Theorem stoweidlem44 46495

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 2739	. . . 4 ⊢ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡)) = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡))
4		eqid 2739	. . . 4 ⊢ (𝑡 ∈ 𝑇 ↦ (1 / 𝑀)) = (𝑡 ∈ 𝑇 ↦ (1 / 𝑀))
5		stoweidlem44.6	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ)
6	5	nnrecred 12220	. . . 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 4012	. . . . . 6 ⊢ {ℎ ∈ 𝐴 ∣ ((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))} ⊆ 𝐴
10	8, 9	eqsstri 3961	. . . . 5 ⊢ 𝑄 ⊆ 𝐴
11		fss 6672	. . . . 5 ⊢ ((𝐺:(1...𝑀)⟶𝑄 ∧ 𝑄 ⊆ 𝐴) → 𝐺:(1...𝑀)⟶𝐴)
12	7, 10, 11	sylancl 592	. . . 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 2739	. . . . 5 ⊢ (𝐽 Cn 𝐾) = (𝐽 Cn 𝐾)
19		stoweidlem44.10	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ (𝐽 Cn 𝐾))
20	19	sselda 3915	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓 ∈ (𝐽 Cn 𝐾))
21	16, 17, 18, 20	fcnre 45482	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ)
22	1, 2, 3, 4, 5, 6, 12, 13, 14, 15, 21	stoweidlem32 46483	. . 3 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
23	8, 2, 5, 7, 21	stoweidlem38 46489	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (0 ≤ (𝑃‘𝑡) ∧ (𝑃‘𝑡) ≤ 1))
24	23	ex 413	. . . . 5 ⊢ (𝜑 → (𝑡 ∈ 𝑇 → (0 ≤ (𝑃‘𝑡) ∧ (𝑃‘𝑡) ≤ 1)))
25	1, 24	ralrimi 3237	. . . 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 46488	. . . 4 ⊢ (𝜑 → (𝑃‘𝑍) = 0)
28		stoweidlem44.1	. . . . . . . . 9 ⊢ Ⅎ𝑗𝜑
29		nfv 1921	. . . . . . . . 9 ⊢ Ⅎ𝑗 𝑡 ∈ (𝑇 ∖ 𝑈)
30	28, 29	nfan 1906	. . . . . . . 8 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈))
31		nfv 1921	. . . . . . . 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 3231	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → ∃𝑗 ∈ (1...𝑀)0 < ((𝐺‘𝑗)‘𝑡))
34		df-rex 3064	. . . . . . . . 9 ⊢ (∃𝑗 ∈ (1...𝑀)0 < ((𝐺‘𝑗)‘𝑡) ↔ ∃𝑗(𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡)))
35	33, 34	sylib 219	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → ∃𝑗(𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡)))
36	6	ad2antrr 732	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → (1 / 𝑀) ∈ ℝ)
37		simpll 772	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → 𝜑)
38		eldifi 4062	. . . . . . . . . . 11 ⊢ (𝑡 ∈ (𝑇 ∖ 𝑈) → 𝑡 ∈ 𝑇)
39	38	ad2antlr 733	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → 𝑡 ∈ 𝑇)
40		fzfid 13927	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (1...𝑀) ∈ Fin)
41	8, 7, 21	stoweidlem15 46466	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ 𝑇) → (((𝐺‘𝑖)‘𝑡) ∈ ℝ ∧ 0 ≤ ((𝐺‘𝑖)‘𝑡) ∧ ((𝐺‘𝑖)‘𝑡) ≤ 1))
42	41	an32s 658	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑀)) → (((𝐺‘𝑖)‘𝑡) ∈ ℝ ∧ 0 ≤ ((𝐺‘𝑖)‘𝑡) ∧ ((𝐺‘𝑖)‘𝑡) ≤ 1))
43	42	simp1d 1148	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐺‘𝑖)‘𝑡) ∈ ℝ)
44	40, 43	fsumrecl 15688	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡) ∈ ℝ)
45	37, 39, 44	syl2anc 590	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡) ∈ ℝ)
46	5	nnred 12181	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℝ)
47	5	nngt0d 12218	. . . . . . . . . . 11 ⊢ (𝜑 → 0 < 𝑀)
48	46, 47	recgt0d 12082	. . . . . . . . . 10 ⊢ (𝜑 → 0 < (1 / 𝑀))
49	48	ad2antrr 732	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → 0 < (1 / 𝑀))
50		0red 11139	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → 0 ∈ ℝ)
51		simprl 776	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → 𝑗 ∈ (1...𝑀))
52	37, 51, 39	3jca 1134	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → (𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇))
53		snfi 8981	. . . . . . . . . . . . . . 15 ⊢ {𝑗} ∈ Fin
54	53	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) → {𝑗} ∈ Fin)
55		simpl1 1198	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ {𝑗}) → 𝜑)
56		simpl3 1200	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ {𝑗}) → 𝑡 ∈ 𝑇)
57		elsni 4573	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ {𝑗} → 𝑖 = 𝑗)
58	57	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ {𝑗}) → 𝑖 = 𝑗)
59		simpl2 1199	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ {𝑗}) → 𝑗 ∈ (1...𝑀))
60	58, 59	eqeltrd 2839	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ {𝑗}) → 𝑖 ∈ (1...𝑀))
61	55, 56, 60, 43	syl21anc 843	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ {𝑗}) → ((𝐺‘𝑖)‘𝑡) ∈ ℝ)
62	54, 61	fsumrecl 15688	. . . . . . . . . . . . 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 11166	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → (0 + Σ𝑖 ∈ {𝑗} ((𝐺‘𝑖)‘𝑡)) ∈ ℝ)
65		fzfi 13926	. . . . . . . . . . . . . . 15 ⊢ (1...𝑀) ∈ Fin
66		diffi 9100	. . . . . . . . . . . . . . 15 ⊢ ((1...𝑀) ∈ Fin → ((1...𝑀) ∖ {𝑗}) ∈ Fin)
67	65, 66	mp1i 13	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((1...𝑀) ∖ {𝑗}) ∈ Fin)
68		eldifi 4062	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ ((1...𝑀) ∖ {𝑗}) → 𝑖 ∈ (1...𝑀))
69	68, 43	sylan2 599	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ ((1...𝑀) ∖ {𝑗})) → ((𝐺‘𝑖)‘𝑡) ∈ ℝ)
70	67, 69	fsumrecl 15688	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → Σ𝑖 ∈ ((1...𝑀) ∖ {𝑗})((𝐺‘𝑖)‘𝑡) ∈ ℝ)
71	37, 39, 70	syl2anc 590	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → Σ𝑖 ∈ ((1...𝑀) ∖ {𝑗})((𝐺‘𝑖)‘𝑡) ∈ ℝ)
72	71, 63	readdcld 11166	. . . . . . . . . . 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 11313	. . . . . . . . . . . 12 ⊢ (0 + 0) = 0
74		simprr 778	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → 0 < ((𝐺‘𝑗)‘𝑡))
75	8, 7, 21	stoweidlem15 46466	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ 𝑡 ∈ 𝑇) → (((𝐺‘𝑗)‘𝑡) ∈ ℝ ∧ 0 ≤ ((𝐺‘𝑗)‘𝑡) ∧ ((𝐺‘𝑗)‘𝑡) ≤ 1))
76	75	simp1d 1148	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ 𝑡 ∈ 𝑇) → ((𝐺‘𝑗)‘𝑡) ∈ ℝ)
77	37, 51, 39, 76	syl21anc 843	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → ((𝐺‘𝑗)‘𝑡) ∈ ℝ)
78	77	recnd 11165	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → ((𝐺‘𝑗)‘𝑡) ∈ ℂ)
79		fveq2 6828	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝑗 → (𝐺‘𝑖) = (𝐺‘𝑗))
80	79	fveq1d 6830	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑗 → ((𝐺‘𝑖)‘𝑡) = ((𝐺‘𝑗)‘𝑡))
81	80	sumsn 15700	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 ∈ (1...𝑀) ∧ ((𝐺‘𝑗)‘𝑡) ∈ ℂ) → Σ𝑖 ∈ {𝑗} ((𝐺‘𝑖)‘𝑡) = ((𝐺‘𝑗)‘𝑡))
82	51, 78, 81	syl2anc 590	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → Σ𝑖 ∈ {𝑗} ((𝐺‘𝑖)‘𝑡) = ((𝐺‘𝑗)‘𝑡))
83	74, 82	breqtrrd 5101	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → 0 < Σ𝑖 ∈ {𝑗} ((𝐺‘𝑖)‘𝑡))
84	50, 63, 50, 83	ltadd2dd 11297	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → (0 + 0) < (0 + Σ𝑖 ∈ {𝑗} ((𝐺‘𝑖)‘𝑡)))
85	73, 84	eqbrtrrid 5109	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → 0 < (0 + Σ𝑖 ∈ {𝑗} ((𝐺‘𝑖)‘𝑡)))
86		0red 11139	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) → 0 ∈ ℝ)
87	70	3adant2 1137	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) → Σ𝑖 ∈ ((1...𝑀) ∖ {𝑗})((𝐺‘𝑖)‘𝑡) ∈ ℝ)
88		simpll 772	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ ((1...𝑀) ∖ {𝑗})) → 𝜑)
89	68	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ ((1...𝑀) ∖ {𝑗})) → 𝑖 ∈ (1...𝑀))
90		simplr 774	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ ((1...𝑀) ∖ {𝑗})) → 𝑡 ∈ 𝑇)
91	88, 89, 90, 41	syl21anc 843	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ ((1...𝑀) ∖ {𝑗})) → (((𝐺‘𝑖)‘𝑡) ∈ ℝ ∧ 0 ≤ ((𝐺‘𝑖)‘𝑡) ∧ ((𝐺‘𝑖)‘𝑡) ≤ 1))
92	91	simp2d 1149	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ ((1...𝑀) ∖ {𝑗})) → 0 ≤ ((𝐺‘𝑖)‘𝑡))
93	67, 69, 92	fsumge0 15750	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ Σ𝑖 ∈ ((1...𝑀) ∖ {𝑗})((𝐺‘𝑖)‘𝑡))
94	93	3adant2 1137	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) → 0 ≤ Σ𝑖 ∈ ((1...𝑀) ∖ {𝑗})((𝐺‘𝑖)‘𝑡))
95	86, 87, 62, 94	leadd1dd 11756	. . . . . . . . . . . 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 11298	. . . . . . . . . 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 4063	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ((1...𝑀) ∖ {𝑗}) → ¬ 𝑥 ∈ {𝑗})
99		imnan 400	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ((1...𝑀) ∖ {𝑗}) → ¬ 𝑥 ∈ {𝑗}) ↔ ¬ (𝑥 ∈ ((1...𝑀) ∖ {𝑗}) ∧ 𝑥 ∈ {𝑗}))
100	98, 99	mpbi 231	. . . . . . . . . . . . . . 15 ⊢ ¬ (𝑥 ∈ ((1...𝑀) ∖ {𝑗}) ∧ 𝑥 ∈ {𝑗})
101		elin 3899	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (((1...𝑀) ∖ {𝑗}) ∩ {𝑗}) ↔ (𝑥 ∈ ((1...𝑀) ∖ {𝑗}) ∧ 𝑥 ∈ {𝑗}))
102	100, 101	mtbir 324	. . . . . . . . . . . . . 14 ⊢ ¬ 𝑥 ∈ (((1...𝑀) ∖ {𝑗}) ∩ {𝑗})
103	102	nel0 4283	. . . . . . . . . . . . 13 ⊢ (((1...𝑀) ∖ {𝑗}) ∩ {𝑗}) = ∅
104	103	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) → (((1...𝑀) ∖ {𝑗}) ∩ {𝑗}) = ∅)
105		undif1 4405	. . . . . . . . . . . . 13 ⊢ (((1...𝑀) ∖ {𝑗}) ∪ {𝑗}) = ((1...𝑀) ∪ {𝑗})
106		snssi 4718	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (1...𝑀) → {𝑗} ⊆ (1...𝑀))
107	106	3ad2ant2 1140	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) → {𝑗} ⊆ (1...𝑀))
108		ssequn2 4119	. . . . . . . . . . . . . 14 ⊢ ({𝑗} ⊆ (1...𝑀) ↔ ((1...𝑀) ∪ {𝑗}) = (1...𝑀))
109	107, 108	sylib 219	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) → ((1...𝑀) ∪ {𝑗}) = (1...𝑀))
110	105, 109	eqtr2id 2787	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) → (1...𝑀) = (((1...𝑀) ∖ {𝑗}) ∪ {𝑗}))
111		fzfid 13927	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) → (1...𝑀) ∈ Fin)
112	43	3adantl2 1174	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐺‘𝑖)‘𝑡) ∈ ℝ)
113	112	recnd 11165	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐺‘𝑖)‘𝑡) ∈ ℂ)
114	104, 110, 111, 113	fsumsplit 15695	. . . . . . . . . . 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 5101	. . . . . . . . 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 11293	. . . . . . . 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 2232	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → 0 < ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡)))
119	8, 2, 5, 7, 21	stoweidlem30 46481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑃‘𝑡) = ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡)))
120	38, 119	sylan2 599	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → (𝑃‘𝑡) = ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡)))
121	118, 120	breqtrrd 5101	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → 0 < (𝑃‘𝑡))
122	121	ex 413	. . . . 5 ⊢ (𝜑 → (𝑡 ∈ (𝑇 ∖ 𝑈) → 0 < (𝑃‘𝑡)))
123	1, 122	ralrimi 3237	. . . 4 ⊢ (𝜑 → ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑃‘𝑡))
124	25, 27, 123	3jca 1134	. . 3 ⊢ (𝜑 → (∀𝑡 ∈ 𝑇 (0 ≤ (𝑃‘𝑡) ∧ (𝑃‘𝑡) ≤ 1) ∧ (𝑃‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑃‘𝑡)))
125		eleq1 2827	. . . . . 6 ⊢ (𝑝 = 𝑃 → (𝑝 ∈ 𝐴 ↔ 𝑃 ∈ 𝐴))
126		nfmpt1 5172	. . . . . . . . . 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 6827	. . . . . . . . . 10 ⊢ (𝑝 = 𝑃 → (𝑝‘𝑡) = (𝑃‘𝑡))
130	129	breq2d 5085	. . . . . . . . 9 ⊢ (𝑝 = 𝑃 → (0 ≤ (𝑝‘𝑡) ↔ 0 ≤ (𝑃‘𝑡)))
131	129	breq1d 5083	. . . . . . . . 9 ⊢ (𝑝 = 𝑃 → ((𝑝‘𝑡) ≤ 1 ↔ (𝑃‘𝑡) ≤ 1))
132	130, 131	anbi12d 638	. . . . . . . 8 ⊢ (𝑝 = 𝑃 → ((0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ↔ (0 ≤ (𝑃‘𝑡) ∧ (𝑃‘𝑡) ≤ 1)))
133	128, 132	ralbid 3252	. . . . . . 7 ⊢ (𝑝 = 𝑃 → (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ↔ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑃‘𝑡) ∧ (𝑃‘𝑡) ≤ 1)))
134		fveq1 6827	. . . . . . . 8 ⊢ (𝑝 = 𝑃 → (𝑝‘𝑍) = (𝑃‘𝑍))
135	134	eqeq1d 2741	. . . . . . 7 ⊢ (𝑝 = 𝑃 → ((𝑝‘𝑍) = 0 ↔ (𝑃‘𝑍) = 0))
136	129	breq2d 5085	. . . . . . . 8 ⊢ (𝑝 = 𝑃 → (0 < (𝑝‘𝑡) ↔ 0 < (𝑃‘𝑡)))
137	128, 136	ralbid 3252	. . . . . . 7 ⊢ (𝑝 = 𝑃 → (∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡) ↔ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑃‘𝑡)))
138	133, 135, 137	3anbi123d 1444	. . . . . 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 638	. . . . 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 3535	. . . 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 705	. 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 3064	. 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 235	1 ⊢ (𝜑 → ∃𝑝 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∧ w3a 1092   = wceq 1547  ∃wex 1786  Ⅎwnf 1790   ∈ wcel 2119  ∀wral 3053  ∃wrex 3063  {crab 3391   ∖ cdif 3880   ∪ cun 3881   ∩ cin 3882   ⊆ wss 3883  ∅c0 4262  {csn 4556  ∪ cuni 4839   class class class wbr 5073   ↦ cmpt 5154  ran crn 5620  ⟶wf 6482  ‘cfv 6486  (class class class)co 7357  Fincfn 8884  ℂcc 11028  ℝcr 11029  0cc0 11030  1c1 11031   + caddc 11033   · cmul 11035   < clt 11171   ≤ cle 11172   / cdiv 11799  ℕcn 12166  (,)cioo 13290  ...cfz 13453  Σcsu 15640  topGenctg 17392   Cn ccn 23208

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5200  ax-sep 5219  ax-nul 5229  ax-pow 5295  ax-pr 5363  ax-un 7679  ax-inf2 9554  ax-cnex 11086  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107  ax-pre-sup 11108

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4263  df-if 4456  df-pw 4532  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-int 4879  df-iun 4924  df-br 5074  df-opab 5136  df-mpt 5155  df-tr 5181  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-riota 7314  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7808  df-1st 7932  df-2nd 7933  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-1o 8396  df-er 8634  df-map 8766  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-sup 9346  df-oi 9416  df-card 9855  df-pnf 11173  df-mnf 11174  df-xr 11175  df-ltxr 11176  df-le 11177  df-sub 11371  df-neg 11372  df-div 11800  df-nn 12167  df-2 12236  df-3 12237  df-n0 12430  df-z 12517  df-uz 12781  df-rp 12935  df-ioo 13294  df-ico 13296  df-fz 13454  df-fzo 13601  df-seq 13956  df-exp 14016  df-hash 14285  df-cj 15053  df-re 15054  df-im 15055  df-sqrt 15189  df-abs 15190  df-clim 15442  df-sum 15641  df-topgen 17398  df-top 22878  df-topon 22895  df-bases 22930  df-cn 23211

This theorem is referenced by:  stoweidlem53  46504