Theorem stoweidlem36 42328

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 2823	. . . . . . . . . . . 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 2997	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑡 𝑓 = 𝐹
11	9	nfeq2 2997	. . . . . . . . . . . . . . . 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 42298	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐹 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) · (𝐹‘𝑡))) ∈ 𝐴)
14	8, 8, 13	mpd3an23 1459	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) · (𝐹‘𝑡))) ∈ 𝐴)
15	7, 14	eqeltrid 2919	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺 ∈ 𝐴)
16	6, 15	sseldd 3970	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾))
17	3, 4, 5, 16	fcnre 41289	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺:𝑇⟶ℝ)
18	17	ffvelrnda 6853	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑡) ∈ ℝ)
19	18	recnd 10671	. . . . . . . . 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 4303	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑇 ≠ ∅)
24	4, 3, 21, 16, 23	cncmpmax 41296	. . . . . . . . . . . . 13 ⊢ (𝜑 → (sup(ran 𝐺, ℝ, < ) ∈ ran 𝐺 ∧ sup(ran 𝐺, ℝ, < ) ∈ ℝ ∧ ∀𝑠 ∈ 𝑇 (𝐺‘𝑠) ≤ sup(ran 𝐺, ℝ, < )))
25	24	simp2d 1139	. . . . . . . . . . . 12 ⊢ (𝜑 → sup(ran 𝐺, ℝ, < ) ∈ ℝ)
26	20, 25	eqeltrid 2919	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℝ)
27	26	recnd 10671	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℂ)
28	27	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑁 ∈ ℂ)
29		0red 10646	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ∈ ℝ)
30	17, 22	ffvelrnd 6854	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐺‘𝑆) ∈ ℝ)
31	6, 8	sseldd 3970	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))
32	3, 4, 5, 31	fcnre 41289	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐹:𝑇⟶ℝ)
33	32, 22	ffvelrnd 6854	. . . . . . . . . . . . . . 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 3087	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐹‘𝑆) ≠ 0)
37	33, 36	msqgt0d 11209	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < ((𝐹‘𝑆) · (𝐹‘𝑆)))
38	33, 33	remulcld 10673	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐹‘𝑆) · (𝐹‘𝑆)) ∈ ℝ)
39		nfcv 2979	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑡𝑆
40	9, 39	nffv 6682	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑡(𝐹‘𝑆)
41		nfcv 2979	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑡 ·
42	40, 41, 40	nfov 7188	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑡((𝐹‘𝑆) · (𝐹‘𝑆))
43		fveq2 6672	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = 𝑆 → (𝐹‘𝑡) = (𝐹‘𝑆))
44	43, 43	oveq12d 7176	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑆 → ((𝐹‘𝑡) · (𝐹‘𝑡)) = ((𝐹‘𝑆) · (𝐹‘𝑆)))
45	39, 42, 44, 7	fvmptf 6791	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ∈ 𝑇 ∧ ((𝐹‘𝑆) · (𝐹‘𝑆)) ∈ ℝ) → (𝐺‘𝑆) = ((𝐹‘𝑆) · (𝐹‘𝑆)))
46	22, 38, 45	syl2anc 586	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐺‘𝑆) = ((𝐹‘𝑆) · (𝐹‘𝑆)))
47	37, 46	breqtrrd 5096	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 < (𝐺‘𝑆))
48	24	simp3d 1140	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑠 ∈ 𝑇 (𝐺‘𝑠) ≤ sup(ran 𝐺, ℝ, < ))
49		fveq2 6672	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = 𝑆 → (𝐺‘𝑠) = (𝐺‘𝑆))
50	49	breq1d 5078	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = 𝑆 → ((𝐺‘𝑠) ≤ sup(ran 𝐺, ℝ, < ) ↔ (𝐺‘𝑆) ≤ sup(ran 𝐺, ℝ, < )))
51	50	rspccva 3624	. . . . . . . . . . . . . 14 ⊢ ((∀𝑠 ∈ 𝑇 (𝐺‘𝑠) ≤ sup(ran 𝐺, ℝ, < ) ∧ 𝑆 ∈ 𝑇) → (𝐺‘𝑆) ≤ sup(ran 𝐺, ℝ, < ))
52	48, 22, 51	syl2anc 586	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐺‘𝑆) ≤ sup(ran 𝐺, ℝ, < ))
53	29, 30, 25, 47, 52	ltletrd 10802	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < sup(ran 𝐺, ℝ, < ))
54	53	gt0ne0d 11206	. . . . . . . . . . 11 ⊢ (𝜑 → sup(ran 𝐺, ℝ, < ) ≠ 0)
55	20	neeq1i 3082	. . . . . . . . . . 11 ⊢ (𝑁 ≠ 0 ↔ sup(ran 𝐺, ℝ, < ) ≠ 0)
56	54, 55	sylibr 236	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ≠ 0)
57	56	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑁 ≠ 0)
58	19, 28, 57	divrecd 11421	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐺‘𝑡) / 𝑁) = ((𝐺‘𝑡) · (1 / 𝑁)))
59		simpr 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇)
60	26, 56	rereccld 11469	. . . . . . . . . . 11 ⊢ (𝜑 → (1 / 𝑁) ∈ ℝ)
61	60	adantr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (1 / 𝑁) ∈ ℝ)
62		eqid 2823	. . . . . . . . . . 11 ⊢ (𝑡 ∈ 𝑇 ↦ (1 / 𝑁)) = (𝑡 ∈ 𝑇 ↦ (1 / 𝑁))
63	62	fvmpt2 6781	. . . . . . . . . 10 ⊢ ((𝑡 ∈ 𝑇 ∧ (1 / 𝑁) ∈ ℝ) → ((𝑡 ∈ 𝑇 ↦ (1 / 𝑁))‘𝑡) = (1 / 𝑁))
64	59, 61, 63	syl2anc 586	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝑡 ∈ 𝑇 ↦ (1 / 𝑁))‘𝑡) = (1 / 𝑁))
65	64	oveq2d 7174	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐺‘𝑡) · ((𝑡 ∈ 𝑇 ↦ (1 / 𝑁))‘𝑡)) = ((𝐺‘𝑡) · (1 / 𝑁)))
66	58, 65	eqtr4d 2861	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐺‘𝑡) / 𝑁) = ((𝐺‘𝑡) · ((𝑡 ∈ 𝑇 ↦ (1 / 𝑁))‘𝑡)))
67	2, 66	mpteq2da 5162	. . . . . 6 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) / 𝑁)) = (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) · ((𝑡 ∈ 𝑇 ↦ (1 / 𝑁))‘𝑡))))
68	1, 67	syl5eq 2870	. . . . 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 42296	. . . . . . 7 ⊢ ((𝜑 ∧ (1 / 𝑁) ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ (1 / 𝑁)) ∈ 𝐴)
71	60, 70	mpdan 685	. . . . . 6 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ (1 / 𝑁)) ∈ 𝐴)
72		stoweidlem36.4	. . . . . . . 8 ⊢ Ⅎ𝑡𝐺
73	72	nfeq2 2997	. . . . . . 7 ⊢ Ⅎ𝑡 𝑓 = 𝐺
74		nfmpt1 5166	. . . . . . . 8 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ (1 / 𝑁))
75	74	nfeq2 2997	. . . . . . 7 ⊢ Ⅎ𝑡 𝑔 = (𝑡 ∈ 𝑇 ↦ (1 / 𝑁))
76	73, 75, 12	stoweidlem6 42298	. . . . . 6 ⊢ ((𝜑 ∧ 𝐺 ∈ 𝐴 ∧ (𝑡 ∈ 𝑇 ↦ (1 / 𝑁)) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) · ((𝑡 ∈ 𝑇 ↦ (1 / 𝑁))‘𝑡))) ∈ 𝐴)
77	15, 71, 76	mpd3an23 1459	. . . . 5 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) · ((𝑡 ∈ 𝑇 ↦ (1 / 𝑁))‘𝑡))) ∈ 𝐴)
78	68, 77	eqeltrd 2915	. . . 4 ⊢ (𝜑 → 𝐻 ∈ 𝐴)
79		stoweidlem36.17	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝑇)
80	17, 79	ffvelrnd 6854	. . . . . . . 8 ⊢ (𝜑 → (𝐺‘𝑍) ∈ ℝ)
81	80, 26, 56	redivcld 11470	. . . . . . 7 ⊢ (𝜑 → ((𝐺‘𝑍) / 𝑁) ∈ ℝ)
82		nfcv 2979	. . . . . . . 8 ⊢ Ⅎ𝑡𝑍
83	72, 82	nffv 6682	. . . . . . . . 9 ⊢ Ⅎ𝑡(𝐺‘𝑍)
84		nfcv 2979	. . . . . . . . 9 ⊢ Ⅎ𝑡 /
85		nfcv 2979	. . . . . . . . 9 ⊢ Ⅎ𝑡𝑁
86	83, 84, 85	nfov 7188	. . . . . . . 8 ⊢ Ⅎ𝑡((𝐺‘𝑍) / 𝑁)
87		fveq2 6672	. . . . . . . . 9 ⊢ (𝑡 = 𝑍 → (𝐺‘𝑡) = (𝐺‘𝑍))
88	87	oveq1d 7173	. . . . . . . 8 ⊢ (𝑡 = 𝑍 → ((𝐺‘𝑡) / 𝑁) = ((𝐺‘𝑍) / 𝑁))
89	82, 86, 88, 1	fvmptf 6791	. . . . . . 7 ⊢ ((𝑍 ∈ 𝑇 ∧ ((𝐺‘𝑍) / 𝑁) ∈ ℝ) → (𝐻‘𝑍) = ((𝐺‘𝑍) / 𝑁))
90	79, 81, 89	syl2anc 586	. . . . . 6 ⊢ (𝜑 → (𝐻‘𝑍) = ((𝐺‘𝑍) / 𝑁))
91		0re 10645	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
92	35, 91	eqeltrdi 2923	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹‘𝑍) ∈ ℝ)
93	92, 92	remulcld 10673	. . . . . . . . 9 ⊢ (𝜑 → ((𝐹‘𝑍) · (𝐹‘𝑍)) ∈ ℝ)
94	9, 82	nffv 6682	. . . . . . . . . . 11 ⊢ Ⅎ𝑡(𝐹‘𝑍)
95	94, 41, 94	nfov 7188	. . . . . . . . . 10 ⊢ Ⅎ𝑡((𝐹‘𝑍) · (𝐹‘𝑍))
96		fveq2 6672	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑍 → (𝐹‘𝑡) = (𝐹‘𝑍))
97	96, 96	oveq12d 7176	. . . . . . . . . 10 ⊢ (𝑡 = 𝑍 → ((𝐹‘𝑡) · (𝐹‘𝑡)) = ((𝐹‘𝑍) · (𝐹‘𝑍)))
98	82, 95, 97, 7	fvmptf 6791	. . . . . . . . 9 ⊢ ((𝑍 ∈ 𝑇 ∧ ((𝐹‘𝑍) · (𝐹‘𝑍)) ∈ ℝ) → (𝐺‘𝑍) = ((𝐹‘𝑍) · (𝐹‘𝑍)))
99	79, 93, 98	syl2anc 586	. . . . . . . 8 ⊢ (𝜑 → (𝐺‘𝑍) = ((𝐹‘𝑍) · (𝐹‘𝑍)))
100	35, 35	oveq12d 7176	. . . . . . . . 9 ⊢ (𝜑 → ((𝐹‘𝑍) · (𝐹‘𝑍)) = (0 · 0))
101		0cn 10635	. . . . . . . . . 10 ⊢ 0 ∈ ℂ
102	101	mul02i 10831	. . . . . . . . 9 ⊢ (0 · 0) = 0
103	100, 102	syl6eq 2874	. . . . . . . 8 ⊢ (𝜑 → ((𝐹‘𝑍) · (𝐹‘𝑍)) = 0)
104	99, 103	eqtrd 2858	. . . . . . 7 ⊢ (𝜑 → (𝐺‘𝑍) = 0)
105	104	oveq1d 7173	. . . . . 6 ⊢ (𝜑 → ((𝐺‘𝑍) / 𝑁) = (0 / 𝑁))
106	27, 56	div0d 11417	. . . . . 6 ⊢ (𝜑 → (0 / 𝑁) = 0)
107	90, 105, 106	3eqtrd 2862	. . . . 5 ⊢ (𝜑 → (𝐻‘𝑍) = 0)
108	32	ffvelrnda 6853	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℝ)
109	108	msqge0d 11210	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ ((𝐹‘𝑡) · (𝐹‘𝑡)))
110	108, 108	remulcld 10673	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐹‘𝑡) · (𝐹‘𝑡)) ∈ ℝ)
111	7	fvmpt2 6781	. . . . . . . . . . . 12 ⊢ ((𝑡 ∈ 𝑇 ∧ ((𝐹‘𝑡) · (𝐹‘𝑡)) ∈ ℝ) → (𝐺‘𝑡) = ((𝐹‘𝑡) · (𝐹‘𝑡)))
112	59, 110, 111	syl2anc 586	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑡) = ((𝐹‘𝑡) · (𝐹‘𝑡)))
113	109, 112	breqtrrd 5096	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ (𝐺‘𝑡))
114	26	adantr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑁 ∈ ℝ)
115	53, 20	breqtrrdi 5110	. . . . . . . . . . 11 ⊢ (𝜑 → 0 < 𝑁)
116	115	adantr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 < 𝑁)
117		divge0 11511	. . . . . . . . . 10 ⊢ ((((𝐺‘𝑡) ∈ ℝ ∧ 0 ≤ (𝐺‘𝑡)) ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → 0 ≤ ((𝐺‘𝑡) / 𝑁))
118	18, 113, 114, 116, 117	syl22anc 836	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ ((𝐺‘𝑡) / 𝑁))
119	18, 114, 57	redivcld 11470	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐺‘𝑡) / 𝑁) ∈ ℝ)
120	1	fvmpt2 6781	. . . . . . . . . 10 ⊢ ((𝑡 ∈ 𝑇 ∧ ((𝐺‘𝑡) / 𝑁) ∈ ℝ) → (𝐻‘𝑡) = ((𝐺‘𝑡) / 𝑁))
121	59, 119, 120	syl2anc 586	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐻‘𝑡) = ((𝐺‘𝑡) / 𝑁))
122	118, 121	breqtrrd 5096	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ (𝐻‘𝑡))
123	19	div1d 11410	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐺‘𝑡) / 1) = (𝐺‘𝑡))
124		fveq2 6672	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = 𝑡 → (𝐺‘𝑠) = (𝐺‘𝑡))
125	124	breq1d 5078	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑡 → ((𝐺‘𝑠) ≤ sup(ran 𝐺, ℝ, < ) ↔ (𝐺‘𝑡) ≤ sup(ran 𝐺, ℝ, < )))
126	125	rspccva 3624	. . . . . . . . . . . . 13 ⊢ ((∀𝑠 ∈ 𝑇 (𝐺‘𝑠) ≤ sup(ran 𝐺, ℝ, < ) ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑡) ≤ sup(ran 𝐺, ℝ, < ))
127	48, 126	sylan 582	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑡) ≤ sup(ran 𝐺, ℝ, < ))
128	127, 20	breqtrrdi 5110	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑡) ≤ 𝑁)
129	123, 128	eqbrtrd 5090	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐺‘𝑡) / 1) ≤ 𝑁)
130		1red 10644	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 1 ∈ ℝ)
131		0lt1 11164	. . . . . . . . . . . 12 ⊢ 0 < 1
132	131	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 < 1)
133		lediv23 11534	. . . . . . . . . . 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 1375	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (((𝐺‘𝑡) / 𝑁) ≤ 1 ↔ ((𝐺‘𝑡) / 1) ≤ 𝑁))
135	129, 134	mpbird 259	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐺‘𝑡) / 𝑁) ≤ 1)
136	121, 135	eqbrtrd 5090	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐻‘𝑡) ≤ 1)
137	122, 136	jca 514	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (0 ≤ (𝐻‘𝑡) ∧ (𝐻‘𝑡) ≤ 1))
138	137	ex 415	. . . . . 6 ⊢ (𝜑 → (𝑡 ∈ 𝑇 → (0 ≤ (𝐻‘𝑡) ∧ (𝐻‘𝑡) ≤ 1)))
139	2, 138	ralrimi 3218	. . . . 5 ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 (0 ≤ (𝐻‘𝑡) ∧ (𝐻‘𝑡) ≤ 1))
140	107, 139	jca 514	. . . 4 ⊢ (𝜑 → ((𝐻‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (𝐻‘𝑡) ∧ (𝐻‘𝑡) ≤ 1)))
141		fveq1 6671	. . . . . . 7 ⊢ (ℎ = 𝐻 → (ℎ‘𝑍) = (𝐻‘𝑍))
142	141	eqeq1d 2825	. . . . . 6 ⊢ (ℎ = 𝐻 → ((ℎ‘𝑍) = 0 ↔ (𝐻‘𝑍) = 0))
143		stoweidlem36.2	. . . . . . . 8 ⊢ Ⅎ𝑡𝐻
144	143	nfeq2 2997	. . . . . . 7 ⊢ Ⅎ𝑡 ℎ = 𝐻
145		fveq1 6671	. . . . . . . . 9 ⊢ (ℎ = 𝐻 → (ℎ‘𝑡) = (𝐻‘𝑡))
146	145	breq2d 5080	. . . . . . . 8 ⊢ (ℎ = 𝐻 → (0 ≤ (ℎ‘𝑡) ↔ 0 ≤ (𝐻‘𝑡)))
147	145	breq1d 5078	. . . . . . . 8 ⊢ (ℎ = 𝐻 → ((ℎ‘𝑡) ≤ 1 ↔ (𝐻‘𝑡) ≤ 1))
148	146, 147	anbi12d 632	. . . . . . 7 ⊢ (ℎ = 𝐻 → ((0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ (0 ≤ (𝐻‘𝑡) ∧ (𝐻‘𝑡) ≤ 1)))
149	144, 148	ralbid 3233	. . . . . 6 ⊢ (ℎ = 𝐻 → (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ ∀𝑡 ∈ 𝑇 (0 ≤ (𝐻‘𝑡) ∧ (𝐻‘𝑡) ≤ 1)))
150	142, 149	anbi12d 632	. . . . 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 3682	. . . 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 585	. . 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 2926	. 2 ⊢ (𝜑 → 𝐻 ∈ 𝑄)
155	30, 26, 47, 115	divgt0d 11577	. . 3 ⊢ (𝜑 → 0 < ((𝐺‘𝑆) / 𝑁))
156	30, 26, 56	redivcld 11470	. . . 4 ⊢ (𝜑 → ((𝐺‘𝑆) / 𝑁) ∈ ℝ)
157	72, 39	nffv 6682	. . . . . 6 ⊢ Ⅎ𝑡(𝐺‘𝑆)
158	157, 84, 85	nfov 7188	. . . . 5 ⊢ Ⅎ𝑡((𝐺‘𝑆) / 𝑁)
159		fveq2 6672	. . . . . 6 ⊢ (𝑡 = 𝑆 → (𝐺‘𝑡) = (𝐺‘𝑆))
160	159	oveq1d 7173	. . . . 5 ⊢ (𝑡 = 𝑆 → ((𝐺‘𝑡) / 𝑁) = ((𝐺‘𝑆) / 𝑁))
161	39, 158, 160, 1	fvmptf 6791	. . . 4 ⊢ ((𝑆 ∈ 𝑇 ∧ ((𝐺‘𝑆) / 𝑁) ∈ ℝ) → (𝐻‘𝑆) = ((𝐺‘𝑆) / 𝑁))
162	22, 156, 161	syl2anc 586	. . 3 ⊢ (𝜑 → (𝐻‘𝑆) = ((𝐺‘𝑆) / 𝑁))
163	155, 162	breqtrrd 5096	. 2 ⊢ (𝜑 → 0 < (𝐻‘𝑆))
164		nfcv 2979	. . . 4 ⊢ Ⅎℎ𝐻
165		stoweidlem36.1	. . . . . 6 ⊢ Ⅎℎ𝑄
166	165	nfel2 2998	. . . . 5 ⊢ Ⅎℎ 𝐻 ∈ 𝑄
167		nfv 1915	. . . . 5 ⊢ Ⅎℎ0 < (𝐻‘𝑆)
168	166, 167	nfan 1900	. . . 4 ⊢ Ⅎℎ(𝐻 ∈ 𝑄 ∧ 0 < (𝐻‘𝑆))
169		eleq1 2902	. . . . 5 ⊢ (ℎ = 𝐻 → (ℎ ∈ 𝑄 ↔ 𝐻 ∈ 𝑄))
170		fveq1 6671	. . . . . 6 ⊢ (ℎ = 𝐻 → (ℎ‘𝑆) = (𝐻‘𝑆))
171	170	breq2d 5080	. . . . 5 ⊢ (ℎ = 𝐻 → (0 < (ℎ‘𝑆) ↔ 0 < (𝐻‘𝑆)))
172	169, 171	anbi12d 632	. . . 4 ⊢ (ℎ = 𝐻 → ((ℎ ∈ 𝑄 ∧ 0 < (ℎ‘𝑆)) ↔ (𝐻 ∈ 𝑄 ∧ 0 < (𝐻‘𝑆))))
173	164, 168, 172	spcegf 3593	. . 3 ⊢ (𝐻 ∈ 𝑄 → ((𝐻 ∈ 𝑄 ∧ 0 < (𝐻‘𝑆)) → ∃ℎ(ℎ ∈ 𝑄 ∧ 0 < (ℎ‘𝑆))))
174	173	anabsi5 667	. 2 ⊢ ((𝐻 ∈ 𝑄 ∧ 0 < (𝐻‘𝑆)) → ∃ℎ(ℎ ∈ 𝑄 ∧ 0 < (ℎ‘𝑆)))
175	154, 163, 174	syl2anc 586	1 ⊢ (𝜑 → ∃ℎ(ℎ ∈ 𝑄 ∧ 0 < (ℎ‘𝑆)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537  ∃wex 1780  Ⅎwnf 1784   ∈ wcel 2114  Ⅎwnfc 2963   ≠ wne 3018  ∀wral 3140  {crab 3144   ⊆ wss 3938  ∪ cuni 4840   class class class wbr 5068   ↦ cmpt 5148  ran crn 5558  ‘cfv 6357  (class class class)co 7158  supcsup 8906  ℂcc 10537  ℝcr 10538  0cc0 10539  1c1 10540   · cmul 10544   < clt 10677   ≤ cle 10678   / cdiv 11299  (,)cioo 12741  topGenctg 16713   Cn ccn 21834  Compccmp 21996

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616  ax-pre-sup 10617  ax-mulf 10619

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-iin 4924  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-se 5517  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-isom 6366  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-of 7411  df-om 7583  df-1st 7691  df-2nd 7692  df-supp 7833  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-2o 8105  df-oadd 8108  df-er 8291  df-map 8410  df-ixp 8464  df-en 8512  df-dom 8513  df-sdom 8514  df-fin 8515  df-fsupp 8836  df-fi 8877  df-sup 8908  df-inf 8909  df-oi 8976  df-card 9370  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-div 11300  df-nn 11641  df-2 11703  df-3 11704  df-4 11705  df-5 11706  df-6 11707  df-7 11708  df-8 11709  df-9 11710  df-n0 11901  df-z 11985  df-dec 12102  df-uz 12247  df-q 12352  df-rp 12393  df-xneg 12510  df-xadd 12511  df-xmul 12512  df-ioo 12745  df-icc 12748  df-fz 12896  df-fzo 13037  df-seq 13373  df-exp 13433  df-hash 13694  df-cj 14460  df-re 14461  df-im 14462  df-sqrt 14596  df-abs 14597  df-struct 16487  df-ndx 16488  df-slot 16489  df-base 16491  df-sets 16492  df-ress 16493  df-plusg 16580  df-mulr 16581  df-starv 16582  df-sca 16583  df-vsca 16584  df-ip 16585  df-tset 16586  df-ple 16587  df-ds 16589  df-unif 16590  df-hom 16591  df-cco 16592  df-rest 16698  df-topn 16699  df-0g 16717  df-gsum 16718  df-topgen 16719  df-pt 16720  df-prds 16723  df-xrs 16777  df-qtop 16782  df-imas 16783  df-xps 16785  df-mre 16859  df-mrc 16860  df-acs 16862  df-mgm 17854  df-sgrp 17903  df-mnd 17914  df-submnd 17959  df-mulg 18227  df-cntz 18449  df-cmn 18910  df-psmet 20539  df-xmet 20540  df-met 20541  df-bl 20542  df-mopn 20543  df-cnfld 20548  df-top 21504  df-topon 21521  df-topsp 21543  df-bases 21556  df-cn 21837  df-cnp 21838  df-cmp 21997  df-tx 22172  df-hmeo 22365  df-xms 22932  df-ms 22933  df-tms 22934

This theorem is referenced by:  stoweidlem43  42335