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Theorem stoweidlem44 46064
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 2736 . . . 4 (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡)) = (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡))
4 eqid 2736 . . . 4 (𝑡𝑇 ↦ (1 / 𝑀)) = (𝑡𝑇 ↦ (1 / 𝑀))
5 stoweidlem44.6 . . . 4 (𝜑𝑀 ∈ ℕ)
65nnrecred 12318 . . . 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 4079 . . . . . 6 {𝐴 ∣ ((𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1))} ⊆ 𝐴
108, 9eqsstri 4029 . . . . 5 𝑄𝐴
11 fss 6751 . . . . 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 2736 . . . . 5 (𝐽 Cn 𝐾) = (𝐽 Cn 𝐾)
19 stoweidlem44.10 . . . . . 6 (𝜑𝐴 ⊆ (𝐽 Cn 𝐾))
2019sselda 3982 . . . . 5 ((𝜑𝑓𝐴) → 𝑓 ∈ (𝐽 Cn 𝐾))
2116, 17, 18, 20fcnre 45035 . . . 4 ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)
221, 2, 3, 4, 5, 6, 12, 13, 14, 15, 21stoweidlem32 46052 . . 3 (𝜑𝑃𝐴)
238, 2, 5, 7, 21stoweidlem38 46058 . . . . . 6 ((𝜑𝑡𝑇) → (0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1))
2423ex 412 . . . . 5 (𝜑 → (𝑡𝑇 → (0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1)))
251, 24ralrimi 3256 . . . 4 (𝜑 → ∀𝑡𝑇 (0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1))
26 stoweidlem44.14 . . . . 5 (𝜑𝑍𝑇)
278, 2, 5, 7, 21, 26stoweidlem37 46057 . . . 4 (𝜑 → (𝑃𝑍) = 0)
28 stoweidlem44.1 . . . . . . . . 9 𝑗𝜑
29 nfv 1913 . . . . . . . . 9 𝑗 𝑡 ∈ (𝑇𝑈)
3028, 29nfan 1898 . . . . . . . 8 𝑗(𝜑𝑡 ∈ (𝑇𝑈))
31 nfv 1913 . . . . . . . 8 𝑗0 < ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡))
32 stoweidlem44.8 . . . . . . . . . 10 (𝜑 → ∀𝑡 ∈ (𝑇𝑈)∃𝑗 ∈ (1...𝑀)0 < ((𝐺𝑗)‘𝑡))
3332r19.21bi 3250 . . . . . . . . 9 ((𝜑𝑡 ∈ (𝑇𝑈)) → ∃𝑗 ∈ (1...𝑀)0 < ((𝐺𝑗)‘𝑡))
34 df-rex 3070 . . . . . . . . 9 (∃𝑗 ∈ (1...𝑀)0 < ((𝐺𝑗)‘𝑡) ↔ ∃𝑗(𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡)))
3533, 34sylib 218 . . . . . . . 8 ((𝜑𝑡 ∈ (𝑇𝑈)) → ∃𝑗(𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡)))
366ad2antrr 726 . . . . . . . . 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 4130 . . . . . . . . . . 11 (𝑡 ∈ (𝑇𝑈) → 𝑡𝑇)
3938ad2antlr 727 . . . . . . . . . 10 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → 𝑡𝑇)
40 fzfid 14015 . . . . . . . . . . 11 ((𝜑𝑡𝑇) → (1...𝑀) ∈ Fin)
418, 7, 21stoweidlem15 46035 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡𝑇) → (((𝐺𝑖)‘𝑡) ∈ ℝ ∧ 0 ≤ ((𝐺𝑖)‘𝑡) ∧ ((𝐺𝑖)‘𝑡) ≤ 1))
4241an32s 652 . . . . . . . . . . . 12 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → (((𝐺𝑖)‘𝑡) ∈ ℝ ∧ 0 ≤ ((𝐺𝑖)‘𝑡) ∧ ((𝐺𝑖)‘𝑡) ≤ 1))
4342simp1d 1142 . . . . . . . . . . 11 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐺𝑖)‘𝑡) ∈ ℝ)
4440, 43fsumrecl 15771 . . . . . . . . . 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 12282 . . . . . . . . . . 11 (𝜑𝑀 ∈ ℝ)
475nngt0d 12316 . . . . . . . . . . 11 (𝜑 → 0 < 𝑀)
4846, 47recgt0d 12203 . . . . . . . . . 10 (𝜑 → 0 < (1 / 𝑀))
4948ad2antrr 726 . . . . . . . . 9 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → 0 < (1 / 𝑀))
50 0red 11265 . . . . . . . . . . 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 1128 . . . . . . . . . . . . 13 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → (𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇))
53 snfi 9084 . . . . . . . . . . . . . . 15 {𝑗} ∈ Fin
5453a1i 11 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) → {𝑗} ∈ Fin)
55 simpl1 1191 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ {𝑗}) → 𝜑)
56 simpl3 1193 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ {𝑗}) → 𝑡𝑇)
57 elsni 4642 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ {𝑗} → 𝑖 = 𝑗)
5857adantl 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ {𝑗}) → 𝑖 = 𝑗)
59 simpl2 1192 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ {𝑗}) → 𝑗 ∈ (1...𝑀))
6058, 59eqeltrd 2840 . . . . . . . . . . . . . . 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 15771 . . . . . . . . . . . . 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 11291 . . . . . . . . . . 11 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → (0 + Σ𝑖 ∈ {𝑗} ((𝐺𝑖)‘𝑡)) ∈ ℝ)
65 fzfi 14014 . . . . . . . . . . . . . . 15 (1...𝑀) ∈ Fin
66 diffi 9216 . . . . . . . . . . . . . . 15 ((1...𝑀) ∈ Fin → ((1...𝑀) ∖ {𝑗}) ∈ Fin)
6765, 66mp1i 13 . . . . . . . . . . . . . 14 ((𝜑𝑡𝑇) → ((1...𝑀) ∖ {𝑗}) ∈ Fin)
68 eldifi 4130 . . . . . . . . . . . . . . 15 (𝑖 ∈ ((1...𝑀) ∖ {𝑗}) → 𝑖 ∈ (1...𝑀))
6968, 43sylan2 593 . . . . . . . . . . . . . 14 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ ((1...𝑀) ∖ {𝑗})) → ((𝐺𝑖)‘𝑡) ∈ ℝ)
7067, 69fsumrecl 15771 . . . . . . . . . . . . 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 11291 . . . . . . . . . . 11 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → (Σ𝑖 ∈ ((1...𝑀) ∖ {𝑗})((𝐺𝑖)‘𝑡) + Σ𝑖 ∈ {𝑗} ((𝐺𝑖)‘𝑡)) ∈ ℝ)
73 00id 11437 . . . . . . . . . . . 12 (0 + 0) = 0
74 simprr 772 . . . . . . . . . . . . . 14 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → 0 < ((𝐺𝑗)‘𝑡))
758, 7, 21stoweidlem15 46035 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ (1...𝑀)) ∧ 𝑡𝑇) → (((𝐺𝑗)‘𝑡) ∈ ℝ ∧ 0 ≤ ((𝐺𝑗)‘𝑡) ∧ ((𝐺𝑗)‘𝑡) ≤ 1))
7675simp1d 1142 . . . . . . . . . . . . . . . . 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 11290 . . . . . . . . . . . . . . 15 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → ((𝐺𝑗)‘𝑡) ∈ ℂ)
79 fveq2 6905 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝑗 → (𝐺𝑖) = (𝐺𝑗))
8079fveq1d 6907 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑗 → ((𝐺𝑖)‘𝑡) = ((𝐺𝑗)‘𝑡))
8180sumsn 15783 . . . . . . . . . . . . . . 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 5170 . . . . . . . . . . . . 13 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → 0 < Σ𝑖 ∈ {𝑗} ((𝐺𝑖)‘𝑡))
8450, 63, 50, 83ltadd2dd 11421 . . . . . . . . . . . 12 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → (0 + 0) < (0 + Σ𝑖 ∈ {𝑗} ((𝐺𝑖)‘𝑡)))
8573, 84eqbrtrrid 5178 . . . . . . . . . . 11 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → 0 < (0 + Σ𝑖 ∈ {𝑗} ((𝐺𝑖)‘𝑡)))
86 0red 11265 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) → 0 ∈ ℝ)
87703adant2 1131 . . . . . . . . . . . . 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 1143 . . . . . . . . . . . . . . 15 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ ((1...𝑀) ∖ {𝑗})) → 0 ≤ ((𝐺𝑖)‘𝑡))
9367, 69, 92fsumge0 15832 . . . . . . . . . . . . . 14 ((𝜑𝑡𝑇) → 0 ≤ Σ𝑖 ∈ ((1...𝑀) ∖ {𝑗})((𝐺𝑖)‘𝑡))
94933adant2 1131 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) → 0 ≤ Σ𝑖 ∈ ((1...𝑀) ∖ {𝑗})((𝐺𝑖)‘𝑡))
9586, 87, 62, 94leadd1dd 11878 . . . . . . . . . . . 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 11422 . . . . . . . . . 10 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → 0 < (Σ𝑖 ∈ ((1...𝑀) ∖ {𝑗})((𝐺𝑖)‘𝑡) + Σ𝑖 ∈ {𝑗} ((𝐺𝑖)‘𝑡)))
98 eldifn 4131 . . . . . . . . . . . . . . . 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 230 . . . . . . . . . . . . . . 15 ¬ (𝑥 ∈ ((1...𝑀) ∖ {𝑗}) ∧ 𝑥 ∈ {𝑗})
101 elin 3966 . . . . . . . . . . . . . . 15 (𝑥 ∈ (((1...𝑀) ∖ {𝑗}) ∩ {𝑗}) ↔ (𝑥 ∈ ((1...𝑀) ∖ {𝑗}) ∧ 𝑥 ∈ {𝑗}))
102100, 101mtbir 323 . . . . . . . . . . . . . 14 ¬ 𝑥 ∈ (((1...𝑀) ∖ {𝑗}) ∩ {𝑗})
103102nel0 4353 . . . . . . . . . . . . 13 (((1...𝑀) ∖ {𝑗}) ∩ {𝑗}) = ∅
104103a1i 11 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) → (((1...𝑀) ∖ {𝑗}) ∩ {𝑗}) = ∅)
105 undif1 4475 . . . . . . . . . . . . 13 (((1...𝑀) ∖ {𝑗}) ∪ {𝑗}) = ((1...𝑀) ∪ {𝑗})
106 snssi 4807 . . . . . . . . . . . . . . 15 (𝑗 ∈ (1...𝑀) → {𝑗} ⊆ (1...𝑀))
1071063ad2ant2 1134 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) → {𝑗} ⊆ (1...𝑀))
108 ssequn2 4188 . . . . . . . . . . . . . 14 ({𝑗} ⊆ (1...𝑀) ↔ ((1...𝑀) ∪ {𝑗}) = (1...𝑀))
109107, 108sylib 218 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) → ((1...𝑀) ∪ {𝑗}) = (1...𝑀))
110105, 109eqtr2id 2789 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) → (1...𝑀) = (((1...𝑀) ∖ {𝑗}) ∪ {𝑗}))
111 fzfid 14015 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) → (1...𝑀) ∈ Fin)
112433adantl2 1167 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐺𝑖)‘𝑡) ∈ ℝ)
113112recnd 11290 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐺𝑖)‘𝑡) ∈ ℂ)
114104, 110, 111, 113fsumsplit 15778 . . . . . . . . . . 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 5170 . . . . . . . . 9 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → 0 < Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡))
11736, 45, 49, 116mulgt0d 11417 . . . . . . . 8 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → 0 < ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡)))
11830, 31, 35, 117exlimdd 2219 . . . . . . 7 ((𝜑𝑡 ∈ (𝑇𝑈)) → 0 < ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡)))
1198, 2, 5, 7, 21stoweidlem30 46050 . . . . . . . 8 ((𝜑𝑡𝑇) → (𝑃𝑡) = ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡)))
12038, 119sylan2 593 . . . . . . 7 ((𝜑𝑡 ∈ (𝑇𝑈)) → (𝑃𝑡) = ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡)))
121118, 120breqtrrd 5170 . . . . . 6 ((𝜑𝑡 ∈ (𝑇𝑈)) → 0 < (𝑃𝑡))
122121ex 412 . . . . 5 (𝜑 → (𝑡 ∈ (𝑇𝑈) → 0 < (𝑃𝑡)))
1231, 122ralrimi 3256 . . . 4 (𝜑 → ∀𝑡 ∈ (𝑇𝑈)0 < (𝑃𝑡))
12425, 27, 1233jca 1128 . . 3 (𝜑 → (∀𝑡𝑇 (0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1) ∧ (𝑃𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑃𝑡)))
125 eleq1 2828 . . . . . 6 (𝑝 = 𝑃 → (𝑝𝐴𝑃𝐴))
126 nfmpt1 5249 . . . . . . . . . 10 𝑡(𝑡𝑇 ↦ ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡)))
1272, 126nfcxfr 2902 . . . . . . . . 9 𝑡𝑃
128127nfeq2 2922 . . . . . . . 8 𝑡 𝑝 = 𝑃
129 fveq1 6904 . . . . . . . . . 10 (𝑝 = 𝑃 → (𝑝𝑡) = (𝑃𝑡))
130129breq2d 5154 . . . . . . . . 9 (𝑝 = 𝑃 → (0 ≤ (𝑝𝑡) ↔ 0 ≤ (𝑃𝑡)))
131129breq1d 5152 . . . . . . . . 9 (𝑝 = 𝑃 → ((𝑝𝑡) ≤ 1 ↔ (𝑃𝑡) ≤ 1))
132130, 131anbi12d 632 . . . . . . . 8 (𝑝 = 𝑃 → ((0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ↔ (0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1)))
133128, 132ralbid 3272 . . . . . . 7 (𝑝 = 𝑃 → (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1)))
134 fveq1 6904 . . . . . . . 8 (𝑝 = 𝑃 → (𝑝𝑍) = (𝑃𝑍))
135134eqeq1d 2738 . . . . . . 7 (𝑝 = 𝑃 → ((𝑝𝑍) = 0 ↔ (𝑃𝑍) = 0))
136129breq2d 5154 . . . . . . . 8 (𝑝 = 𝑃 → (0 < (𝑝𝑡) ↔ 0 < (𝑃𝑡)))
137128, 136ralbid 3272 . . . . . . 7 (𝑝 = 𝑃 → (∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡) ↔ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑃𝑡)))
138133, 135, 1373anbi123d 1437 . . . . . 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 632 . . . . 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 3596 . . . 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 699 . 2 (𝜑 → ∃𝑝(𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡))))
143 df-rex 3070 . 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 234 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 1086   = wceq 1539  wex 1778  wnf 1782  wcel 2107  wral 3060  wrex 3069  {crab 3435  cdif 3947  cun 3948  cin 3949  wss 3950  c0 4332  {csn 4625   cuni 4906   class class class wbr 5142  cmpt 5224  ran crn 5685  wf 6556  cfv 6560  (class class class)co 7432  Fincfn 8986  cc 11154  cr 11155  0cc0 11156  1c1 11157   + caddc 11159   · cmul 11161   < clt 11296  cle 11297   / cdiv 11921  cn 12267  (,)cioo 13388  ...cfz 13548  Σcsu 15723  topGenctg 17483   Cn ccn 23233
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-inf2 9682  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233  ax-pre-sup 11234
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-er 8746  df-map 8869  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-sup 9483  df-oi 9551  df-card 9980  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-div 11922  df-nn 12268  df-2 12330  df-3 12331  df-n0 12529  df-z 12616  df-uz 12880  df-rp 13036  df-ioo 13392  df-ico 13394  df-fz 13549  df-fzo 13696  df-seq 14044  df-exp 14104  df-hash 14371  df-cj 15139  df-re 15140  df-im 15141  df-sqrt 15275  df-abs 15276  df-clim 15525  df-sum 15724  df-topgen 17489  df-top 22901  df-topon 22918  df-bases 22954  df-cn 23236
This theorem is referenced by:  stoweidlem53  46073
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