Theorem stoweidlem36 43206

Description: This lemma is used to prove the existence of a function p_t as in Lemma 1 of [BrosowskiDeutsh] p. 90 (at the beginning of Lemma 1): for all t in T - U, there exists a function p in the subalgebra, such that p_t ( t₀ ) = 0 , p_t ( t ) > 0, and 0 <= p_t <= 1. Z is used for t₀ , S is used for t e. T - U , h is used for p_t . G is used for (h_t)^2 and the final h is a normalized version of G ( divided by its norm, see the variable N ). (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Hypotheses

Ref	Expression
stoweidlem36.1	⊢ Ⅎℎ𝑄
stoweidlem36.2	⊢ Ⅎ𝑡𝐻
stoweidlem36.3	⊢ Ⅎ𝑡𝐹
stoweidlem36.4	⊢ Ⅎ𝑡𝐺
stoweidlem36.5	⊢ Ⅎ𝑡𝜑
stoweidlem36.6	⊢ 𝐾 = (topGen‘ran (,))
stoweidlem36.7	⊢ 𝑄 = {ℎ ∈ 𝐴 ∣ ((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))}
stoweidlem36.8	⊢ 𝑇 = ∪ 𝐽
stoweidlem36.9	⊢ 𝐺 = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) · (𝐹‘𝑡)))
stoweidlem36.10	⊢ 𝑁 = sup(ran 𝐺, ℝ, < )
stoweidlem36.11	⊢ 𝐻 = (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) / 𝑁))
stoweidlem36.12	⊢ (𝜑 → 𝐽 ∈ Comp)
stoweidlem36.13	⊢ (𝜑 → 𝐴 ⊆ (𝐽 Cn 𝐾))
stoweidlem36.14	⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
stoweidlem36.15	⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴)
stoweidlem36.16	⊢ (𝜑 → 𝑆 ∈ 𝑇)
stoweidlem36.17	⊢ (𝜑 → 𝑍 ∈ 𝑇)
stoweidlem36.18	⊢ (𝜑 → 𝐹 ∈ 𝐴)
stoweidlem36.19	⊢ (𝜑 → (𝐹‘𝑆) ≠ (𝐹‘𝑍))
stoweidlem36.20	⊢ (𝜑 → (𝐹‘𝑍) = 0)

Assertion

Ref	Expression
stoweidlem36	⊢ (𝜑 → ∃ℎ(ℎ ∈ 𝑄 ∧ 0 < (ℎ‘𝑆)))

Distinct variable groups:   𝑓,𝑔,𝑡,𝑇   𝐴,𝑓,𝑔   𝑓,𝐹,𝑔   𝑓,𝐺,𝑔   𝜑,𝑓,𝑔   𝑔,𝑁,𝑡   𝑡,ℎ,𝑆   𝐴,ℎ   ℎ,𝐻   𝑇,ℎ   ℎ,𝑍,𝑡   𝑥,𝑡,𝑁   𝑥,𝐴   𝑥,𝑇   𝜑,𝑥

Allowed substitution hints:   𝜑(𝑡,ℎ)   𝐴(𝑡)   𝑄(𝑥,𝑡,𝑓,𝑔,ℎ)   𝑆(𝑥,𝑓,𝑔)   𝐹(𝑥,𝑡,ℎ)   𝐺(𝑥,𝑡,ℎ)   𝐻(𝑥,𝑡,𝑓,𝑔)   𝐽(𝑥,𝑡,𝑓,𝑔,ℎ)   𝐾(𝑥,𝑡,𝑓,𝑔,ℎ)   𝑁(𝑓,ℎ)   𝑍(𝑥,𝑓,𝑔)

Proof of Theorem stoweidlem36

Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		stoweidlem36.11	. . . . . 6 ⊢ 𝐻 = (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) / 𝑁))
2		stoweidlem36.5	. . . . . . 7 ⊢ Ⅎ𝑡𝜑
3		stoweidlem36.6	. . . . . . . . . . . 12 ⊢ 𝐾 = (topGen‘ran (,))
4		stoweidlem36.8	. . . . . . . . . . . 12 ⊢ 𝑇 = ∪ 𝐽
5		eqid 2734	. . . . . . . . . . . 12 ⊢ (𝐽 Cn 𝐾) = (𝐽 Cn 𝐾)
6		stoweidlem36.13	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ⊆ (𝐽 Cn 𝐾))
7		stoweidlem36.9	. . . . . . . . . . . . . 14 ⊢ 𝐺 = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) · (𝐹‘𝑡)))
8		stoweidlem36.18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹 ∈ 𝐴)
9		stoweidlem36.3	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑡𝐹
10	9	nfeq2 2917	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑡 𝑓 = 𝐹
11	9	nfeq2 2917	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑡 𝑔 = 𝐹
12		stoweidlem36.14	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
13	10, 11, 12	stoweidlem6 43176	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐹 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) · (𝐹‘𝑡))) ∈ 𝐴)
14	8, 8, 13	mpd3an23 1465	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) · (𝐹‘𝑡))) ∈ 𝐴)
15	7, 14	eqeltrid 2838	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺 ∈ 𝐴)
16	6, 15	sseldd 3892	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾))
17	3, 4, 5, 16	fcnre 42193	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺:𝑇⟶ℝ)
18	17	ffvelrnda 6893	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑡) ∈ ℝ)
19	18	recnd 10844	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑡) ∈ ℂ)
20		stoweidlem36.10	. . . . . . . . . . . 12 ⊢ 𝑁 = sup(ran 𝐺, ℝ, < )
21		stoweidlem36.12	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐽 ∈ Comp)
22		stoweidlem36.16	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆 ∈ 𝑇)
23	22	ne0d 4240	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑇 ≠ ∅)
24	4, 3, 21, 16, 23	cncmpmax 42200	. . . . . . . . . . . . 13 ⊢ (𝜑 → (sup(ran 𝐺, ℝ, < ) ∈ ran 𝐺 ∧ sup(ran 𝐺, ℝ, < ) ∈ ℝ ∧ ∀𝑠 ∈ 𝑇 (𝐺‘𝑠) ≤ sup(ran 𝐺, ℝ, < )))
25	24	simp2d 1145	. . . . . . . . . . . 12 ⊢ (𝜑 → sup(ran 𝐺, ℝ, < ) ∈ ℝ)
26	20, 25	eqeltrid 2838	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℝ)
27	26	recnd 10844	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℂ)
28	27	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑁 ∈ ℂ)
29		0red 10819	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ∈ ℝ)
30	17, 22	ffvelrnd 6894	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐺‘𝑆) ∈ ℝ)
31	6, 8	sseldd 3892	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))
32	3, 4, 5, 31	fcnre 42193	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐹:𝑇⟶ℝ)
33	32, 22	ffvelrnd 6894	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐹‘𝑆) ∈ ℝ)
34		stoweidlem36.19	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐹‘𝑆) ≠ (𝐹‘𝑍))
35		stoweidlem36.20	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐹‘𝑍) = 0)
36	34, 35	neeqtrd 3004	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐹‘𝑆) ≠ 0)
37	33, 36	msqgt0d 11382	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < ((𝐹‘𝑆) · (𝐹‘𝑆)))
38	33, 33	remulcld 10846	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐹‘𝑆) · (𝐹‘𝑆)) ∈ ℝ)
39		nfcv 2900	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑡𝑆
40	9, 39	nffv 6716	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑡(𝐹‘𝑆)
41		nfcv 2900	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑡 ·
42	40, 41, 40	nfov 7232	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑡((𝐹‘𝑆) · (𝐹‘𝑆))
43		fveq2 6706	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = 𝑆 → (𝐹‘𝑡) = (𝐹‘𝑆))
44	43, 43	oveq12d 7220	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑆 → ((𝐹‘𝑡) · (𝐹‘𝑡)) = ((𝐹‘𝑆) · (𝐹‘𝑆)))
45	39, 42, 44, 7	fvmptf 6828	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ∈ 𝑇 ∧ ((𝐹‘𝑆) · (𝐹‘𝑆)) ∈ ℝ) → (𝐺‘𝑆) = ((𝐹‘𝑆) · (𝐹‘𝑆)))
46	22, 38, 45	syl2anc 587	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐺‘𝑆) = ((𝐹‘𝑆) · (𝐹‘𝑆)))
47	37, 46	breqtrrd 5071	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 < (𝐺‘𝑆))
48	24	simp3d 1146	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑠 ∈ 𝑇 (𝐺‘𝑠) ≤ sup(ran 𝐺, ℝ, < ))
49		fveq2 6706	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = 𝑆 → (𝐺‘𝑠) = (𝐺‘𝑆))
50	49	breq1d 5053	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = 𝑆 → ((𝐺‘𝑠) ≤ sup(ran 𝐺, ℝ, < ) ↔ (𝐺‘𝑆) ≤ sup(ran 𝐺, ℝ, < )))
51	50	rspccva 3529	. . . . . . . . . . . . . 14 ⊢ ((∀𝑠 ∈ 𝑇 (𝐺‘𝑠) ≤ sup(ran 𝐺, ℝ, < ) ∧ 𝑆 ∈ 𝑇) → (𝐺‘𝑆) ≤ sup(ran 𝐺, ℝ, < ))
52	48, 22, 51	syl2anc 587	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐺‘𝑆) ≤ sup(ran 𝐺, ℝ, < ))
53	29, 30, 25, 47, 52	ltletrd 10975	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < sup(ran 𝐺, ℝ, < ))
54	53	gt0ne0d 11379	. . . . . . . . . . 11 ⊢ (𝜑 → sup(ran 𝐺, ℝ, < ) ≠ 0)
55	20	neeq1i 2999	. . . . . . . . . . 11 ⊢ (𝑁 ≠ 0 ↔ sup(ran 𝐺, ℝ, < ) ≠ 0)
56	54, 55	sylibr 237	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ≠ 0)
57	56	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑁 ≠ 0)
58	19, 28, 57	divrecd 11594	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐺‘𝑡) / 𝑁) = ((𝐺‘𝑡) · (1 / 𝑁)))
59		simpr 488	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇)
60	26, 56	rereccld 11642	. . . . . . . . . . 11 ⊢ (𝜑 → (1 / 𝑁) ∈ ℝ)
61	60	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (1 / 𝑁) ∈ ℝ)
62		eqid 2734	. . . . . . . . . . 11 ⊢ (𝑡 ∈ 𝑇 ↦ (1 / 𝑁)) = (𝑡 ∈ 𝑇 ↦ (1 / 𝑁))
63	62	fvmpt2 6818	. . . . . . . . . 10 ⊢ ((𝑡 ∈ 𝑇 ∧ (1 / 𝑁) ∈ ℝ) → ((𝑡 ∈ 𝑇 ↦ (1 / 𝑁))‘𝑡) = (1 / 𝑁))
64	59, 61, 63	syl2anc 587	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝑡 ∈ 𝑇 ↦ (1 / 𝑁))‘𝑡) = (1 / 𝑁))
65	64	oveq2d 7218	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐺‘𝑡) · ((𝑡 ∈ 𝑇 ↦ (1 / 𝑁))‘𝑡)) = ((𝐺‘𝑡) · (1 / 𝑁)))
66	58, 65	eqtr4d 2777	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐺‘𝑡) / 𝑁) = ((𝐺‘𝑡) · ((𝑡 ∈ 𝑇 ↦ (1 / 𝑁))‘𝑡)))
67	2, 66	mpteq2da 5138	. . . . . 6 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) / 𝑁)) = (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) · ((𝑡 ∈ 𝑇 ↦ (1 / 𝑁))‘𝑡))))
68	1, 67	syl5eq 2786	. . . . 5 ⊢ (𝜑 → 𝐻 = (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) · ((𝑡 ∈ 𝑇 ↦ (1 / 𝑁))‘𝑡))))
69		stoweidlem36.15	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴)
70	69	stoweidlem4 43174	. . . . . . 7 ⊢ ((𝜑 ∧ (1 / 𝑁) ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ (1 / 𝑁)) ∈ 𝐴)
71	60, 70	mpdan 687	. . . . . 6 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ (1 / 𝑁)) ∈ 𝐴)
72		stoweidlem36.4	. . . . . . . 8 ⊢ Ⅎ𝑡𝐺
73	72	nfeq2 2917	. . . . . . 7 ⊢ Ⅎ𝑡 𝑓 = 𝐺
74		nfmpt1 5142	. . . . . . . 8 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ (1 / 𝑁))
75	74	nfeq2 2917	. . . . . . 7 ⊢ Ⅎ𝑡 𝑔 = (𝑡 ∈ 𝑇 ↦ (1 / 𝑁))
76	73, 75, 12	stoweidlem6 43176	. . . . . 6 ⊢ ((𝜑 ∧ 𝐺 ∈ 𝐴 ∧ (𝑡 ∈ 𝑇 ↦ (1 / 𝑁)) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) · ((𝑡 ∈ 𝑇 ↦ (1 / 𝑁))‘𝑡))) ∈ 𝐴)
77	15, 71, 76	mpd3an23 1465	. . . . 5 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) · ((𝑡 ∈ 𝑇 ↦ (1 / 𝑁))‘𝑡))) ∈ 𝐴)
78	68, 77	eqeltrd 2834	. . . 4 ⊢ (𝜑 → 𝐻 ∈ 𝐴)
79		stoweidlem36.17	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝑇)
80	17, 79	ffvelrnd 6894	. . . . . . . 8 ⊢ (𝜑 → (𝐺‘𝑍) ∈ ℝ)
81	80, 26, 56	redivcld 11643	. . . . . . 7 ⊢ (𝜑 → ((𝐺‘𝑍) / 𝑁) ∈ ℝ)
82		nfcv 2900	. . . . . . . 8 ⊢ Ⅎ𝑡𝑍
83	72, 82	nffv 6716	. . . . . . . . 9 ⊢ Ⅎ𝑡(𝐺‘𝑍)
84		nfcv 2900	. . . . . . . . 9 ⊢ Ⅎ𝑡 /
85		nfcv 2900	. . . . . . . . 9 ⊢ Ⅎ𝑡𝑁
86	83, 84, 85	nfov 7232	. . . . . . . 8 ⊢ Ⅎ𝑡((𝐺‘𝑍) / 𝑁)
87		fveq2 6706	. . . . . . . . 9 ⊢ (𝑡 = 𝑍 → (𝐺‘𝑡) = (𝐺‘𝑍))
88	87	oveq1d 7217	. . . . . . . 8 ⊢ (𝑡 = 𝑍 → ((𝐺‘𝑡) / 𝑁) = ((𝐺‘𝑍) / 𝑁))
89	82, 86, 88, 1	fvmptf 6828	. . . . . . 7 ⊢ ((𝑍 ∈ 𝑇 ∧ ((𝐺‘𝑍) / 𝑁) ∈ ℝ) → (𝐻‘𝑍) = ((𝐺‘𝑍) / 𝑁))
90	79, 81, 89	syl2anc 587	. . . . . 6 ⊢ (𝜑 → (𝐻‘𝑍) = ((𝐺‘𝑍) / 𝑁))
91		0re 10818	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
92	35, 91	eqeltrdi 2842	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹‘𝑍) ∈ ℝ)
93	92, 92	remulcld 10846	. . . . . . . . 9 ⊢ (𝜑 → ((𝐹‘𝑍) · (𝐹‘𝑍)) ∈ ℝ)
94	9, 82	nffv 6716	. . . . . . . . . . 11 ⊢ Ⅎ𝑡(𝐹‘𝑍)
95	94, 41, 94	nfov 7232	. . . . . . . . . 10 ⊢ Ⅎ𝑡((𝐹‘𝑍) · (𝐹‘𝑍))
96		fveq2 6706	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑍 → (𝐹‘𝑡) = (𝐹‘𝑍))
97	96, 96	oveq12d 7220	. . . . . . . . . 10 ⊢ (𝑡 = 𝑍 → ((𝐹‘𝑡) · (𝐹‘𝑡)) = ((𝐹‘𝑍) · (𝐹‘𝑍)))
98	82, 95, 97, 7	fvmptf 6828	. . . . . . . . 9 ⊢ ((𝑍 ∈ 𝑇 ∧ ((𝐹‘𝑍) · (𝐹‘𝑍)) ∈ ℝ) → (𝐺‘𝑍) = ((𝐹‘𝑍) · (𝐹‘𝑍)))
99	79, 93, 98	syl2anc 587	. . . . . . . 8 ⊢ (𝜑 → (𝐺‘𝑍) = ((𝐹‘𝑍) · (𝐹‘𝑍)))
100	35, 35	oveq12d 7220	. . . . . . . . 9 ⊢ (𝜑 → ((𝐹‘𝑍) · (𝐹‘𝑍)) = (0 · 0))
101		0cn 10808	. . . . . . . . . 10 ⊢ 0 ∈ ℂ
102	101	mul02i 11004	. . . . . . . . 9 ⊢ (0 · 0) = 0
103	100, 102	eqtrdi 2790	. . . . . . . 8 ⊢ (𝜑 → ((𝐹‘𝑍) · (𝐹‘𝑍)) = 0)
104	99, 103	eqtrd 2774	. . . . . . 7 ⊢ (𝜑 → (𝐺‘𝑍) = 0)
105	104	oveq1d 7217	. . . . . 6 ⊢ (𝜑 → ((𝐺‘𝑍) / 𝑁) = (0 / 𝑁))
106	27, 56	div0d 11590	. . . . . 6 ⊢ (𝜑 → (0 / 𝑁) = 0)
107	90, 105, 106	3eqtrd 2778	. . . . 5 ⊢ (𝜑 → (𝐻‘𝑍) = 0)
108	32	ffvelrnda 6893	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℝ)
109	108	msqge0d 11383	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ ((𝐹‘𝑡) · (𝐹‘𝑡)))
110	108, 108	remulcld 10846	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐹‘𝑡) · (𝐹‘𝑡)) ∈ ℝ)
111	7	fvmpt2 6818	. . . . . . . . . . . 12 ⊢ ((𝑡 ∈ 𝑇 ∧ ((𝐹‘𝑡) · (𝐹‘𝑡)) ∈ ℝ) → (𝐺‘𝑡) = ((𝐹‘𝑡) · (𝐹‘𝑡)))
112	59, 110, 111	syl2anc 587	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑡) = ((𝐹‘𝑡) · (𝐹‘𝑡)))
113	109, 112	breqtrrd 5071	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ (𝐺‘𝑡))
114	26	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑁 ∈ ℝ)
115	53, 20	breqtrrdi 5085	. . . . . . . . . . 11 ⊢ (𝜑 → 0 < 𝑁)
116	115	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 < 𝑁)
117		divge0 11684	. . . . . . . . . 10 ⊢ ((((𝐺‘𝑡) ∈ ℝ ∧ 0 ≤ (𝐺‘𝑡)) ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → 0 ≤ ((𝐺‘𝑡) / 𝑁))
118	18, 113, 114, 116, 117	syl22anc 839	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ ((𝐺‘𝑡) / 𝑁))
119	18, 114, 57	redivcld 11643	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐺‘𝑡) / 𝑁) ∈ ℝ)
120	1	fvmpt2 6818	. . . . . . . . . 10 ⊢ ((𝑡 ∈ 𝑇 ∧ ((𝐺‘𝑡) / 𝑁) ∈ ℝ) → (𝐻‘𝑡) = ((𝐺‘𝑡) / 𝑁))
121	59, 119, 120	syl2anc 587	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐻‘𝑡) = ((𝐺‘𝑡) / 𝑁))
122	118, 121	breqtrrd 5071	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ (𝐻‘𝑡))
123	19	div1d 11583	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐺‘𝑡) / 1) = (𝐺‘𝑡))
124		fveq2 6706	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = 𝑡 → (𝐺‘𝑠) = (𝐺‘𝑡))
125	124	breq1d 5053	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑡 → ((𝐺‘𝑠) ≤ sup(ran 𝐺, ℝ, < ) ↔ (𝐺‘𝑡) ≤ sup(ran 𝐺, ℝ, < )))
126	125	rspccva 3529	. . . . . . . . . . . . 13 ⊢ ((∀𝑠 ∈ 𝑇 (𝐺‘𝑠) ≤ sup(ran 𝐺, ℝ, < ) ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑡) ≤ sup(ran 𝐺, ℝ, < ))
127	48, 126	sylan 583	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑡) ≤ sup(ran 𝐺, ℝ, < ))
128	127, 20	breqtrrdi 5085	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑡) ≤ 𝑁)
129	123, 128	eqbrtrd 5065	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐺‘𝑡) / 1) ≤ 𝑁)
130		1red 10817	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 1 ∈ ℝ)
131		0lt1 11337	. . . . . . . . . . . 12 ⊢ 0 < 1
132	131	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 < 1)
133		lediv23 11707	. . . . . . . . . . 11 ⊢ (((𝐺‘𝑡) ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁) ∧ (1 ∈ ℝ ∧ 0 < 1)) → (((𝐺‘𝑡) / 𝑁) ≤ 1 ↔ ((𝐺‘𝑡) / 1) ≤ 𝑁))
134	18, 114, 116, 130, 132, 133	syl122anc 1381	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (((𝐺‘𝑡) / 𝑁) ≤ 1 ↔ ((𝐺‘𝑡) / 1) ≤ 𝑁))
135	129, 134	mpbird 260	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐺‘𝑡) / 𝑁) ≤ 1)
136	121, 135	eqbrtrd 5065	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐻‘𝑡) ≤ 1)
137	122, 136	jca 515	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (0 ≤ (𝐻‘𝑡) ∧ (𝐻‘𝑡) ≤ 1))
138	137	ex 416	. . . . . 6 ⊢ (𝜑 → (𝑡 ∈ 𝑇 → (0 ≤ (𝐻‘𝑡) ∧ (𝐻‘𝑡) ≤ 1)))
139	2, 138	ralrimi 3130	. . . . 5 ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 (0 ≤ (𝐻‘𝑡) ∧ (𝐻‘𝑡) ≤ 1))
140	107, 139	jca 515	. . . 4 ⊢ (𝜑 → ((𝐻‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (𝐻‘𝑡) ∧ (𝐻‘𝑡) ≤ 1)))
141		fveq1 6705	. . . . . . 7 ⊢ (ℎ = 𝐻 → (ℎ‘𝑍) = (𝐻‘𝑍))
142	141	eqeq1d 2736	. . . . . 6 ⊢ (ℎ = 𝐻 → ((ℎ‘𝑍) = 0 ↔ (𝐻‘𝑍) = 0))
143		stoweidlem36.2	. . . . . . . 8 ⊢ Ⅎ𝑡𝐻
144	143	nfeq2 2917	. . . . . . 7 ⊢ Ⅎ𝑡 ℎ = 𝐻
145		fveq1 6705	. . . . . . . . 9 ⊢ (ℎ = 𝐻 → (ℎ‘𝑡) = (𝐻‘𝑡))
146	145	breq2d 5055	. . . . . . . 8 ⊢ (ℎ = 𝐻 → (0 ≤ (ℎ‘𝑡) ↔ 0 ≤ (𝐻‘𝑡)))
147	145	breq1d 5053	. . . . . . . 8 ⊢ (ℎ = 𝐻 → ((ℎ‘𝑡) ≤ 1 ↔ (𝐻‘𝑡) ≤ 1))
148	146, 147	anbi12d 634	. . . . . . 7 ⊢ (ℎ = 𝐻 → ((0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ (0 ≤ (𝐻‘𝑡) ∧ (𝐻‘𝑡) ≤ 1)))
149	144, 148	ralbid 3147	. . . . . 6 ⊢ (ℎ = 𝐻 → (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ ∀𝑡 ∈ 𝑇 (0 ≤ (𝐻‘𝑡) ∧ (𝐻‘𝑡) ≤ 1)))
150	142, 149	anbi12d 634	. . . . 5 ⊢ (ℎ = 𝐻 → (((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)) ↔ ((𝐻‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (𝐻‘𝑡) ∧ (𝐻‘𝑡) ≤ 1))))
151	150	elrab 3595	. . . 4 ⊢ (𝐻 ∈ {ℎ ∈ 𝐴 ∣ ((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))} ↔ (𝐻 ∈ 𝐴 ∧ ((𝐻‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (𝐻‘𝑡) ∧ (𝐻‘𝑡) ≤ 1))))
152	78, 140, 151	sylanbrc 586	. . 3 ⊢ (𝜑 → 𝐻 ∈ {ℎ ∈ 𝐴 ∣ ((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))})
153		stoweidlem36.7	. . 3 ⊢ 𝑄 = {ℎ ∈ 𝐴 ∣ ((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))}
154	152, 153	eleqtrrdi 2845	. 2 ⊢ (𝜑 → 𝐻 ∈ 𝑄)
155	30, 26, 47, 115	divgt0d 11750	. . 3 ⊢ (𝜑 → 0 < ((𝐺‘𝑆) / 𝑁))
156	30, 26, 56	redivcld 11643	. . . 4 ⊢ (𝜑 → ((𝐺‘𝑆) / 𝑁) ∈ ℝ)
157	72, 39	nffv 6716	. . . . . 6 ⊢ Ⅎ𝑡(𝐺‘𝑆)
158	157, 84, 85	nfov 7232	. . . . 5 ⊢ Ⅎ𝑡((𝐺‘𝑆) / 𝑁)
159		fveq2 6706	. . . . . 6 ⊢ (𝑡 = 𝑆 → (𝐺‘𝑡) = (𝐺‘𝑆))
160	159	oveq1d 7217	. . . . 5 ⊢ (𝑡 = 𝑆 → ((𝐺‘𝑡) / 𝑁) = ((𝐺‘𝑆) / 𝑁))
161	39, 158, 160, 1	fvmptf 6828	. . . 4 ⊢ ((𝑆 ∈ 𝑇 ∧ ((𝐺‘𝑆) / 𝑁) ∈ ℝ) → (𝐻‘𝑆) = ((𝐺‘𝑆) / 𝑁))
162	22, 156, 161	syl2anc 587	. . 3 ⊢ (𝜑 → (𝐻‘𝑆) = ((𝐺‘𝑆) / 𝑁))
163	155, 162	breqtrrd 5071	. 2 ⊢ (𝜑 → 0 < (𝐻‘𝑆))
164		nfcv 2900	. . . 4 ⊢ Ⅎℎ𝐻
165		stoweidlem36.1	. . . . . 6 ⊢ Ⅎℎ𝑄
166	165	nfel2 2918	. . . . 5 ⊢ Ⅎℎ 𝐻 ∈ 𝑄
167		nfv 1922	. . . . 5 ⊢ Ⅎℎ0 < (𝐻‘𝑆)
168	166, 167	nfan 1907	. . . 4 ⊢ Ⅎℎ(𝐻 ∈ 𝑄 ∧ 0 < (𝐻‘𝑆))
169		eleq1 2821	. . . . 5 ⊢ (ℎ = 𝐻 → (ℎ ∈ 𝑄 ↔ 𝐻 ∈ 𝑄))
170		fveq1 6705	. . . . . 6 ⊢ (ℎ = 𝐻 → (ℎ‘𝑆) = (𝐻‘𝑆))
171	170	breq2d 5055	. . . . 5 ⊢ (ℎ = 𝐻 → (0 < (ℎ‘𝑆) ↔ 0 < (𝐻‘𝑆)))
172	169, 171	anbi12d 634	. . . 4 ⊢ (ℎ = 𝐻 → ((ℎ ∈ 𝑄 ∧ 0 < (ℎ‘𝑆)) ↔ (𝐻 ∈ 𝑄 ∧ 0 < (𝐻‘𝑆))))
173	164, 168, 172	spcegf 3500	. . 3 ⊢ (𝐻 ∈ 𝑄 → ((𝐻 ∈ 𝑄 ∧ 0 < (𝐻‘𝑆)) → ∃ℎ(ℎ ∈ 𝑄 ∧ 0 < (ℎ‘𝑆))))
174	173	anabsi5 669	. 2 ⊢ ((𝐻 ∈ 𝑄 ∧ 0 < (𝐻‘𝑆)) → ∃ℎ(ℎ ∈ 𝑄 ∧ 0 < (ℎ‘𝑆)))
175	154, 163, 174	syl2anc 587	1 ⊢ (𝜑 → ∃ℎ(ℎ ∈ 𝑄 ∧ 0 < (ℎ‘𝑆)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1089   = wceq 1543  ∃wex 1787  Ⅎwnf 1791   ∈ wcel 2110  Ⅎwnfc 2880   ≠ wne 2935  ∀wral 3054  {crab 3058   ⊆ wss 3857  ∪ cuni 4809   class class class wbr 5043   ↦ cmpt 5124  ran crn 5541  ‘cfv 6369  (class class class)co 7202  supcsup 9045  ℂcc 10710  ℝcr 10711  0cc0 10712  1c1 10713   · cmul 10717   < clt 10850   ≤ cle 10851   / cdiv 11472  (,)cioo 12918  topGenctg 16914   Cn ccn 22093  Compccmp 22255

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2706  ax-rep 5168  ax-sep 5181  ax-nul 5188  ax-pow 5247  ax-pr 5311  ax-un 7512  ax-cnex 10768  ax-resscn 10769  ax-1cn 10770  ax-icn 10771  ax-addcl 10772  ax-addrcl 10773  ax-mulcl 10774  ax-mulrcl 10775  ax-mulcom 10776  ax-addass 10777  ax-mulass 10778  ax-distr 10779  ax-i2m1 10780  ax-1ne0 10781  ax-1rid 10782  ax-rnegex 10783  ax-rrecex 10784  ax-cnre 10785  ax-pre-lttri 10786  ax-pre-lttrn 10787  ax-pre-ltadd 10788  ax-pre-mulgt0 10789  ax-pre-sup 10790  ax-mulf 10792

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2537  df-eu 2566  df-clab 2713  df-cleq 2726  df-clel 2812  df-nfc 2882  df-ne 2936  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3403  df-sbc 3688  df-csb 3803  df-dif 3860  df-un 3862  df-in 3864  df-ss 3874  df-pss 3876  df-nul 4228  df-if 4430  df-pw 4505  df-sn 4532  df-pr 4534  df-tp 4536  df-op 4538  df-uni 4810  df-int 4850  df-iun 4896  df-iin 4897  df-br 5044  df-opab 5106  df-mpt 5125  df-tr 5151  df-id 5444  df-eprel 5449  df-po 5457  df-so 5458  df-fr 5498  df-se 5499  df-we 5500  df-xp 5546  df-rel 5547  df-cnv 5548  df-co 5549  df-dm 5550  df-rn 5551  df-res 5552  df-ima 5553  df-pred 6149  df-ord 6205  df-on 6206  df-lim 6207  df-suc 6208  df-iota 6327  df-fun 6371  df-fn 6372  df-f 6373  df-f1 6374  df-fo 6375  df-f1o 6376  df-fv 6377  df-isom 6378  df-riota 7159  df-ov 7205  df-oprab 7206  df-mpo 7207  df-of 7458  df-om 7634  df-1st 7750  df-2nd 7751  df-supp 7893  df-wrecs 8036  df-recs 8097  df-rdg 8135  df-1o 8191  df-2o 8192  df-er 8380  df-map 8499  df-ixp 8568  df-en 8616  df-dom 8617  df-sdom 8618  df-fin 8619  df-fsupp 8975  df-fi 9016  df-sup 9047  df-inf 9048  df-oi 9115  df-card 9538  df-pnf 10852  df-mnf 10853  df-xr 10854  df-ltxr 10855  df-le 10856  df-sub 11047  df-neg 11048  df-div 11473  df-nn 11814  df-2 11876  df-3 11877  df-4 11878  df-5 11879  df-6 11880  df-7 11881  df-8 11882  df-9 11883  df-n0 12074  df-z 12160  df-dec 12277  df-uz 12422  df-q 12528  df-rp 12570  df-xneg 12687  df-xadd 12688  df-xmul 12689  df-ioo 12922  df-icc 12925  df-fz 13079  df-fzo 13222  df-seq 13558  df-exp 13619  df-hash 13880  df-cj 14645  df-re 14646  df-im 14647  df-sqrt 14781  df-abs 14782  df-struct 16686  df-ndx 16687  df-slot 16688  df-base 16690  df-sets 16691  df-ress 16692  df-plusg 16780  df-mulr 16781  df-starv 16782  df-sca 16783  df-vsca 16784  df-ip 16785  df-tset 16786  df-ple 16787  df-ds 16789  df-unif 16790  df-hom 16791  df-cco 16792  df-rest 16899  df-topn 16900  df-0g 16918  df-gsum 16919  df-topgen 16920  df-pt 16921  df-prds 16924  df-xrs 16979  df-qtop 16984  df-imas 16985  df-xps 16987  df-mre 17061  df-mrc 17062  df-acs 17064  df-mgm 18086  df-sgrp 18135  df-mnd 18146  df-submnd 18191  df-mulg 18461  df-cntz 18683  df-cmn 19144  df-psmet 20327  df-xmet 20328  df-met 20329  df-bl 20330  df-mopn 20331  df-cnfld 20336  df-top 21763  df-topon 21780  df-topsp 21802  df-bases 21815  df-cn 22096  df-cnp 22097  df-cmp 22256  df-tx 22431  df-hmeo 22624  df-xms 23190  df-ms 23191  df-tms 23192

This theorem is referenced by:  stoweidlem43  43213