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Theorem stoweidlem44 40579
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(t_0) = 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 2651 . . . 4 (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡)) = (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡))
4 eqid 2651 . . . 4 (𝑡𝑇 ↦ (1 / 𝑀)) = (𝑡𝑇 ↦ (1 / 𝑀))
5 stoweidlem44.6 . . . 4 (𝜑𝑀 ∈ ℕ)
65nnrecred 11104 . . . 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 3720 . . . . . 6 {𝐴 ∣ ((𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1))} ⊆ 𝐴
108, 9eqsstri 3668 . . . . 5 𝑄𝐴
11 fss 6094 . . . . 5 ((𝐺:(1...𝑀)⟶𝑄𝑄𝐴) → 𝐺:(1...𝑀)⟶𝐴)
127, 10, 11sylancl 695 . . . 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 2651 . . . . 5 (𝐽 Cn 𝐾) = (𝐽 Cn 𝐾)
19 stoweidlem44.10 . . . . . 6 (𝜑𝐴 ⊆ (𝐽 Cn 𝐾))
2019sselda 3636 . . . . 5 ((𝜑𝑓𝐴) → 𝑓 ∈ (𝐽 Cn 𝐾))
2116, 17, 18, 20fcnre 39498 . . . 4 ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)
221, 2, 3, 4, 5, 6, 12, 13, 14, 15, 21stoweidlem32 40567 . . 3 (𝜑𝑃𝐴)
238, 2, 5, 7, 21stoweidlem38 40573 . . . . . 6 ((𝜑𝑡𝑇) → (0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1))
2423ex 449 . . . . 5 (𝜑 → (𝑡𝑇 → (0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1)))
251, 24ralrimi 2986 . . . 4 (𝜑 → ∀𝑡𝑇 (0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1))
26 stoweidlem44.14 . . . . 5 (𝜑𝑍𝑇)
278, 2, 5, 7, 21, 26stoweidlem37 40572 . . . 4 (𝜑 → (𝑃𝑍) = 0)
28 stoweidlem44.1 . . . . . . . . 9 𝑗𝜑
29 nfv 1883 . . . . . . . . 9 𝑗 𝑡 ∈ (𝑇𝑈)
3028, 29nfan 1868 . . . . . . . 8 𝑗(𝜑𝑡 ∈ (𝑇𝑈))
31 nfv 1883 . . . . . . . 8 𝑗0 < ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡))
32 stoweidlem44.8 . . . . . . . . . 10 (𝜑 → ∀𝑡 ∈ (𝑇𝑈)∃𝑗 ∈ (1...𝑀)0 < ((𝐺𝑗)‘𝑡))
3332r19.21bi 2961 . . . . . . . . 9 ((𝜑𝑡 ∈ (𝑇𝑈)) → ∃𝑗 ∈ (1...𝑀)0 < ((𝐺𝑗)‘𝑡))
34 df-rex 2947 . . . . . . . . 9 (∃𝑗 ∈ (1...𝑀)0 < ((𝐺𝑗)‘𝑡) ↔ ∃𝑗(𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡)))
3533, 34sylib 208 . . . . . . . 8 ((𝜑𝑡 ∈ (𝑇𝑈)) → ∃𝑗(𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡)))
366ad2antrr 762 . . . . . . . . 9 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → (1 / 𝑀) ∈ ℝ)
37 simpll 805 . . . . . . . . . 10 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → 𝜑)
38 eldifi 3765 . . . . . . . . . . 11 (𝑡 ∈ (𝑇𝑈) → 𝑡𝑇)
3938ad2antlr 763 . . . . . . . . . 10 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → 𝑡𝑇)
40 fzfid 12812 . . . . . . . . . . 11 ((𝜑𝑡𝑇) → (1...𝑀) ∈ Fin)
418, 7, 21stoweidlem15 40550 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡𝑇) → (((𝐺𝑖)‘𝑡) ∈ ℝ ∧ 0 ≤ ((𝐺𝑖)‘𝑡) ∧ ((𝐺𝑖)‘𝑡) ≤ 1))
4241an32s 863 . . . . . . . . . . . 12 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → (((𝐺𝑖)‘𝑡) ∈ ℝ ∧ 0 ≤ ((𝐺𝑖)‘𝑡) ∧ ((𝐺𝑖)‘𝑡) ≤ 1))
4342simp1d 1093 . . . . . . . . . . 11 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐺𝑖)‘𝑡) ∈ ℝ)
4440, 43fsumrecl 14509 . . . . . . . . . 10 ((𝜑𝑡𝑇) → Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡) ∈ ℝ)
4537, 39, 44syl2anc 694 . . . . . . . . 9 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡) ∈ ℝ)
465nnred 11073 . . . . . . . . . . 11 (𝜑𝑀 ∈ ℝ)
475nngt0d 11102 . . . . . . . . . . 11 (𝜑 → 0 < 𝑀)
4846, 47recgt0d 10996 . . . . . . . . . 10 (𝜑 → 0 < (1 / 𝑀))
4948ad2antrr 762 . . . . . . . . 9 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → 0 < (1 / 𝑀))
50 0red 10079 . . . . . . . . . . 11 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → 0 ∈ ℝ)
51 simprl 809 . . . . . . . . . . . . . 14 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → 𝑗 ∈ (1...𝑀))
5237, 51, 393jca 1261 . . . . . . . . . . . . 13 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → (𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇))
53 snfi 8079 . . . . . . . . . . . . . . 15 {𝑗} ∈ Fin
5453a1i 11 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) → {𝑗} ∈ Fin)
55 simpl1 1084 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ {𝑗}) → 𝜑)
56 simpl3 1086 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ {𝑗}) → 𝑡𝑇)
57 elsni 4227 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ {𝑗} → 𝑖 = 𝑗)
5857adantl 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ {𝑗}) → 𝑖 = 𝑗)
59 simpl2 1085 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ {𝑗}) → 𝑗 ∈ (1...𝑀))
6058, 59eqeltrd 2730 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ {𝑗}) → 𝑖 ∈ (1...𝑀))
6155, 56, 60, 43syl21anc 1365 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ {𝑗}) → ((𝐺𝑖)‘𝑡) ∈ ℝ)
6254, 61fsumrecl 14509 . . . . . . . . . . . . 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 10107 . . . . . . . . . . 11 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → (0 + Σ𝑖 ∈ {𝑗} ((𝐺𝑖)‘𝑡)) ∈ ℝ)
65 fzfi 12811 . . . . . . . . . . . . . . 15 (1...𝑀) ∈ Fin
66 diffi 8233 . . . . . . . . . . . . . . 15 ((1...𝑀) ∈ Fin → ((1...𝑀) ∖ {𝑗}) ∈ Fin)
6765, 66mp1i 13 . . . . . . . . . . . . . 14 ((𝜑𝑡𝑇) → ((1...𝑀) ∖ {𝑗}) ∈ Fin)
68 eldifi 3765 . . . . . . . . . . . . . . 15 (𝑖 ∈ ((1...𝑀) ∖ {𝑗}) → 𝑖 ∈ (1...𝑀))
6968, 43sylan2 490 . . . . . . . . . . . . . 14 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ ((1...𝑀) ∖ {𝑗})) → ((𝐺𝑖)‘𝑡) ∈ ℝ)
7067, 69fsumrecl 14509 . . . . . . . . . . . . 13 ((𝜑𝑡𝑇) → Σ𝑖 ∈ ((1...𝑀) ∖ {𝑗})((𝐺𝑖)‘𝑡) ∈ ℝ)
7137, 39, 70syl2anc 694 . . . . . . . . . . . 12 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → Σ𝑖 ∈ ((1...𝑀) ∖ {𝑗})((𝐺𝑖)‘𝑡) ∈ ℝ)
7271, 63readdcld 10107 . . . . . . . . . . 11 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → (Σ𝑖 ∈ ((1...𝑀) ∖ {𝑗})((𝐺𝑖)‘𝑡) + Σ𝑖 ∈ {𝑗} ((𝐺𝑖)‘𝑡)) ∈ ℝ)
73 00id 10249 . . . . . . . . . . . 12 (0 + 0) = 0
74 simprr 811 . . . . . . . . . . . . . 14 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → 0 < ((𝐺𝑗)‘𝑡))
758, 7, 21stoweidlem15 40550 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ (1...𝑀)) ∧ 𝑡𝑇) → (((𝐺𝑗)‘𝑡) ∈ ℝ ∧ 0 ≤ ((𝐺𝑗)‘𝑡) ∧ ((𝐺𝑗)‘𝑡) ≤ 1))
7675simp1d 1093 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ (1...𝑀)) ∧ 𝑡𝑇) → ((𝐺𝑗)‘𝑡) ∈ ℝ)
7737, 51, 39, 76syl21anc 1365 . . . . . . . . . . . . . . . 16 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → ((𝐺𝑗)‘𝑡) ∈ ℝ)
7877recnd 10106 . . . . . . . . . . . . . . 15 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → ((𝐺𝑗)‘𝑡) ∈ ℂ)
79 fveq2 6229 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝑗 → (𝐺𝑖) = (𝐺𝑗))
8079fveq1d 6231 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑗 → ((𝐺𝑖)‘𝑡) = ((𝐺𝑗)‘𝑡))
8180sumsn 14519 . . . . . . . . . . . . . . 15 ((𝑗 ∈ (1...𝑀) ∧ ((𝐺𝑗)‘𝑡) ∈ ℂ) → Σ𝑖 ∈ {𝑗} ((𝐺𝑖)‘𝑡) = ((𝐺𝑗)‘𝑡))
8251, 78, 81syl2anc 694 . . . . . . . . . . . . . 14 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → Σ𝑖 ∈ {𝑗} ((𝐺𝑖)‘𝑡) = ((𝐺𝑗)‘𝑡))
8374, 82breqtrrd 4713 . . . . . . . . . . . . 13 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → 0 < Σ𝑖 ∈ {𝑗} ((𝐺𝑖)‘𝑡))
8450, 63, 50, 83ltadd2dd 10234 . . . . . . . . . . . 12 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → (0 + 0) < (0 + Σ𝑖 ∈ {𝑗} ((𝐺𝑖)‘𝑡)))
8573, 84syl5eqbrr 4721 . . . . . . . . . . 11 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → 0 < (0 + Σ𝑖 ∈ {𝑗} ((𝐺𝑖)‘𝑡)))
86 0red 10079 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) → 0 ∈ ℝ)
87703adant2 1100 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) → Σ𝑖 ∈ ((1...𝑀) ∖ {𝑗})((𝐺𝑖)‘𝑡) ∈ ℝ)
88 simpll 805 . . . . . . . . . . . . . . . . 17 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ ((1...𝑀) ∖ {𝑗})) → 𝜑)
8968adantl 481 . . . . . . . . . . . . . . . . 17 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ ((1...𝑀) ∖ {𝑗})) → 𝑖 ∈ (1...𝑀))
90 simplr 807 . . . . . . . . . . . . . . . . 17 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ ((1...𝑀) ∖ {𝑗})) → 𝑡𝑇)
9188, 89, 90, 41syl21anc 1365 . . . . . . . . . . . . . . . 16 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ ((1...𝑀) ∖ {𝑗})) → (((𝐺𝑖)‘𝑡) ∈ ℝ ∧ 0 ≤ ((𝐺𝑖)‘𝑡) ∧ ((𝐺𝑖)‘𝑡) ≤ 1))
9291simp2d 1094 . . . . . . . . . . . . . . 15 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ ((1...𝑀) ∖ {𝑗})) → 0 ≤ ((𝐺𝑖)‘𝑡))
9367, 69, 92fsumge0 14571 . . . . . . . . . . . . . 14 ((𝜑𝑡𝑇) → 0 ≤ Σ𝑖 ∈ ((1...𝑀) ∖ {𝑗})((𝐺𝑖)‘𝑡))
94933adant2 1100 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) → 0 ≤ Σ𝑖 ∈ ((1...𝑀) ∖ {𝑗})((𝐺𝑖)‘𝑡))
9586, 87, 62, 94leadd1dd 10679 . . . . . . . . . . . 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 10235 . . . . . . . . . 10 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → 0 < (Σ𝑖 ∈ ((1...𝑀) ∖ {𝑗})((𝐺𝑖)‘𝑡) + Σ𝑖 ∈ {𝑗} ((𝐺𝑖)‘𝑡)))
98 eldifn 3766 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ((1...𝑀) ∖ {𝑗}) → ¬ 𝑥 ∈ {𝑗})
99 imnan 437 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ((1...𝑀) ∖ {𝑗}) → ¬ 𝑥 ∈ {𝑗}) ↔ ¬ (𝑥 ∈ ((1...𝑀) ∖ {𝑗}) ∧ 𝑥 ∈ {𝑗}))
10098, 99mpbi 220 . . . . . . . . . . . . . . 15 ¬ (𝑥 ∈ ((1...𝑀) ∖ {𝑗}) ∧ 𝑥 ∈ {𝑗})
101 elin 3829 . . . . . . . . . . . . . . 15 (𝑥 ∈ (((1...𝑀) ∖ {𝑗}) ∩ {𝑗}) ↔ (𝑥 ∈ ((1...𝑀) ∖ {𝑗}) ∧ 𝑥 ∈ {𝑗}))
102100, 101mtbir 312 . . . . . . . . . . . . . 14 ¬ 𝑥 ∈ (((1...𝑀) ∖ {𝑗}) ∩ {𝑗})
103102nel0 3965 . . . . . . . . . . . . 13 (((1...𝑀) ∖ {𝑗}) ∩ {𝑗}) = ∅
104103a1i 11 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) → (((1...𝑀) ∖ {𝑗}) ∩ {𝑗}) = ∅)
105 undif1 4076 . . . . . . . . . . . . 13 (((1...𝑀) ∖ {𝑗}) ∪ {𝑗}) = ((1...𝑀) ∪ {𝑗})
106 snssi 4371 . . . . . . . . . . . . . . 15 (𝑗 ∈ (1...𝑀) → {𝑗} ⊆ (1...𝑀))
1071063ad2ant2 1103 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) → {𝑗} ⊆ (1...𝑀))
108 ssequn2 3819 . . . . . . . . . . . . . 14 ({𝑗} ⊆ (1...𝑀) ↔ ((1...𝑀) ∪ {𝑗}) = (1...𝑀))
109107, 108sylib 208 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) → ((1...𝑀) ∪ {𝑗}) = (1...𝑀))
110105, 109syl5req 2698 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) → (1...𝑀) = (((1...𝑀) ∖ {𝑗}) ∪ {𝑗}))
111 fzfid 12812 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) → (1...𝑀) ∈ Fin)
112433adantl2 1238 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐺𝑖)‘𝑡) ∈ ℝ)
113112recnd 10106 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐺𝑖)‘𝑡) ∈ ℂ)
114104, 110, 111, 113fsumsplit 14515 . . . . . . . . . . 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 4713 . . . . . . . . 9 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → 0 < Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡))
11736, 45, 49, 116mulgt0d 10230 . . . . . . . 8 (((𝜑𝑡 ∈ (𝑇𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺𝑗)‘𝑡))) → 0 < ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡)))
11830, 31, 35, 117exlimdd 2126 . . . . . . 7 ((𝜑𝑡 ∈ (𝑇𝑈)) → 0 < ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡)))
1198, 2, 5, 7, 21stoweidlem30 40565 . . . . . . . 8 ((𝜑𝑡𝑇) → (𝑃𝑡) = ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡)))
12038, 119sylan2 490 . . . . . . 7 ((𝜑𝑡 ∈ (𝑇𝑈)) → (𝑃𝑡) = ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡)))
121118, 120breqtrrd 4713 . . . . . 6 ((𝜑𝑡 ∈ (𝑇𝑈)) → 0 < (𝑃𝑡))
122121ex 449 . . . . 5 (𝜑 → (𝑡 ∈ (𝑇𝑈) → 0 < (𝑃𝑡)))
1231, 122ralrimi 2986 . . . 4 (𝜑 → ∀𝑡 ∈ (𝑇𝑈)0 < (𝑃𝑡))
12425, 27, 1233jca 1261 . . 3 (𝜑 → (∀𝑡𝑇 (0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1) ∧ (𝑃𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑃𝑡)))
125 eleq1 2718 . . . . . 6 (𝑝 = 𝑃 → (𝑝𝐴𝑃𝐴))
126 nfmpt1 4780 . . . . . . . . . 10 𝑡(𝑡𝑇 ↦ ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡)))
1272, 126nfcxfr 2791 . . . . . . . . 9 𝑡𝑃
128127nfeq2 2809 . . . . . . . 8 𝑡 𝑝 = 𝑃
129 fveq1 6228 . . . . . . . . . 10 (𝑝 = 𝑃 → (𝑝𝑡) = (𝑃𝑡))
130129breq2d 4697 . . . . . . . . 9 (𝑝 = 𝑃 → (0 ≤ (𝑝𝑡) ↔ 0 ≤ (𝑃𝑡)))
131129breq1d 4695 . . . . . . . . 9 (𝑝 = 𝑃 → ((𝑝𝑡) ≤ 1 ↔ (𝑃𝑡) ≤ 1))
132130, 131anbi12d 747 . . . . . . . 8 (𝑝 = 𝑃 → ((0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ↔ (0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1)))
133128, 132ralbid 3012 . . . . . . 7 (𝑝 = 𝑃 → (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1)))
134 fveq1 6228 . . . . . . . 8 (𝑝 = 𝑃 → (𝑝𝑍) = (𝑃𝑍))
135134eqeq1d 2653 . . . . . . 7 (𝑝 = 𝑃 → ((𝑝𝑍) = 0 ↔ (𝑃𝑍) = 0))
136129breq2d 4697 . . . . . . . 8 (𝑝 = 𝑃 → (0 < (𝑝𝑡) ↔ 0 < (𝑃𝑡)))
137128, 136ralbid 3012 . . . . . . 7 (𝑝 = 𝑃 → (∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡) ↔ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑃𝑡)))
138133, 135, 1373anbi123d 1439 . . . . . 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 747 . . . . 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 3325 . . . 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 715 . 2 (𝜑 → ∃𝑝(𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡))))
143 df-rex 2947 . 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 224 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 383  w3a 1054   = wceq 1523  wex 1744  wnf 1748  wcel 2030  wral 2941  wrex 2942  {crab 2945  cdif 3604  cun 3605  cin 3606  wss 3607  c0 3948  {csn 4210   cuni 4468   class class class wbr 4685  cmpt 4762  ran crn 5144  wf 5922  cfv 5926  (class class class)co 6690  Fincfn 7997  cc 9972  cr 9973  0cc0 9974  1c1 9975   + caddc 9977   · cmul 9979   < clt 10112  cle 10113   / cdiv 10722  cn 11058  (,)cioo 12213  ...cfz 12364  Σcsu 14460  topGenctg 16145   Cn ccn 21076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-rp 11871  df-ioo 12217  df-ico 12219  df-fz 12365  df-fzo 12505  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-sum 14461  df-topgen 16151  df-top 20747  df-topon 20764  df-bases 20798  df-cn 21079
This theorem is referenced by:  stoweidlem53  40588
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