stoweidlem44 - Mathbox for Glauco Siliprandi

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Theorem stoweidlem44 45058

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	⊢ Ⅎ𝑗𝜑
stoweidlem44.2	⊢ Ⅎ𝑡𝜑
stoweidlem44.3	⊢ 𝐾 = (topGen‘ran (,))
stoweidlem44.4	⊢ 𝑄 = {ℎ ∈ 𝐴 ∣ ((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))}
stoweidlem44.5	⊢ 𝑃 = (𝑡 ∈ 𝑇 ↦ ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡)))
stoweidlem44.6	⊢ (𝜑 → 𝑀 ∈ ℕ)
stoweidlem44.7	⊢ (𝜑 → 𝐺:(1...𝑀)⟶𝑄)
stoweidlem44.8	⊢ (𝜑 → ∀𝑡 ∈ (𝑇 ∖ 𝑈)∃𝑗 ∈ (1...𝑀)0 < ((𝐺‘𝑗)‘𝑡))
stoweidlem44.9	⊢ 𝑇 = ∪ 𝐽
stoweidlem44.10	⊢ (𝜑 → 𝐴 ⊆ (𝐽 Cn 𝐾))
stoweidlem44.11	⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)
stoweidlem44.12	⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
stoweidlem44.13	⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴)
stoweidlem44.14	⊢ (𝜑 → 𝑍 ∈ 𝑇)

**Assertion**
Ref	Expression
stoweidlem44	⊢ (𝜑 → ∃𝑝 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡)))

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 ℎ,𝑖,𝑗,𝑡,𝐺 𝐴,ℎ 𝑇,ℎ,𝑗 ℎ,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,ℎ,𝑗,𝑝) 𝐴(𝑡,𝑖,𝑗) 𝑃(𝑥,𝑡,𝑓,𝑔,ℎ,𝑖,𝑗) 𝑄(𝑥,𝑡,𝑓,𝑔,ℎ,𝑖,𝑗,𝑝) 𝑈(𝑥,𝑡,𝑓,𝑔,ℎ,𝑖) 𝐺(𝑥,𝑝) 𝐽(𝑥,𝑡,𝑓,𝑔,ℎ,𝑖,𝑗,𝑝) 𝐾(𝑥,𝑡,𝑓,𝑔,ℎ,𝑖,𝑗,𝑝) 𝑀(ℎ,𝑝) 𝑍(𝑥,𝑓,𝑔,𝑗)

**Proof of Theorem stoweidlem44**
Step	Hyp	Ref	Expression
1		stoweidlem44.2	. . . 4 ⊢ Ⅎ𝑡𝜑
2		stoweidlem44.5	. . . 4 ⊢ 𝑃 = (𝑡 ∈ 𝑇 ↦ ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡)))
3		eqid 2730	. . . 4 ⊢ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡)) = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡))
4		eqid 2730	. . . 4 ⊢ (𝑡 ∈ 𝑇 ↦ (1 / 𝑀)) = (𝑡 ∈ 𝑇 ↦ (1 / 𝑀))
5		stoweidlem44.6	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ)
6	5	nnrecred 12267	. . . 4 ⊢ (𝜑 → (1 / 𝑀) ∈ ℝ)
7		stoweidlem44.7	. . . . 5 ⊢ (𝜑 → 𝐺:(1...𝑀)⟶𝑄)
8		stoweidlem44.4	. . . . . 6 ⊢ 𝑄 = {ℎ ∈ 𝐴 ∣ ((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))}
9		ssrab2 4076	. . . . . 6 ⊢ {ℎ ∈ 𝐴 ∣ ((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))} ⊆ 𝐴
10	8, 9	eqsstri 4015	. . . . 5 ⊢ 𝑄 ⊆ 𝐴
11		fss 6733	. . . . 5 ⊢ ((𝐺:(1...𝑀)⟶𝑄 ∧ 𝑄 ⊆ 𝐴) → 𝐺:(1...𝑀)⟶𝐴)
12	7, 10, 11	sylancl 584	. . . 4 ⊢ (𝜑 → 𝐺:(1...𝑀)⟶𝐴)
13		stoweidlem44.11	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)
14		stoweidlem44.12	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
15		stoweidlem44.13	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴)
16		stoweidlem44.3	. . . . 5 ⊢ 𝐾 = (topGen‘ran (,))
17		stoweidlem44.9	. . . . 5 ⊢ 𝑇 = ∪ 𝐽
18		eqid 2730	. . . . 5 ⊢ (𝐽 Cn 𝐾) = (𝐽 Cn 𝐾)
19		stoweidlem44.10	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ (𝐽 Cn 𝐾))
20	19	sselda 3981	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓 ∈ (𝐽 Cn 𝐾))
21	16, 17, 18, 20	fcnre 44011	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ)
22	1, 2, 3, 4, 5, 6, 12, 13, 14, 15, 21	stoweidlem32 45046	. . 3 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
23	8, 2, 5, 7, 21	stoweidlem38 45052	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (0 ≤ (𝑃‘𝑡) ∧ (𝑃‘𝑡) ≤ 1))
24	23	ex 411	. . . . 5 ⊢ (𝜑 → (𝑡 ∈ 𝑇 → (0 ≤ (𝑃‘𝑡) ∧ (𝑃‘𝑡) ≤ 1)))
25	1, 24	ralrimi 3252	. . . 4 ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 (0 ≤ (𝑃‘𝑡) ∧ (𝑃‘𝑡) ≤ 1))
26		stoweidlem44.14	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝑇)
27	8, 2, 5, 7, 21, 26	stoweidlem37 45051	. . . 4 ⊢ (𝜑 → (𝑃‘𝑍) = 0)
28		stoweidlem44.1	. . . . . . . . 9 ⊢ Ⅎ𝑗𝜑
29		nfv 1915	. . . . . . . . 9 ⊢ Ⅎ𝑗 𝑡 ∈ (𝑇 ∖ 𝑈)
30	28, 29	nfan 1900	. . . . . . . 8 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈))
31		nfv 1915	. . . . . . . 8 ⊢ Ⅎ𝑗0 < ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡))
32		stoweidlem44.8	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑡 ∈ (𝑇 ∖ 𝑈)∃𝑗 ∈ (1...𝑀)0 < ((𝐺‘𝑗)‘𝑡))
33	32	r19.21bi 3246	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → ∃𝑗 ∈ (1...𝑀)0 < ((𝐺‘𝑗)‘𝑡))
34		df-rex 3069	. . . . . . . . 9 ⊢ (∃𝑗 ∈ (1...𝑀)0 < ((𝐺‘𝑗)‘𝑡) ↔ ∃𝑗(𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡)))
35	33, 34	sylib 217	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → ∃𝑗(𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡)))
36	6	ad2antrr 722	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → (1 / 𝑀) ∈ ℝ)
37		simpll 763	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → 𝜑)
38		eldifi 4125	. . . . . . . . . . 11 ⊢ (𝑡 ∈ (𝑇 ∖ 𝑈) → 𝑡 ∈ 𝑇)
39	38	ad2antlr 723	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → 𝑡 ∈ 𝑇)
40		fzfid 13942	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (1...𝑀) ∈ Fin)
41	8, 7, 21	stoweidlem15 45029	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ 𝑇) → (((𝐺‘𝑖)‘𝑡) ∈ ℝ ∧ 0 ≤ ((𝐺‘𝑖)‘𝑡) ∧ ((𝐺‘𝑖)‘𝑡) ≤ 1))
42	41	an32s 648	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑀)) → (((𝐺‘𝑖)‘𝑡) ∈ ℝ ∧ 0 ≤ ((𝐺‘𝑖)‘𝑡) ∧ ((𝐺‘𝑖)‘𝑡) ≤ 1))
43	42	simp1d 1140	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐺‘𝑖)‘𝑡) ∈ ℝ)
44	40, 43	fsumrecl 15684	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡) ∈ ℝ)
45	37, 39, 44	syl2anc 582	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡) ∈ ℝ)
46	5	nnred 12231	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℝ)
47	5	nngt0d 12265	. . . . . . . . . . 11 ⊢ (𝜑 → 0 < 𝑀)
48	46, 47	recgt0d 12152	. . . . . . . . . 10 ⊢ (𝜑 → 0 < (1 / 𝑀))
49	48	ad2antrr 722	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → 0 < (1 / 𝑀))
50		0red 11221	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → 0 ∈ ℝ)
51		simprl 767	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → 𝑗 ∈ (1...𝑀))
52	37, 51, 39	3jca 1126	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → (𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇))
53		snfi 9046	. . . . . . . . . . . . . . 15 ⊢ {𝑗} ∈ Fin
54	53	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) → {𝑗} ∈ Fin)
55		simpl1 1189	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ {𝑗}) → 𝜑)
56		simpl3 1191	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ {𝑗}) → 𝑡 ∈ 𝑇)
57		elsni 4644	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ {𝑗} → 𝑖 = 𝑗)
58	57	adantl 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ {𝑗}) → 𝑖 = 𝑗)
59		simpl2 1190	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ {𝑗}) → 𝑗 ∈ (1...𝑀))
60	58, 59	eqeltrd 2831	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ {𝑗}) → 𝑖 ∈ (1...𝑀))
61	55, 56, 60, 43	syl21anc 834	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ {𝑗}) → ((𝐺‘𝑖)‘𝑡) ∈ ℝ)
62	54, 61	fsumrecl 15684	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) → Σ𝑖 ∈ {𝑗} ((𝐺‘𝑖)‘𝑡) ∈ ℝ)
63	52, 62	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → Σ𝑖 ∈ {𝑗} ((𝐺‘𝑖)‘𝑡) ∈ ℝ)
64	50, 63	readdcld 11247	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → (0 + Σ𝑖 ∈ {𝑗} ((𝐺‘𝑖)‘𝑡)) ∈ ℝ)
65		fzfi 13941	. . . . . . . . . . . . . . 15 ⊢ (1...𝑀) ∈ Fin
66		diffi 9181	. . . . . . . . . . . . . . 15 ⊢ ((1...𝑀) ∈ Fin → ((1...𝑀) ∖ {𝑗}) ∈ Fin)
67	65, 66	mp1i 13	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((1...𝑀) ∖ {𝑗}) ∈ Fin)
68		eldifi 4125	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ ((1...𝑀) ∖ {𝑗}) → 𝑖 ∈ (1...𝑀))
69	68, 43	sylan2 591	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ ((1...𝑀) ∖ {𝑗})) → ((𝐺‘𝑖)‘𝑡) ∈ ℝ)
70	67, 69	fsumrecl 15684	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → Σ𝑖 ∈ ((1...𝑀) ∖ {𝑗})((𝐺‘𝑖)‘𝑡) ∈ ℝ)
71	37, 39, 70	syl2anc 582	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → Σ𝑖 ∈ ((1...𝑀) ∖ {𝑗})((𝐺‘𝑖)‘𝑡) ∈ ℝ)
72	71, 63	readdcld 11247	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → (Σ𝑖 ∈ ((1...𝑀) ∖ {𝑗})((𝐺‘𝑖)‘𝑡) + Σ𝑖 ∈ {𝑗} ((𝐺‘𝑖)‘𝑡)) ∈ ℝ)
73		00id 11393	. . . . . . . . . . . 12 ⊢ (0 + 0) = 0
74		simprr 769	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → 0 < ((𝐺‘𝑗)‘𝑡))
75	8, 7, 21	stoweidlem15 45029	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ 𝑡 ∈ 𝑇) → (((𝐺‘𝑗)‘𝑡) ∈ ℝ ∧ 0 ≤ ((𝐺‘𝑗)‘𝑡) ∧ ((𝐺‘𝑗)‘𝑡) ≤ 1))
76	75	simp1d 1140	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ 𝑡 ∈ 𝑇) → ((𝐺‘𝑗)‘𝑡) ∈ ℝ)
77	37, 51, 39, 76	syl21anc 834	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → ((𝐺‘𝑗)‘𝑡) ∈ ℝ)
78	77	recnd 11246	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → ((𝐺‘𝑗)‘𝑡) ∈ ℂ)
79		fveq2 6890	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝑗 → (𝐺‘𝑖) = (𝐺‘𝑗))
80	79	fveq1d 6892	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑗 → ((𝐺‘𝑖)‘𝑡) = ((𝐺‘𝑗)‘𝑡))
81	80	sumsn 15696	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 ∈ (1...𝑀) ∧ ((𝐺‘𝑗)‘𝑡) ∈ ℂ) → Σ𝑖 ∈ {𝑗} ((𝐺‘𝑖)‘𝑡) = ((𝐺‘𝑗)‘𝑡))
82	51, 78, 81	syl2anc 582	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → Σ𝑖 ∈ {𝑗} ((𝐺‘𝑖)‘𝑡) = ((𝐺‘𝑗)‘𝑡))
83	74, 82	breqtrrd 5175	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → 0 < Σ𝑖 ∈ {𝑗} ((𝐺‘𝑖)‘𝑡))
84	50, 63, 50, 83	ltadd2dd 11377	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → (0 + 0) < (0 + Σ𝑖 ∈ {𝑗} ((𝐺‘𝑖)‘𝑡)))
85	73, 84	eqbrtrrid 5183	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → 0 < (0 + Σ𝑖 ∈ {𝑗} ((𝐺‘𝑖)‘𝑡)))
86		0red 11221	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) → 0 ∈ ℝ)
87	70	3adant2 1129	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) → Σ𝑖 ∈ ((1...𝑀) ∖ {𝑗})((𝐺‘𝑖)‘𝑡) ∈ ℝ)
88		simpll 763	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ ((1...𝑀) ∖ {𝑗})) → 𝜑)
89	68	adantl 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ ((1...𝑀) ∖ {𝑗})) → 𝑖 ∈ (1...𝑀))
90		simplr 765	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ ((1...𝑀) ∖ {𝑗})) → 𝑡 ∈ 𝑇)
91	88, 89, 90, 41	syl21anc 834	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ ((1...𝑀) ∖ {𝑗})) → (((𝐺‘𝑖)‘𝑡) ∈ ℝ ∧ 0 ≤ ((𝐺‘𝑖)‘𝑡) ∧ ((𝐺‘𝑖)‘𝑡) ≤ 1))
92	91	simp2d 1141	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ ((1...𝑀) ∖ {𝑗})) → 0 ≤ ((𝐺‘𝑖)‘𝑡))
93	67, 69, 92	fsumge0 15745	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ Σ𝑖 ∈ ((1...𝑀) ∖ {𝑗})((𝐺‘𝑖)‘𝑡))
94	93	3adant2 1129	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) → 0 ≤ Σ𝑖 ∈ ((1...𝑀) ∖ {𝑗})((𝐺‘𝑖)‘𝑡))
95	86, 87, 62, 94	leadd1dd 11832	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) → (0 + Σ𝑖 ∈ {𝑗} ((𝐺‘𝑖)‘𝑡)) ≤ (Σ𝑖 ∈ ((1...𝑀) ∖ {𝑗})((𝐺‘𝑖)‘𝑡) + Σ𝑖 ∈ {𝑗} ((𝐺‘𝑖)‘𝑡)))
96	52, 95	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → (0 + Σ𝑖 ∈ {𝑗} ((𝐺‘𝑖)‘𝑡)) ≤ (Σ𝑖 ∈ ((1...𝑀) ∖ {𝑗})((𝐺‘𝑖)‘𝑡) + Σ𝑖 ∈ {𝑗} ((𝐺‘𝑖)‘𝑡)))
97	50, 64, 72, 85, 96	ltletrd 11378	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → 0 < (Σ𝑖 ∈ ((1...𝑀) ∖ {𝑗})((𝐺‘𝑖)‘𝑡) + Σ𝑖 ∈ {𝑗} ((𝐺‘𝑖)‘𝑡)))
98		eldifn 4126	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ((1...𝑀) ∖ {𝑗}) → ¬ 𝑥 ∈ {𝑗})
99		imnan 398	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ((1...𝑀) ∖ {𝑗}) → ¬ 𝑥 ∈ {𝑗}) ↔ ¬ (𝑥 ∈ ((1...𝑀) ∖ {𝑗}) ∧ 𝑥 ∈ {𝑗}))
100	98, 99	mpbi 229	. . . . . . . . . . . . . . 15 ⊢ ¬ (𝑥 ∈ ((1...𝑀) ∖ {𝑗}) ∧ 𝑥 ∈ {𝑗})
101		elin 3963	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (((1...𝑀) ∖ {𝑗}) ∩ {𝑗}) ↔ (𝑥 ∈ ((1...𝑀) ∖ {𝑗}) ∧ 𝑥 ∈ {𝑗}))
102	100, 101	mtbir 322	. . . . . . . . . . . . . 14 ⊢ ¬ 𝑥 ∈ (((1...𝑀) ∖ {𝑗}) ∩ {𝑗})
103	102	nel0 4349	. . . . . . . . . . . . 13 ⊢ (((1...𝑀) ∖ {𝑗}) ∩ {𝑗}) = ∅
104	103	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) → (((1...𝑀) ∖ {𝑗}) ∩ {𝑗}) = ∅)
105		undif1 4474	. . . . . . . . . . . . 13 ⊢ (((1...𝑀) ∖ {𝑗}) ∪ {𝑗}) = ((1...𝑀) ∪ {𝑗})
106		snssi 4810	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (1...𝑀) → {𝑗} ⊆ (1...𝑀))
107	106	3ad2ant2 1132	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) → {𝑗} ⊆ (1...𝑀))
108		ssequn2 4182	. . . . . . . . . . . . . 14 ⊢ ({𝑗} ⊆ (1...𝑀) ↔ ((1...𝑀) ∪ {𝑗}) = (1...𝑀))
109	107, 108	sylib 217	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) → ((1...𝑀) ∪ {𝑗}) = (1...𝑀))
110	105, 109	eqtr2id 2783	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) → (1...𝑀) = (((1...𝑀) ∖ {𝑗}) ∪ {𝑗}))
111		fzfid 13942	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) → (1...𝑀) ∈ Fin)
112	43	3adantl2 1165	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐺‘𝑖)‘𝑡) ∈ ℝ)
113	112	recnd 11246	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐺‘𝑖)‘𝑡) ∈ ℂ)
114	104, 110, 111, 113	fsumsplit 15691	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) → Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡) = (Σ𝑖 ∈ ((1...𝑀) ∖ {𝑗})((𝐺‘𝑖)‘𝑡) + Σ𝑖 ∈ {𝑗} ((𝐺‘𝑖)‘𝑡)))
115	52, 114	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡) = (Σ𝑖 ∈ ((1...𝑀) ∖ {𝑗})((𝐺‘𝑖)‘𝑡) + Σ𝑖 ∈ {𝑗} ((𝐺‘𝑖)‘𝑡)))
116	97, 115	breqtrrd 5175	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → 0 < Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡))
117	36, 45, 49, 116	mulgt0d 11373	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → 0 < ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡)))
118	30, 31, 35, 117	exlimdd 2211	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → 0 < ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡)))
119	8, 2, 5, 7, 21	stoweidlem30 45044	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑃‘𝑡) = ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡)))
120	38, 119	sylan2 591	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → (𝑃‘𝑡) = ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡)))
121	118, 120	breqtrrd 5175	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → 0 < (𝑃‘𝑡))
122	121	ex 411	. . . . 5 ⊢ (𝜑 → (𝑡 ∈ (𝑇 ∖ 𝑈) → 0 < (𝑃‘𝑡)))
123	1, 122	ralrimi 3252	. . . 4 ⊢ (𝜑 → ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑃‘𝑡))
124	25, 27, 123	3jca 1126	. . 3 ⊢ (𝜑 → (∀𝑡 ∈ 𝑇 (0 ≤ (𝑃‘𝑡) ∧ (𝑃‘𝑡) ≤ 1) ∧ (𝑃‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑃‘𝑡)))
125		eleq1 2819	. . . . . 6 ⊢ (𝑝 = 𝑃 → (𝑝 ∈ 𝐴 ↔ 𝑃 ∈ 𝐴))
126		nfmpt1 5255	. . . . . . . . . 10 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡)))
127	2, 126	nfcxfr 2899	. . . . . . . . 9 ⊢ Ⅎ𝑡𝑃
128	127	nfeq2 2918	. . . . . . . 8 ⊢ Ⅎ𝑡 𝑝 = 𝑃
129		fveq1 6889	. . . . . . . . . 10 ⊢ (𝑝 = 𝑃 → (𝑝‘𝑡) = (𝑃‘𝑡))
130	129	breq2d 5159	. . . . . . . . 9 ⊢ (𝑝 = 𝑃 → (0 ≤ (𝑝‘𝑡) ↔ 0 ≤ (𝑃‘𝑡)))
131	129	breq1d 5157	. . . . . . . . 9 ⊢ (𝑝 = 𝑃 → ((𝑝‘𝑡) ≤ 1 ↔ (𝑃‘𝑡) ≤ 1))
132	130, 131	anbi12d 629	. . . . . . . 8 ⊢ (𝑝 = 𝑃 → ((0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ↔ (0 ≤ (𝑃‘𝑡) ∧ (𝑃‘𝑡) ≤ 1)))
133	128, 132	ralbid 3268	. . . . . . 7 ⊢ (𝑝 = 𝑃 → (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ↔ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑃‘𝑡) ∧ (𝑃‘𝑡) ≤ 1)))
134		fveq1 6889	. . . . . . . 8 ⊢ (𝑝 = 𝑃 → (𝑝‘𝑍) = (𝑃‘𝑍))
135	134	eqeq1d 2732	. . . . . . 7 ⊢ (𝑝 = 𝑃 → ((𝑝‘𝑍) = 0 ↔ (𝑃‘𝑍) = 0))
136	129	breq2d 5159	. . . . . . . 8 ⊢ (𝑝 = 𝑃 → (0 < (𝑝‘𝑡) ↔ 0 < (𝑃‘𝑡)))
137	128, 136	ralbid 3268	. . . . . . 7 ⊢ (𝑝 = 𝑃 → (∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡) ↔ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑃‘𝑡)))
138	133, 135, 137	3anbi123d 1434	. . . . . 6 ⊢ (𝑝 = 𝑃 → ((∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡)) ↔ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑃‘𝑡) ∧ (𝑃‘𝑡) ≤ 1) ∧ (𝑃‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑃‘𝑡))))
139	125, 138	anbi12d 629	. . . . 5 ⊢ (𝑝 = 𝑃 → ((𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡))) ↔ (𝑃 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑃‘𝑡) ∧ (𝑃‘𝑡) ≤ 1) ∧ (𝑃‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑃‘𝑡)))))
140	139	spcegv 3586	. . . 4 ⊢ (𝑃 ∈ 𝐴 → ((𝑃 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑃‘𝑡) ∧ (𝑃‘𝑡) ≤ 1) ∧ (𝑃‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑃‘𝑡))) → ∃𝑝(𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡)))))
141	22, 140	syl 17	. . 3 ⊢ (𝜑 → ((𝑃 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑃‘𝑡) ∧ (𝑃‘𝑡) ≤ 1) ∧ (𝑃‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑃‘𝑡))) → ∃𝑝(𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡)))))
142	22, 124, 141	mp2and 695	. 2 ⊢ (𝜑 → ∃𝑝(𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡))))
143		df-rex 3069	. 2 ⊢ (∃𝑝 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡)) ↔ ∃𝑝(𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡))))
144	142, 143	sylibr 233	1 ⊢ (𝜑 → ∃𝑝 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 ∧ w3a 1085 = wceq 1539 ∃wex 1779 Ⅎwnf 1783 ∈ wcel 2104 ∀wral 3059 ∃wrex 3068 {crab 3430 ∖ cdif 3944 ∪ cun 3945 ∩ cin 3946 ⊆ wss 3947 ∅c0 4321 {csn 4627 ∪ cuni 4907 class class class wbr 5147 ↦ cmpt 5230 ran crn 5676 ⟶wf 6538 ‘cfv 6542 (class class class)co 7411 Fincfn 8941 ℂcc 11110 ℝcr 11111 0cc0 11112 1c1 11113 + caddc 11115 · cmul 11117 < clt 11252 ≤ cle 11253 / cdiv 11875 ℕcn 12216 (,)cioo 13328 ...cfz 13488 Σcsu 15636 topGenctg 17387 Cn ccn 22948

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12979 df-ioo 13332 df-ico 13334 df-fz 13489 df-fzo 13632 df-seq 13971 df-exp 14032 df-hash 14295 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-clim 15436 df-sum 15637 df-topgen 17393 df-top 22616 df-topon 22633 df-bases 22669 df-cn 22951

This theorem is referenced by: stoweidlem53 45067

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