Theorem stoweidlem46 46037

Description: This lemma proves that sets U(t) as defined in Lemma 1 of [BrosowskiDeutsh] p. 90, are a cover of T \ U. Using this lemma, in a later theorem we will prove that a finite subcover exists. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Hypotheses

Ref	Expression
stoweidlem46.1	⊢ Ⅎ𝑡𝑈
stoweidlem46.2	⊢ Ⅎℎ𝑄
stoweidlem46.3	⊢ Ⅎ𝑞𝜑
stoweidlem46.4	⊢ Ⅎ𝑡𝜑
stoweidlem46.5	⊢ 𝐾 = (topGen‘ran (,))
stoweidlem46.6	⊢ 𝑄 = {ℎ ∈ 𝐴 ∣ ((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))}
stoweidlem46.7	⊢ 𝑊 = {𝑤 ∈ 𝐽 ∣ ∃ℎ ∈ 𝑄 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}}
stoweidlem46.8	⊢ 𝑇 = ∪ 𝐽
stoweidlem46.9	⊢ (𝜑 → 𝐽 ∈ Comp)
stoweidlem46.10	⊢ (𝜑 → 𝐴 ⊆ (𝐽 Cn 𝐾))
stoweidlem46.11	⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)
stoweidlem46.12	⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
stoweidlem46.13	⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴)
stoweidlem46.14	⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑡 ∈ 𝑇 ∧ 𝑟 ≠ 𝑡)) → ∃𝑞 ∈ 𝐴 (𝑞‘𝑟) ≠ (𝑞‘𝑡))
stoweidlem46.15	⊢ (𝜑 → 𝑈 ∈ 𝐽)
stoweidlem46.16	⊢ (𝜑 → 𝑍 ∈ 𝑈)
stoweidlem46.17	⊢ (𝜑 → 𝑇 ∈ V)

Assertion

Ref	Expression
stoweidlem46	⊢ (𝜑 → (𝑇 ∖ 𝑈) ⊆ ∪ 𝑊)

Distinct variable groups:   𝑓,𝑔,ℎ,𝑡,𝑇   𝑓,𝑞,𝑔,𝑡,𝑇   𝑓,𝑟,𝑞,𝑡,𝑇   𝑥,𝑓,𝑞,𝑡,𝑇   𝐴,𝑓,𝑔,ℎ,𝑡   𝑄,𝑓,𝑔   𝑈,𝑓,𝑔,𝑞   𝑓,𝑍,𝑔,ℎ,𝑡   𝜑,𝑓,𝑔   𝑤,𝑔,ℎ,𝑡,𝑇   𝑔,𝑊   𝐴,𝑞,𝑟   𝑍,𝑞,𝑥   𝑈,𝑟   𝜑,𝑟   𝑡,𝐽,𝑤   𝑡,𝐾   𝑤,𝑄   𝑥,𝐴   𝑥,𝑈   𝜑,𝑥

Allowed substitution hints:   𝜑(𝑤,𝑡,ℎ,𝑞)   𝐴(𝑤)   𝑄(𝑥,𝑡,ℎ,𝑟,𝑞)   𝑈(𝑤,𝑡,ℎ)   𝐽(𝑥,𝑓,𝑔,ℎ,𝑟,𝑞)   𝐾(𝑥,𝑤,𝑓,𝑔,ℎ,𝑟,𝑞)   𝑊(𝑥,𝑤,𝑡,𝑓,ℎ,𝑟,𝑞)   𝑍(𝑤,𝑟)

Proof of Theorem stoweidlem46

Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		stoweidlem46.3	. . . . . . . 8 ⊢ Ⅎ𝑞𝜑
2		nfv 1914	. . . . . . . 8 ⊢ Ⅎ𝑞 𝑠 ∈ (𝑇 ∖ 𝑈)
3	1, 2	nfan 1899	. . . . . . 7 ⊢ Ⅎ𝑞(𝜑 ∧ 𝑠 ∈ (𝑇 ∖ 𝑈))
4		stoweidlem46.4	. . . . . . . 8 ⊢ Ⅎ𝑡𝜑
5		nfcv 2891	. . . . . . . . . 10 ⊢ Ⅎ𝑡𝑇
6		stoweidlem46.1	. . . . . . . . . 10 ⊢ Ⅎ𝑡𝑈
7	5, 6	nfdif 4080	. . . . . . . . 9 ⊢ Ⅎ𝑡(𝑇 ∖ 𝑈)
8	7	nfel2 2910	. . . . . . . 8 ⊢ Ⅎ𝑡 𝑠 ∈ (𝑇 ∖ 𝑈)
9	4, 8	nfan 1899	. . . . . . 7 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑠 ∈ (𝑇 ∖ 𝑈))
10		stoweidlem46.2	. . . . . . 7 ⊢ Ⅎℎ𝑄
11		stoweidlem46.5	. . . . . . 7 ⊢ 𝐾 = (topGen‘ran (,))
12		stoweidlem46.6	. . . . . . 7 ⊢ 𝑄 = {ℎ ∈ 𝐴 ∣ ((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))}
13		stoweidlem46.8	. . . . . . 7 ⊢ 𝑇 = ∪ 𝐽
14		stoweidlem46.9	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ Comp)
15	14	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝑇 ∖ 𝑈)) → 𝐽 ∈ Comp)
16		stoweidlem46.10	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ (𝐽 Cn 𝐾))
17	16	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝑇 ∖ 𝑈)) → 𝐴 ⊆ (𝐽 Cn 𝐾))
18		stoweidlem46.11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)
19	18	3adant1r 1178	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑇 ∖ 𝑈)) ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)
20		stoweidlem46.12	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
21	20	3adant1r 1178	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑇 ∖ 𝑈)) ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
22		stoweidlem46.13	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴)
23	22	adantlr 715	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑇 ∖ 𝑈)) ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴)
24		stoweidlem46.14	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑡 ∈ 𝑇 ∧ 𝑟 ≠ 𝑡)) → ∃𝑞 ∈ 𝐴 (𝑞‘𝑟) ≠ (𝑞‘𝑡))
25	24	adantlr 715	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑟 ∈ 𝑇 ∧ 𝑡 ∈ 𝑇 ∧ 𝑟 ≠ 𝑡)) → ∃𝑞 ∈ 𝐴 (𝑞‘𝑟) ≠ (𝑞‘𝑡))
26		stoweidlem46.15	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ 𝐽)
27	26	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝑇 ∖ 𝑈)) → 𝑈 ∈ 𝐽)
28		stoweidlem46.16	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ 𝑈)
29	28	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝑇 ∖ 𝑈)) → 𝑍 ∈ 𝑈)
30		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝑇 ∖ 𝑈)) → 𝑠 ∈ (𝑇 ∖ 𝑈))
31	3, 9, 10, 11, 12, 13, 15, 17, 19, 21, 23, 25, 27, 29, 30	stoweidlem43 46034	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝑇 ∖ 𝑈)) → ∃ℎ(ℎ ∈ 𝑄 ∧ 0 < (ℎ‘𝑠)))
32		nfv 1914	. . . . . . 7 ⊢ Ⅎ𝑔(ℎ ∈ 𝑄 ∧ 0 < (ℎ‘𝑠))
33	10	nfel2 2910	. . . . . . . 8 ⊢ Ⅎℎ 𝑔 ∈ 𝑄
34		nfv 1914	. . . . . . . 8 ⊢ Ⅎℎ0 < (𝑔‘𝑠)
35	33, 34	nfan 1899	. . . . . . 7 ⊢ Ⅎℎ(𝑔 ∈ 𝑄 ∧ 0 < (𝑔‘𝑠))
36		eleq1 2816	. . . . . . . 8 ⊢ (ℎ = 𝑔 → (ℎ ∈ 𝑄 ↔ 𝑔 ∈ 𝑄))
37		fveq1 6821	. . . . . . . . 9 ⊢ (ℎ = 𝑔 → (ℎ‘𝑠) = (𝑔‘𝑠))
38	37	breq2d 5104	. . . . . . . 8 ⊢ (ℎ = 𝑔 → (0 < (ℎ‘𝑠) ↔ 0 < (𝑔‘𝑠)))
39	36, 38	anbi12d 632	. . . . . . 7 ⊢ (ℎ = 𝑔 → ((ℎ ∈ 𝑄 ∧ 0 < (ℎ‘𝑠)) ↔ (𝑔 ∈ 𝑄 ∧ 0 < (𝑔‘𝑠))))
40	32, 35, 39	cbvexv1 2340	. . . . . 6 ⊢ (∃ℎ(ℎ ∈ 𝑄 ∧ 0 < (ℎ‘𝑠)) ↔ ∃𝑔(𝑔 ∈ 𝑄 ∧ 0 < (𝑔‘𝑠)))
41	31, 40	sylib 218	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝑇 ∖ 𝑈)) → ∃𝑔(𝑔 ∈ 𝑄 ∧ 0 < (𝑔‘𝑠)))
42		stoweidlem46.17	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ V)
43		rabexg 5276	. . . . . . . 8 ⊢ (𝑇 ∈ V → {𝑡 ∈ 𝑇 ∣ 0 < (𝑔‘𝑡)} ∈ V)
44	42, 43	syl 17	. . . . . . 7 ⊢ (𝜑 → {𝑡 ∈ 𝑇 ∣ 0 < (𝑔‘𝑡)} ∈ V)
45	44	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑔 ∈ 𝑄 ∧ 0 < (𝑔‘𝑠))) → {𝑡 ∈ 𝑇 ∣ 0 < (𝑔‘𝑡)} ∈ V)
46		eldifi 4082	. . . . . . . 8 ⊢ (𝑠 ∈ (𝑇 ∖ 𝑈) → 𝑠 ∈ 𝑇)
47	46	ad2antlr 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑔 ∈ 𝑄 ∧ 0 < (𝑔‘𝑠))) → 𝑠 ∈ 𝑇)
48		simprr 772	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑔 ∈ 𝑄 ∧ 0 < (𝑔‘𝑠))) → 0 < (𝑔‘𝑠))
49		fveq2 6822	. . . . . . . . 9 ⊢ (𝑡 = 𝑠 → (𝑔‘𝑡) = (𝑔‘𝑠))
50	49	breq2d 5104	. . . . . . . 8 ⊢ (𝑡 = 𝑠 → (0 < (𝑔‘𝑡) ↔ 0 < (𝑔‘𝑠)))
51	50	elrab 3648	. . . . . . 7 ⊢ (𝑠 ∈ {𝑡 ∈ 𝑇 ∣ 0 < (𝑔‘𝑡)} ↔ (𝑠 ∈ 𝑇 ∧ 0 < (𝑔‘𝑠)))
52	47, 48, 51	sylanbrc 583	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑔 ∈ 𝑄 ∧ 0 < (𝑔‘𝑠))) → 𝑠 ∈ {𝑡 ∈ 𝑇 ∣ 0 < (𝑔‘𝑡)})
53		simpll 766	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑔 ∈ 𝑄 ∧ 0 < (𝑔‘𝑠))) → 𝜑)
54	16	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑄) → 𝐴 ⊆ (𝐽 Cn 𝐾))
55		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑄) → 𝑔 ∈ 𝑄)
56	55, 12	eleqtrdi 2838	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑄) → 𝑔 ∈ {ℎ ∈ 𝐴 ∣ ((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))})
57		fveq1 6821	. . . . . . . . . . . . . . . 16 ⊢ (ℎ = 𝑔 → (ℎ‘𝑍) = (𝑔‘𝑍))
58	57	eqeq1d 2731	. . . . . . . . . . . . . . 15 ⊢ (ℎ = 𝑔 → ((ℎ‘𝑍) = 0 ↔ (𝑔‘𝑍) = 0))
59		fveq1 6821	. . . . . . . . . . . . . . . . . 18 ⊢ (ℎ = 𝑔 → (ℎ‘𝑡) = (𝑔‘𝑡))
60	59	breq2d 5104	. . . . . . . . . . . . . . . . 17 ⊢ (ℎ = 𝑔 → (0 ≤ (ℎ‘𝑡) ↔ 0 ≤ (𝑔‘𝑡)))
61	59	breq1d 5102	. . . . . . . . . . . . . . . . 17 ⊢ (ℎ = 𝑔 → ((ℎ‘𝑡) ≤ 1 ↔ (𝑔‘𝑡) ≤ 1))
62	60, 61	anbi12d 632	. . . . . . . . . . . . . . . 16 ⊢ (ℎ = 𝑔 → ((0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ (0 ≤ (𝑔‘𝑡) ∧ (𝑔‘𝑡) ≤ 1)))
63	62	ralbidv 3152	. . . . . . . . . . . . . . 15 ⊢ (ℎ = 𝑔 → (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑔‘𝑡) ∧ (𝑔‘𝑡) ≤ 1)))
64	58, 63	anbi12d 632	. . . . . . . . . . . . . 14 ⊢ (ℎ = 𝑔 → (((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)) ↔ ((𝑔‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑔‘𝑡) ∧ (𝑔‘𝑡) ≤ 1))))
65	64	elrab 3648	. . . . . . . . . . . . 13 ⊢ (𝑔 ∈ {ℎ ∈ 𝐴 ∣ ((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))} ↔ (𝑔 ∈ 𝐴 ∧ ((𝑔‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑔‘𝑡) ∧ (𝑔‘𝑡) ≤ 1))))
66	56, 65	sylib 218	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑄) → (𝑔 ∈ 𝐴 ∧ ((𝑔‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑔‘𝑡) ∧ (𝑔‘𝑡) ≤ 1))))
67	66	simpld 494	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑄) → 𝑔 ∈ 𝐴)
68	54, 67	sseldd 3936	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑄) → 𝑔 ∈ (𝐽 Cn 𝐾))
69	68	ad2ant2r 747	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑔 ∈ 𝑄 ∧ 0 < (𝑔‘𝑠))) → 𝑔 ∈ (𝐽 Cn 𝐾))
70		nfcv 2891	. . . . . . . . . 10 ⊢ Ⅎ𝑡0
71		nfcv 2891	. . . . . . . . . 10 ⊢ Ⅎ𝑡𝑔
72		nfv 1914	. . . . . . . . . . 11 ⊢ Ⅎ𝑡 𝑔 ∈ (𝐽 Cn 𝐾)
73	4, 72	nfan 1899	. . . . . . . . . 10 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑔 ∈ (𝐽 Cn 𝐾))
74		eqid 2729	. . . . . . . . . 10 ⊢ {𝑡 ∈ 𝑇 ∣ 0 < (𝑔‘𝑡)} = {𝑡 ∈ 𝑇 ∣ 0 < (𝑔‘𝑡)}
75		0xr 11162	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ*
76	75	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐽 Cn 𝐾)) → 0 ∈ ℝ*)
77		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐽 Cn 𝐾)) → 𝑔 ∈ (𝐽 Cn 𝐾))
78	70, 71, 73, 11, 13, 74, 76, 77	rfcnpre1 45007	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐽 Cn 𝐾)) → {𝑡 ∈ 𝑇 ∣ 0 < (𝑔‘𝑡)} ∈ 𝐽)
79	53, 69, 78	syl2anc 584	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑔 ∈ 𝑄 ∧ 0 < (𝑔‘𝑠))) → {𝑡 ∈ 𝑇 ∣ 0 < (𝑔‘𝑡)} ∈ 𝐽)
80		eqidd 2730	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑄) → {𝑡 ∈ 𝑇 ∣ 0 < (𝑔‘𝑡)} = {𝑡 ∈ 𝑇 ∣ 0 < (𝑔‘𝑡)})
81		nfv 1914	. . . . . . . . . . 11 ⊢ Ⅎℎ{𝑡 ∈ 𝑇 ∣ 0 < (𝑔‘𝑡)} = {𝑡 ∈ 𝑇 ∣ 0 < (𝑔‘𝑡)}
82		nfcv 2891	. . . . . . . . . . 11 ⊢ Ⅎℎ𝑔
83	59	breq2d 5104	. . . . . . . . . . . . 13 ⊢ (ℎ = 𝑔 → (0 < (ℎ‘𝑡) ↔ 0 < (𝑔‘𝑡)))
84	83	rabbidv 3402	. . . . . . . . . . . 12 ⊢ (ℎ = 𝑔 → {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)} = {𝑡 ∈ 𝑇 ∣ 0 < (𝑔‘𝑡)})
85	84	eqeq2d 2740	. . . . . . . . . . 11 ⊢ (ℎ = 𝑔 → ({𝑡 ∈ 𝑇 ∣ 0 < (𝑔‘𝑡)} = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)} ↔ {𝑡 ∈ 𝑇 ∣ 0 < (𝑔‘𝑡)} = {𝑡 ∈ 𝑇 ∣ 0 < (𝑔‘𝑡)}))
86	81, 82, 10, 85	rspcegf 45011	. . . . . . . . . 10 ⊢ ((𝑔 ∈ 𝑄 ∧ {𝑡 ∈ 𝑇 ∣ 0 < (𝑔‘𝑡)} = {𝑡 ∈ 𝑇 ∣ 0 < (𝑔‘𝑡)}) → ∃ℎ ∈ 𝑄 {𝑡 ∈ 𝑇 ∣ 0 < (𝑔‘𝑡)} = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)})
87	55, 80, 86	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑄) → ∃ℎ ∈ 𝑄 {𝑡 ∈ 𝑇 ∣ 0 < (𝑔‘𝑡)} = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)})
88	87	ad2ant2r 747	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑔 ∈ 𝑄 ∧ 0 < (𝑔‘𝑠))) → ∃ℎ ∈ 𝑄 {𝑡 ∈ 𝑇 ∣ 0 < (𝑔‘𝑡)} = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)})
89		eqeq1 2733	. . . . . . . . . 10 ⊢ (𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (𝑔‘𝑡)} → (𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)} ↔ {𝑡 ∈ 𝑇 ∣ 0 < (𝑔‘𝑡)} = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}))
90	89	rexbidv 3153	. . . . . . . . 9 ⊢ (𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (𝑔‘𝑡)} → (∃ℎ ∈ 𝑄 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)} ↔ ∃ℎ ∈ 𝑄 {𝑡 ∈ 𝑇 ∣ 0 < (𝑔‘𝑡)} = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}))
91	90	elrab 3648	. . . . . . . 8 ⊢ ({𝑡 ∈ 𝑇 ∣ 0 < (𝑔‘𝑡)} ∈ {𝑤 ∈ 𝐽 ∣ ∃ℎ ∈ 𝑄 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}} ↔ ({𝑡 ∈ 𝑇 ∣ 0 < (𝑔‘𝑡)} ∈ 𝐽 ∧ ∃ℎ ∈ 𝑄 {𝑡 ∈ 𝑇 ∣ 0 < (𝑔‘𝑡)} = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}))
92	79, 88, 91	sylanbrc 583	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑔 ∈ 𝑄 ∧ 0 < (𝑔‘𝑠))) → {𝑡 ∈ 𝑇 ∣ 0 < (𝑔‘𝑡)} ∈ {𝑤 ∈ 𝐽 ∣ ∃ℎ ∈ 𝑄 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}})
93		stoweidlem46.7	. . . . . . 7 ⊢ 𝑊 = {𝑤 ∈ 𝐽 ∣ ∃ℎ ∈ 𝑄 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}}
94	92, 93	eleqtrrdi 2839	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑔 ∈ 𝑄 ∧ 0 < (𝑔‘𝑠))) → {𝑡 ∈ 𝑇 ∣ 0 < (𝑔‘𝑡)} ∈ 𝑊)
95		nfcv 2891	. . . . . . . 8 ⊢ Ⅎ𝑤{𝑡 ∈ 𝑇 ∣ 0 < (𝑔‘𝑡)}
96		nfv 1914	. . . . . . . . 9 ⊢ Ⅎ𝑤 𝑠 ∈ {𝑡 ∈ 𝑇 ∣ 0 < (𝑔‘𝑡)}
97		nfrab1 3415	. . . . . . . . . . 11 ⊢ Ⅎ𝑤{𝑤 ∈ 𝐽 ∣ ∃ℎ ∈ 𝑄 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}}
98	93, 97	nfcxfr 2889	. . . . . . . . . 10 ⊢ Ⅎ𝑤𝑊
99	98	nfel2 2910	. . . . . . . . 9 ⊢ Ⅎ𝑤{𝑡 ∈ 𝑇 ∣ 0 < (𝑔‘𝑡)} ∈ 𝑊
100	96, 99	nfan 1899	. . . . . . . 8 ⊢ Ⅎ𝑤(𝑠 ∈ {𝑡 ∈ 𝑇 ∣ 0 < (𝑔‘𝑡)} ∧ {𝑡 ∈ 𝑇 ∣ 0 < (𝑔‘𝑡)} ∈ 𝑊)
101		eleq2 2817	. . . . . . . . 9 ⊢ (𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (𝑔‘𝑡)} → (𝑠 ∈ 𝑤 ↔ 𝑠 ∈ {𝑡 ∈ 𝑇 ∣ 0 < (𝑔‘𝑡)}))
102		eleq1 2816	. . . . . . . . 9 ⊢ (𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (𝑔‘𝑡)} → (𝑤 ∈ 𝑊 ↔ {𝑡 ∈ 𝑇 ∣ 0 < (𝑔‘𝑡)} ∈ 𝑊))
103	101, 102	anbi12d 632	. . . . . . . 8 ⊢ (𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (𝑔‘𝑡)} → ((𝑠 ∈ 𝑤 ∧ 𝑤 ∈ 𝑊) ↔ (𝑠 ∈ {𝑡 ∈ 𝑇 ∣ 0 < (𝑔‘𝑡)} ∧ {𝑡 ∈ 𝑇 ∣ 0 < (𝑔‘𝑡)} ∈ 𝑊)))
104	95, 100, 103	spcegf 3547	. . . . . . 7 ⊢ ({𝑡 ∈ 𝑇 ∣ 0 < (𝑔‘𝑡)} ∈ V → ((𝑠 ∈ {𝑡 ∈ 𝑇 ∣ 0 < (𝑔‘𝑡)} ∧ {𝑡 ∈ 𝑇 ∣ 0 < (𝑔‘𝑡)} ∈ 𝑊) → ∃𝑤(𝑠 ∈ 𝑤 ∧ 𝑤 ∈ 𝑊)))
105	104	imp 406	. . . . . 6 ⊢ (({𝑡 ∈ 𝑇 ∣ 0 < (𝑔‘𝑡)} ∈ V ∧ (𝑠 ∈ {𝑡 ∈ 𝑇 ∣ 0 < (𝑔‘𝑡)} ∧ {𝑡 ∈ 𝑇 ∣ 0 < (𝑔‘𝑡)} ∈ 𝑊)) → ∃𝑤(𝑠 ∈ 𝑤 ∧ 𝑤 ∈ 𝑊))
106	45, 52, 94, 105	syl12anc 836	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑇 ∖ 𝑈)) ∧ (𝑔 ∈ 𝑄 ∧ 0 < (𝑔‘𝑠))) → ∃𝑤(𝑠 ∈ 𝑤 ∧ 𝑤 ∈ 𝑊))
107	41, 106	exlimddv 1935	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝑇 ∖ 𝑈)) → ∃𝑤(𝑠 ∈ 𝑤 ∧ 𝑤 ∈ 𝑊))
108		nfcv 2891	. . . . 5 ⊢ Ⅎ𝑤𝑠
109	108, 98	elunif 45004	. . . 4 ⊢ (𝑠 ∈ ∪ 𝑊 ↔ ∃𝑤(𝑠 ∈ 𝑤 ∧ 𝑤 ∈ 𝑊))
110	107, 109	sylibr 234	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝑇 ∖ 𝑈)) → 𝑠 ∈ ∪ 𝑊)
111	110	ex 412	. 2 ⊢ (𝜑 → (𝑠 ∈ (𝑇 ∖ 𝑈) → 𝑠 ∈ ∪ 𝑊))
112	111	ssrdv 3941	1 ⊢ (𝜑 → (𝑇 ∖ 𝑈) ⊆ ∪ 𝑊)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779  Ⅎwnf 1783   ∈ wcel 2109  Ⅎwnfc 2876   ≠ wne 2925  ∀wral 3044  ∃wrex 3053  {crab 3394  Vcvv 3436   ∖ cdif 3900   ⊆ wss 3903  ∪ cuni 4858   class class class wbr 5092   ↦ cmpt 5173  ran crn 5620  ‘cfv 6482  (class class class)co 7349  ℝcr 11008  0cc0 11009  1c1 11010   + caddc 11012   · cmul 11014  ℝ^*cxr 11148   < clt 11149   ≤ cle 11150  (,)cioo 13248  topGenctg 17341   Cn ccn 23109  Compccmp 23271

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086  ax-pre-sup 11087

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-iin 4944  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  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 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-isom 6491  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-of 7613  df-om 7800  df-1st 7924  df-2nd 7925  df-supp 8094  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-er 8625  df-map 8755  df-ixp 8825  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-fsupp 9252  df-fi 9301  df-sup 9332  df-inf 9333  df-oi 9402  df-card 9835  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-div 11778  df-nn 12129  df-2 12191  df-3 12192  df-4 12193  df-5 12194  df-6 12195  df-7 12196  df-8 12197  df-9 12198  df-n0 12385  df-z 12472  df-dec 12592  df-uz 12736  df-q 12850  df-rp 12894  df-xneg 13014  df-xadd 13015  df-xmul 13016  df-ioo 13252  df-icc 13255  df-fz 13411  df-fzo 13558  df-seq 13909  df-exp 13969  df-hash 14238  df-cj 15006  df-re 15007  df-im 15008  df-sqrt 15142  df-abs 15143  df-struct 17058  df-sets 17075  df-slot 17093  df-ndx 17105  df-base 17121  df-ress 17142  df-plusg 17174  df-mulr 17175  df-starv 17176  df-sca 17177  df-vsca 17178  df-ip 17179  df-tset 17180  df-ple 17181  df-ds 17183  df-unif 17184  df-hom 17185  df-cco 17186  df-rest 17326  df-topn 17327  df-0g 17345  df-gsum 17346  df-topgen 17347  df-pt 17348  df-prds 17351  df-xrs 17406  df-qtop 17411  df-imas 17412  df-xps 17414  df-mre 17488  df-mrc 17489  df-acs 17491  df-mgm 18514  df-sgrp 18593  df-mnd 18609  df-submnd 18658  df-mulg 18947  df-cntz 19196  df-cmn 19661  df-psmet 21253  df-xmet 21254  df-met 21255  df-bl 21256  df-mopn 21257  df-cnfld 21262  df-top 22779  df-topon 22796  df-topsp 22818  df-bases 22831  df-cn 23112  df-cnp 23113  df-cmp 23272  df-tx 23447  df-hmeo 23640  df-xms 24206  df-ms 24207  df-tms 24208

This theorem is referenced by:  stoweidlem50  46041