Theorem stoweidlem44 46494

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 2737	. . . 4 ⊢ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡)) = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡))
4		eqid 2737	. . . 4 ⊢ (𝑡 ∈ 𝑇 ↦ (1 / 𝑀)) = (𝑡 ∈ 𝑇 ↦ (1 / 𝑀))
5		stoweidlem44.6	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ)
6	5	nnrecred 12223	. . . 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 4021	. . . . . 6 ⊢ {ℎ ∈ 𝐴 ∣ ((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))} ⊆ 𝐴
10	8, 9	eqsstri 3969	. . . . 5 ⊢ 𝑄 ⊆ 𝐴
11		fss 6680	. . . . 5 ⊢ ((𝐺:(1...𝑀)⟶𝑄 ∧ 𝑄 ⊆ 𝐴) → 𝐺:(1...𝑀)⟶𝐴)
12	7, 10, 11	sylancl 587	. . . 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 2737	. . . . 5 ⊢ (𝐽 Cn 𝐾) = (𝐽 Cn 𝐾)
19		stoweidlem44.10	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ (𝐽 Cn 𝐾))
20	19	sselda 3922	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓 ∈ (𝐽 Cn 𝐾))
21	16, 17, 18, 20	fcnre 45478	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ)
22	1, 2, 3, 4, 5, 6, 12, 13, 14, 15, 21	stoweidlem32 46482	. . 3 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
23	8, 2, 5, 7, 21	stoweidlem38 46488	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (0 ≤ (𝑃‘𝑡) ∧ (𝑃‘𝑡) ≤ 1))
24	23	ex 412	. . . . 5 ⊢ (𝜑 → (𝑡 ∈ 𝑇 → (0 ≤ (𝑃‘𝑡) ∧ (𝑃‘𝑡) ≤ 1)))
25	1, 24	ralrimi 3236	. . . 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 46487	. . . 4 ⊢ (𝜑 → (𝑃‘𝑍) = 0)
28		stoweidlem44.1	. . . . . . . . 9 ⊢ Ⅎ𝑗𝜑
29		nfv 1916	. . . . . . . . 9 ⊢ Ⅎ𝑗 𝑡 ∈ (𝑇 ∖ 𝑈)
30	28, 29	nfan 1901	. . . . . . . 8 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈))
31		nfv 1916	. . . . . . . 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 3230	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → ∃𝑗 ∈ (1...𝑀)0 < ((𝐺‘𝑗)‘𝑡))
34		df-rex 3063	. . . . . . . . 9 ⊢ (∃𝑗 ∈ (1...𝑀)0 < ((𝐺‘𝑗)‘𝑡) ↔ ∃𝑗(𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡)))
35	33, 34	sylib 218	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → ∃𝑗(𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡)))
36	6	ad2antrr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → (1 / 𝑀) ∈ ℝ)
37		simpll 767	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → 𝜑)
38		eldifi 4072	. . . . . . . . . . 11 ⊢ (𝑡 ∈ (𝑇 ∖ 𝑈) → 𝑡 ∈ 𝑇)
39	38	ad2antlr 728	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → 𝑡 ∈ 𝑇)
40		fzfid 13930	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (1...𝑀) ∈ Fin)
41	8, 7, 21	stoweidlem15 46465	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ 𝑇) → (((𝐺‘𝑖)‘𝑡) ∈ ℝ ∧ 0 ≤ ((𝐺‘𝑖)‘𝑡) ∧ ((𝐺‘𝑖)‘𝑡) ≤ 1))
42	41	an32s 653	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑀)) → (((𝐺‘𝑖)‘𝑡) ∈ ℝ ∧ 0 ≤ ((𝐺‘𝑖)‘𝑡) ∧ ((𝐺‘𝑖)‘𝑡) ≤ 1))
43	42	simp1d 1143	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐺‘𝑖)‘𝑡) ∈ ℝ)
44	40, 43	fsumrecl 15691	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡) ∈ ℝ)
45	37, 39, 44	syl2anc 585	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡) ∈ ℝ)
46	5	nnred 12184	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℝ)
47	5	nngt0d 12221	. . . . . . . . . . 11 ⊢ (𝜑 → 0 < 𝑀)
48	46, 47	recgt0d 12085	. . . . . . . . . 10 ⊢ (𝜑 → 0 < (1 / 𝑀))
49	48	ad2antrr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → 0 < (1 / 𝑀))
50		0red 11142	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → 0 ∈ ℝ)
51		simprl 771	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → 𝑗 ∈ (1...𝑀))
52	37, 51, 39	3jca 1129	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → (𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇))
53		snfi 8985	. . . . . . . . . . . . . . 15 ⊢ {𝑗} ∈ Fin
54	53	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) → {𝑗} ∈ Fin)
55		simpl1 1193	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ {𝑗}) → 𝜑)
56		simpl3 1195	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ {𝑗}) → 𝑡 ∈ 𝑇)
57		elsni 4585	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ {𝑗} → 𝑖 = 𝑗)
58	57	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ {𝑗}) → 𝑖 = 𝑗)
59		simpl2 1194	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ {𝑗}) → 𝑗 ∈ (1...𝑀))
60	58, 59	eqeltrd 2837	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ {𝑗}) → 𝑖 ∈ (1...𝑀))
61	55, 56, 60, 43	syl21anc 838	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ {𝑗}) → ((𝐺‘𝑖)‘𝑡) ∈ ℝ)
62	54, 61	fsumrecl 15691	. . . . . . . . . . . . 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 11169	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → (0 + Σ𝑖 ∈ {𝑗} ((𝐺‘𝑖)‘𝑡)) ∈ ℝ)
65		fzfi 13929	. . . . . . . . . . . . . . 15 ⊢ (1...𝑀) ∈ Fin
66		diffi 9104	. . . . . . . . . . . . . . 15 ⊢ ((1...𝑀) ∈ Fin → ((1...𝑀) ∖ {𝑗}) ∈ Fin)
67	65, 66	mp1i 13	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((1...𝑀) ∖ {𝑗}) ∈ Fin)
68		eldifi 4072	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ ((1...𝑀) ∖ {𝑗}) → 𝑖 ∈ (1...𝑀))
69	68, 43	sylan2 594	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ ((1...𝑀) ∖ {𝑗})) → ((𝐺‘𝑖)‘𝑡) ∈ ℝ)
70	67, 69	fsumrecl 15691	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → Σ𝑖 ∈ ((1...𝑀) ∖ {𝑗})((𝐺‘𝑖)‘𝑡) ∈ ℝ)
71	37, 39, 70	syl2anc 585	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → Σ𝑖 ∈ ((1...𝑀) ∖ {𝑗})((𝐺‘𝑖)‘𝑡) ∈ ℝ)
72	71, 63	readdcld 11169	. . . . . . . . . . 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 11316	. . . . . . . . . . . 12 ⊢ (0 + 0) = 0
74		simprr 773	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → 0 < ((𝐺‘𝑗)‘𝑡))
75	8, 7, 21	stoweidlem15 46465	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ 𝑡 ∈ 𝑇) → (((𝐺‘𝑗)‘𝑡) ∈ ℝ ∧ 0 ≤ ((𝐺‘𝑗)‘𝑡) ∧ ((𝐺‘𝑗)‘𝑡) ≤ 1))
76	75	simp1d 1143	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ 𝑡 ∈ 𝑇) → ((𝐺‘𝑗)‘𝑡) ∈ ℝ)
77	37, 51, 39, 76	syl21anc 838	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → ((𝐺‘𝑗)‘𝑡) ∈ ℝ)
78	77	recnd 11168	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → ((𝐺‘𝑗)‘𝑡) ∈ ℂ)
79		fveq2 6836	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝑗 → (𝐺‘𝑖) = (𝐺‘𝑗))
80	79	fveq1d 6838	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑗 → ((𝐺‘𝑖)‘𝑡) = ((𝐺‘𝑗)‘𝑡))
81	80	sumsn 15703	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 ∈ (1...𝑀) ∧ ((𝐺‘𝑗)‘𝑡) ∈ ℂ) → Σ𝑖 ∈ {𝑗} ((𝐺‘𝑖)‘𝑡) = ((𝐺‘𝑗)‘𝑡))
82	51, 78, 81	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → Σ𝑖 ∈ {𝑗} ((𝐺‘𝑖)‘𝑡) = ((𝐺‘𝑗)‘𝑡))
83	74, 82	breqtrrd 5114	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → 0 < Σ𝑖 ∈ {𝑗} ((𝐺‘𝑖)‘𝑡))
84	50, 63, 50, 83	ltadd2dd 11300	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → (0 + 0) < (0 + Σ𝑖 ∈ {𝑗} ((𝐺‘𝑖)‘𝑡)))
85	73, 84	eqbrtrrid 5122	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑗 ∈ (1...𝑀) ∧ 0 < ((𝐺‘𝑗)‘𝑡))) → 0 < (0 + Σ𝑖 ∈ {𝑗} ((𝐺‘𝑖)‘𝑡)))
86		0red 11142	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) → 0 ∈ ℝ)
87	70	3adant2 1132	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) → Σ𝑖 ∈ ((1...𝑀) ∖ {𝑗})((𝐺‘𝑖)‘𝑡) ∈ ℝ)
88		simpll 767	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ ((1...𝑀) ∖ {𝑗})) → 𝜑)
89	68	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ ((1...𝑀) ∖ {𝑗})) → 𝑖 ∈ (1...𝑀))
90		simplr 769	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ ((1...𝑀) ∖ {𝑗})) → 𝑡 ∈ 𝑇)
91	88, 89, 90, 41	syl21anc 838	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ ((1...𝑀) ∖ {𝑗})) → (((𝐺‘𝑖)‘𝑡) ∈ ℝ ∧ 0 ≤ ((𝐺‘𝑖)‘𝑡) ∧ ((𝐺‘𝑖)‘𝑡) ≤ 1))
92	91	simp2d 1144	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ ((1...𝑀) ∖ {𝑗})) → 0 ≤ ((𝐺‘𝑖)‘𝑡))
93	67, 69, 92	fsumge0 15753	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ Σ𝑖 ∈ ((1...𝑀) ∖ {𝑗})((𝐺‘𝑖)‘𝑡))
94	93	3adant2 1132	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) → 0 ≤ Σ𝑖 ∈ ((1...𝑀) ∖ {𝑗})((𝐺‘𝑖)‘𝑡))
95	86, 87, 62, 94	leadd1dd 11759	. . . . . . . . . . . 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 11301	. . . . . . . . . 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 4073	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ((1...𝑀) ∖ {𝑗}) → ¬ 𝑥 ∈ {𝑗})
99		imnan 399	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ((1...𝑀) ∖ {𝑗}) → ¬ 𝑥 ∈ {𝑗}) ↔ ¬ (𝑥 ∈ ((1...𝑀) ∖ {𝑗}) ∧ 𝑥 ∈ {𝑗}))
100	98, 99	mpbi 230	. . . . . . . . . . . . . . 15 ⊢ ¬ (𝑥 ∈ ((1...𝑀) ∖ {𝑗}) ∧ 𝑥 ∈ {𝑗})
101		elin 3906	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (((1...𝑀) ∖ {𝑗}) ∩ {𝑗}) ↔ (𝑥 ∈ ((1...𝑀) ∖ {𝑗}) ∧ 𝑥 ∈ {𝑗}))
102	100, 101	mtbir 323	. . . . . . . . . . . . . 14 ⊢ ¬ 𝑥 ∈ (((1...𝑀) ∖ {𝑗}) ∩ {𝑗})
103	102	nel0 4295	. . . . . . . . . . . . 13 ⊢ (((1...𝑀) ∖ {𝑗}) ∩ {𝑗}) = ∅
104	103	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) → (((1...𝑀) ∖ {𝑗}) ∩ {𝑗}) = ∅)
105		undif1 4417	. . . . . . . . . . . . 13 ⊢ (((1...𝑀) ∖ {𝑗}) ∪ {𝑗}) = ((1...𝑀) ∪ {𝑗})
106		snssi 4752	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (1...𝑀) → {𝑗} ⊆ (1...𝑀))
107	106	3ad2ant2 1135	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) → {𝑗} ⊆ (1...𝑀))
108		ssequn2 4130	. . . . . . . . . . . . . 14 ⊢ ({𝑗} ⊆ (1...𝑀) ↔ ((1...𝑀) ∪ {𝑗}) = (1...𝑀))
109	107, 108	sylib 218	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) → ((1...𝑀) ∪ {𝑗}) = (1...𝑀))
110	105, 109	eqtr2id 2785	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) → (1...𝑀) = (((1...𝑀) ∖ {𝑗}) ∪ {𝑗}))
111		fzfid 13930	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) → (1...𝑀) ∈ Fin)
112	43	3adantl2 1169	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐺‘𝑖)‘𝑡) ∈ ℝ)
113	112	recnd 11168	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐺‘𝑖)‘𝑡) ∈ ℂ)
114	104, 110, 111, 113	fsumsplit 15698	. . . . . . . . . . 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 5114	. . . . . . . . 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 11296	. . . . . . . 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 2228	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → 0 < ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡)))
119	8, 2, 5, 7, 21	stoweidlem30 46480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑃‘𝑡) = ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡)))
120	38, 119	sylan2 594	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → (𝑃‘𝑡) = ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡)))
121	118, 120	breqtrrd 5114	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → 0 < (𝑃‘𝑡))
122	121	ex 412	. . . . 5 ⊢ (𝜑 → (𝑡 ∈ (𝑇 ∖ 𝑈) → 0 < (𝑃‘𝑡)))
123	1, 122	ralrimi 3236	. . . 4 ⊢ (𝜑 → ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑃‘𝑡))
124	25, 27, 123	3jca 1129	. . 3 ⊢ (𝜑 → (∀𝑡 ∈ 𝑇 (0 ≤ (𝑃‘𝑡) ∧ (𝑃‘𝑡) ≤ 1) ∧ (𝑃‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑃‘𝑡)))
125		eleq1 2825	. . . . . 6 ⊢ (𝑝 = 𝑃 → (𝑝 ∈ 𝐴 ↔ 𝑃 ∈ 𝐴))
126		nfmpt1 5185	. . . . . . . . . 10 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡)))
127	2, 126	nfcxfr 2897	. . . . . . . . 9 ⊢ Ⅎ𝑡𝑃
128	127	nfeq2 2917	. . . . . . . 8 ⊢ Ⅎ𝑡 𝑝 = 𝑃
129		fveq1 6835	. . . . . . . . . 10 ⊢ (𝑝 = 𝑃 → (𝑝‘𝑡) = (𝑃‘𝑡))
130	129	breq2d 5098	. . . . . . . . 9 ⊢ (𝑝 = 𝑃 → (0 ≤ (𝑝‘𝑡) ↔ 0 ≤ (𝑃‘𝑡)))
131	129	breq1d 5096	. . . . . . . . 9 ⊢ (𝑝 = 𝑃 → ((𝑝‘𝑡) ≤ 1 ↔ (𝑃‘𝑡) ≤ 1))
132	130, 131	anbi12d 633	. . . . . . . 8 ⊢ (𝑝 = 𝑃 → ((0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ↔ (0 ≤ (𝑃‘𝑡) ∧ (𝑃‘𝑡) ≤ 1)))
133	128, 132	ralbid 3251	. . . . . . 7 ⊢ (𝑝 = 𝑃 → (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ↔ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑃‘𝑡) ∧ (𝑃‘𝑡) ≤ 1)))
134		fveq1 6835	. . . . . . . 8 ⊢ (𝑝 = 𝑃 → (𝑝‘𝑍) = (𝑃‘𝑍))
135	134	eqeq1d 2739	. . . . . . 7 ⊢ (𝑝 = 𝑃 → ((𝑝‘𝑍) = 0 ↔ (𝑃‘𝑍) = 0))
136	129	breq2d 5098	. . . . . . . 8 ⊢ (𝑝 = 𝑃 → (0 < (𝑝‘𝑡) ↔ 0 < (𝑃‘𝑡)))
137	128, 136	ralbid 3251	. . . . . . 7 ⊢ (𝑝 = 𝑃 → (∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡) ↔ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑃‘𝑡)))
138	133, 135, 137	3anbi123d 1439	. . . . . 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 633	. . . . 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 3540	. . . 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 700	. 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 3063	. 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 234	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 395   ∧ w3a 1087   = wceq 1542  ∃wex 1781  Ⅎwnf 1785   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  {crab 3390   ∖ cdif 3887   ∪ cun 3888   ∩ cin 3889   ⊆ wss 3890  ∅c0 4274  {csn 4568  ∪ cuni 4851   class class class wbr 5086   ↦ cmpt 5167  ran crn 5627  ⟶wf 6490  ‘cfv 6494  (class class class)co 7362  Fincfn 8888  ℂcc 11031  ℝcr 11032  0cc0 11033  1c1 11034   + caddc 11036   · cmul 11038   < clt 11174   ≤ cle 11175   / cdiv 11802  ℕcn 12169  (,)cioo 13293  ...cfz 13456  Σcsu 15643  topGenctg 17395   Cn ccn 23203

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684  ax-inf2 9557  ax-cnex 11089  ax-resscn 11090  ax-1cn 11091  ax-icn 11092  ax-addcl 11093  ax-addrcl 11094  ax-mulcl 11095  ax-mulrcl 11096  ax-mulcom 11097  ax-addass 11098  ax-mulass 11099  ax-distr 11100  ax-i2m1 11101  ax-1ne0 11102  ax-1rid 11103  ax-rnegex 11104  ax-rrecex 11105  ax-cnre 11106  ax-pre-lttri 11107  ax-pre-lttrn 11108  ax-pre-ltadd 11109  ax-pre-mulgt0 11110  ax-pre-sup 11111

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-se 5580  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-pred 6261  df-ord 6322  df-on 6323  df-lim 6324  df-suc 6325  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-isom 6503  df-riota 7319  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7813  df-1st 7937  df-2nd 7938  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-1o 8400  df-er 8638  df-map 8770  df-en 8889  df-dom 8890  df-sdom 8891  df-fin 8892  df-sup 9350  df-oi 9420  df-card 9858  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-sub 11374  df-neg 11375  df-div 11803  df-nn 12170  df-2 12239  df-3 12240  df-n0 12433  df-z 12520  df-uz 12784  df-rp 12938  df-ioo 13297  df-ico 13299  df-fz 13457  df-fzo 13604  df-seq 13959  df-exp 14019  df-hash 14288  df-cj 15056  df-re 15057  df-im 15058  df-sqrt 15192  df-abs 15193  df-clim 15445  df-sum 15644  df-topgen 17401  df-top 22873  df-topon 22890  df-bases 22925  df-cn 23206

This theorem is referenced by:  stoweidlem53  46503