ∈ (1...𝑀))
6155, 56, 60, 43syl21anc 833 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ {𝑗}) → ((𝐺𝑖)‘𝑡) ∈ ℝ)
6254, 61fsumrecl 15079 . . . . . . . . . . . . 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 10658 . . . . . . . . . . 11 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → (0 + Σ𝑖 ∈ {𝑗} ((𝐺𝑖)‘𝑡)) ∈ ℝ)
65 fzfi 13328 . . . . . . . . . . . . . . 15 (1...𝑀) ∈ Fin
66 diffi 8738 . . . . . . . . . . . . . . 15 ((1...𝑀) ∈ Fin → ((1...𝑀) ∖ {𝑗}) ∈ Fin)
6765, 66mp1i 13 . . . . . . . . . . . . . 14 ((𝜑𝑡𝑇) → ((1...𝑀) ∖ {𝑗}) ∈ Fin)
68 eldifi 4100 . . . . . . . . . . . . . . 15 (𝑖 ∈ ((1...𝑀) ∖ {𝑗}) → 𝑖 ∈ (1...𝑀))
6968, 43sylan2 592 . . . . . . . . . . . . . 14 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ ((1...𝑀) ∖ {𝑗})) → ((𝐺𝑖)‘𝑡) ∈ ℝ)
7067, 69fsumrecl 15079 . . . . . . . . . . . . 13 ((𝜑𝑡𝑇) → Σ𝑖 ∈ ((1...𝑀) ∖ {𝑗})((𝐺𝑖)‘𝑡) ∈ ℝ)
7137, 39, 70syl2anc 584 . . . . . . . . . . . 12 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → Σ𝑖 ∈ ((1...𝑀) ∖ {𝑗})((𝐺𝑖)‘𝑡) ∈ ℝ)
7271, 63readdcld 10658 . . . . . . . . . . 11 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → (Σ𝑖 ∈ ((1...𝑀) ∖ {𝑗})((𝐺𝑖)‘𝑡) + Σ𝑖 ∈ {𝑗} ((𝐺𝑖)‘𝑡)) ∈ ℝ)
73 00id 10803 . . . . . . . . . . . 12 (0 + 0) = 0
74 simprr 769 . . . . . . . . . . . . . 14 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → 0 < ((𝐺𝑗)‘𝑡))
758, 7, 21stoweidlem15 42177 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ (1...𝑀)) ∧ 𝑡𝑇) → (((𝐺𝑗)‘𝑡) ∈ ℝ ∧ 0 ≤ ((𝐺𝑗)‘𝑡) ∧ ((𝐺𝑗)‘𝑡) ≤ 1))
7675simp1d 1134 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ (1...𝑀)) ∧ 𝑡𝑇) → ((𝐺𝑗)‘𝑡) ∈ ℝ)
7737, 51, 39, 76syl21anc 833 . . . . . . . . . . . . . . . 16 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → ((𝐺𝑗)‘𝑡) ∈ ℝ)
7877recnd 10657 . . . . . . . . . . . . . . 15 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → ((𝐺𝑗)‘𝑡) ∈ ℂ)
79 fveq2 6663 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝑗 → (𝐺𝑖) = (𝐺𝑗))
8079fveq1d 6665 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑗 → ((𝐺𝑖)‘𝑡) = ((𝐺𝑗)‘𝑡))
8180sumsn 15089 . . . . . . . . . . . . . . 15 ((𝑗 ∈ (1...𝑀) ∧ ((𝐺𝑗)‘𝑡) ∈ ℂ) → Σ𝑖 ∈ {𝑗} ((𝐺𝑖)‘𝑡) = ((𝐺𝑗)‘𝑡))
8251, 78, 81syl2anc 584 . . . . . . . . . . . . . 14 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → Σ𝑖 ∈ {𝑗} ((𝐺𝑖)‘𝑡) = ((𝐺𝑗)‘𝑡))
8374, 82breqtrrd 5085 . . . . . . . . . . . . 13 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → 0 < Σ𝑖 ∈ {𝑗} ((𝐺𝑖)‘𝑡))
8450, 63, 50, 83ltadd2dd 10787 . . . . . . . . . . . 12 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → (0 + 0) < (0 + Σ𝑖 ∈ {𝑗} ((𝐺𝑖)‘𝑡)))
8573, 84eqbrtrrid 5093 . . . . . . . . . . 11 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → 0 < (0 + Σ𝑖 ∈ {𝑗} ((𝐺𝑖)‘𝑡)))
86 0red 10632 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) → 0 ∈ ℝ)
87703adant2 1123 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) → Σ𝑖 ∈ ((1...𝑀) ∖ {𝑗})((𝐺𝑖)‘𝑡) ∈ ℝ)
88 simpll 763 . . . . . . . . . . . . . . . . 17 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ ((1...𝑀) ∖ {𝑗})) → 𝜑)
8968adantl 482 . . . . . . . . . . . . . . . . 17 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ ((1...𝑀) ∖ {𝑗})) → 𝑖 ∈ (1...𝑀))
90 simplr 765 . . . . . . . . . . . . . . . . 17 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ ((1...𝑀) ∖ {𝑗})) → 𝑡𝑇)
9188, 89, 90, 41syl21anc 833 . . . . . . . . . . . . . . . 16 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ ((1...𝑀) ∖ {𝑗})) → (((𝐺𝑖)‘𝑡) ∈ ℝ ∧ 0 ≤ ((𝐺𝑖)‘𝑡) ∧ ((𝐺𝑖)‘𝑡) ≤ 1))
9291simp2d 1135 . . . . . . . . . . . . . . 15 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ ((1...𝑀) ∖ {𝑗})) → 0 ≤ ((𝐺𝑖)‘𝑡))
9367, 69, 92fsumge0 15138 . . . . . . . . . . . . . 14 ((𝜑𝑡𝑇) → 0 ≤ Σ𝑖 ∈ ((1...𝑀) ∖ {𝑗})((𝐺𝑖)‘𝑡))
94933adant2 1123 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) → 0 ≤ Σ𝑖 ∈ ((1...𝑀) ∖ {𝑗})((𝐺𝑖)‘𝑡))
9586, 87, 62, 94leadd1dd 11242 . . . . . . . . . . . 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 10788 . . . . . . . . . 10 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → 0 < (Σ𝑖 ∈ ((1...𝑀) ∖ {𝑗})((𝐺𝑖)‘𝑡) + Σ𝑖 ∈ {𝑗} ((𝐺𝑖)‘𝑡)))
98 eldifn 4101 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ((1...𝑀) ∖ {𝑗}) → ¬ 𝑥 ∈ {𝑗})
99 imnan 400 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ((1...𝑀) ∖ {𝑗}) → ¬ 𝑥 ∈ {𝑗}) ↔ ¬ (𝑥 ∈ ((1...𝑀) ∖ {𝑗}) ∧ 𝑥 ∈ {𝑗}))
10098, 99mpbi 231 . . . . . . . . . . . . . . 15 ¬ (𝑥 ∈ ((1...𝑀) ∖ {𝑗}) ∧ 𝑥 ∈ {𝑗})
101 elin 4166 . . . . . . . . . . . . . . 15 (𝑥 ∈ (((1...𝑀) ∖ {𝑗}) ∩ {𝑗}) ↔ (𝑥 ∈ ((1...𝑀) ∖ {𝑗}) ∧ 𝑥 ∈ {𝑗}))
102100, 101mtbir 324 . . . . . . . . . . . . . 14 ¬ 𝑥 ∈ (((1...𝑀) ∖ {𝑗}) ∩ {𝑗})
103102nel0 4308 . . . . . . . . . . . . 13 (((1...𝑀) ∖ {𝑗}) ∩ {𝑗}) = ∅
104103a1i 11 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) → (((1...𝑀) ∖ {𝑗}) ∩ {𝑗}) = ∅)
105 undif1 4420 . . . . . . . . . . . . 13 (((1...𝑀) ∖ {𝑗}) ∪ {𝑗}) = ((1...𝑀) ∪ {𝑗})
106 snssi 4733 . . . . . . . . . . . . . . 15 (𝑗 ∈ (1...𝑀) → {𝑗} ⊆ (1...𝑀))
1071063ad2ant2 1126 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) → {𝑗} ⊆ (1...𝑀))
108 ssequn2 4156 . . . . . . . . . . . . . 14 ({𝑗} ⊆ (1...𝑀) ↔ ((1...𝑀) ∪ {𝑗}) = (1...𝑀))
109107, 108sylib 219 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) → ((1...𝑀) ∪ {𝑗}) = (1...𝑀))
110105, 109syl5req 2866 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) → (1...𝑀) = (((1...𝑀) ∖ {𝑗}) ∪ {𝑗}))
111 fzfid 13329 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) → (1...𝑀) ∈ Fin)
112433adantl2 1159 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐺𝑖)‘𝑡) ∈ ℝ)
113112recnd 10657 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐺𝑖)‘𝑡) ∈ ℂ)
114104, 110, 111, 113fsumsplit 15085 . . . . . . . . . . 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 5085 . . . . . . . . 9 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → 0 < Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡))
11736, 45, 49, 116mulgt0d 10783 . . . . . . . 8 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → 0 < ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡)))
11830, 31, 35, 117exlimdd 2210 . . . . . . 7 ((𝜑𝑡 ∈ (𝑇𝑈)) → 0 < ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡)))
1198, 2, 5, 7, 21stoweidlem30 42192 . . . . . . . 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 5085 . . . . . 6 ((𝜑𝑡 ∈ (𝑇𝑈)) → 0 < (𝑃𝑡))
122121ex 413 . . . . 5 (𝜑 → (𝑡 ∈ (𝑇𝑈) → 0 < (𝑃𝑡)))
1231, 122ralrimi 3213 . . . 4 (𝜑 → ∀𝑡 ∈ (𝑇𝑈)0 < (𝑃𝑡))
12425, 27, 1233jca 1120 . . 3 (𝜑 → (∀𝑡𝑇 (0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1) ∧ (𝑃𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑃𝑡)))
125 eleq1 2897 . . . . . 6 (𝑝 = 𝑃 → (𝑝𝐴𝑃𝐴))
126 nfmpt1 5155 . . . . . . . . . 10 𝑡(𝑡𝑇 ↦ ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡)))
1272, 126nfcxfr 2972 . . . . . . . . 9 𝑡𝑃
128127nfeq2 2992 . . . . . . . 8 𝑡 𝑝 = 𝑃
129 fveq1 6662 . . . . . . . . . 10 (𝑝 = 𝑃 → (𝑝𝑡) = (𝑃𝑡))
130129breq2d 5069 . . . . . . . . 9 (𝑝 = 𝑃 → (0 ≤ (𝑝𝑡) ↔ 0 ≤ (𝑃𝑡)))
131129breq1d 5067 . . . . . . . . 9 (𝑝 = 𝑃 → ((𝑝𝑡) ≤ 1 ↔ (𝑃𝑡) ≤ 1))
132130, 131anbi12d 630 . . . . . . . 8 (𝑝 = 𝑃 → ((0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ↔ (0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1)))
133128, 132ralbid 3228 . . . . . . 7 (𝑝 = 𝑃 → (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1)))
134 fveq1 6662 . . . . . . . 8 (𝑝 = 𝑃 → (𝑝𝑍) = (𝑃𝑍))
135134eqeq1d 2820 . . . . . . 7 (𝑝 = 𝑃 → ((𝑝𝑍) = 0 ↔ (𝑃𝑍) = 0))
136129breq2d 5069 . . . . . . . 8 (𝑝 = 𝑃 → (0 < (𝑝𝑡) ↔ 0 < (𝑃𝑡)))
137128, 136ralbid 3228 . . . . . . 7 (𝑝 = 𝑃 → (∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡) ↔ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑃𝑡)))
138133, 135, 1373anbi123d 1427 . . . . . 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 630 . . . . 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 3594 . . . 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 695 . 2 (𝜑 → ∃𝑝(𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡))))
143 df-rex 3141 . 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 235 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 396  w3a 1079   = wceq 1528  wex 1771  wnf 1775  wcel 2105  wral 3135  wrex 3136  {crab 3139  cdif 3930  cun 3931  cin 3932  wss 3933  c0 4288  {csn 4557   cuni 4830   class class class wbr 5057  cmpt 5137  ran crn 5549  wf 6344  cfv 6348  (class class class)co 7145  Fincfn 8497  cc 10523  cr 10524  0cc0 10525  1c1 10526   + caddc 10528   · cmul 10530   < clt 10663  cle 10664   / cdiv 11285  cn 11626  (,)cioo 12726  ...cfz 12880  Σcsu 15030  topGenctg 16699   Cn ccn 21760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-sup 8894  df-oi 8962  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-ioo 12730  df-ico 12732  df-fz 12881  df-fzo 13022  df-seq 13358  df-exp 13418  df-hash 13679  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-clim 14833  df-sum 15031  df-topgen 16705  df-top 21430  df-topon 21447  df-bases 21482  df-cn 21763
This theorem is referenced by:  stoweidlem53  42215
  Copyright terms: Public domain W3C validator