Theorem stoweidlem16 46014

Description: Lemma for stoweid 46061. The subset 𝑌 of functions in the algebra 𝐴, with values in [ 0 , 1 ], is closed under multiplication. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Hypotheses

Ref	Expression
stoweidlem16.1	⊢ Ⅎ𝑡𝜑
stoweidlem16.2	⊢ 𝑌 = {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)}
stoweidlem16.3	⊢ 𝐻 = (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡)))
stoweidlem16.4	⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ)
stoweidlem16.5	⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)

Assertion

Ref	Expression
stoweidlem16	⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → 𝐻 ∈ 𝑌)

Distinct variable groups:   𝑓,𝑔,ℎ,𝑡,𝐴   𝑇,𝑓,ℎ,𝑡   𝜑,𝑓   ℎ,𝐻

Allowed substitution hints:   𝜑(𝑡,𝑔,ℎ)   𝑇(𝑔)   𝐻(𝑡,𝑓,𝑔)   𝑌(𝑡,𝑓,𝑔,ℎ)

Proof of Theorem stoweidlem16

Step	Hyp	Ref	Expression
1		stoweidlem16.3	. . . 4 ⊢ 𝐻 = (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡)))
2		simp1 1136	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → 𝜑)
3		fveq1 6857	. . . . . . . . . . 11 ⊢ (ℎ = 𝑓 → (ℎ‘𝑡) = (𝑓‘𝑡))
4	3	breq2d 5119	. . . . . . . . . 10 ⊢ (ℎ = 𝑓 → (0 ≤ (ℎ‘𝑡) ↔ 0 ≤ (𝑓‘𝑡)))
5	3	breq1d 5117	. . . . . . . . . 10 ⊢ (ℎ = 𝑓 → ((ℎ‘𝑡) ≤ 1 ↔ (𝑓‘𝑡) ≤ 1))
6	4, 5	anbi12d 632	. . . . . . . . 9 ⊢ (ℎ = 𝑓 → ((0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ (0 ≤ (𝑓‘𝑡) ∧ (𝑓‘𝑡) ≤ 1)))
7	6	ralbidv 3156	. . . . . . . 8 ⊢ (ℎ = 𝑓 → (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑓‘𝑡) ∧ (𝑓‘𝑡) ≤ 1)))
8		stoweidlem16.2	. . . . . . . 8 ⊢ 𝑌 = {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)}
9	7, 8	elrab2 3662	. . . . . . 7 ⊢ (𝑓 ∈ 𝑌 ↔ (𝑓 ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑓‘𝑡) ∧ (𝑓‘𝑡) ≤ 1)))
10	9	simplbi 497	. . . . . 6 ⊢ (𝑓 ∈ 𝑌 → 𝑓 ∈ 𝐴)
11	10	3ad2ant2 1134	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → 𝑓 ∈ 𝐴)
12		fveq1 6857	. . . . . . . . . . 11 ⊢ (ℎ = 𝑔 → (ℎ‘𝑡) = (𝑔‘𝑡))
13	12	breq2d 5119	. . . . . . . . . 10 ⊢ (ℎ = 𝑔 → (0 ≤ (ℎ‘𝑡) ↔ 0 ≤ (𝑔‘𝑡)))
14	12	breq1d 5117	. . . . . . . . . 10 ⊢ (ℎ = 𝑔 → ((ℎ‘𝑡) ≤ 1 ↔ (𝑔‘𝑡) ≤ 1))
15	13, 14	anbi12d 632	. . . . . . . . 9 ⊢ (ℎ = 𝑔 → ((0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ (0 ≤ (𝑔‘𝑡) ∧ (𝑔‘𝑡) ≤ 1)))
16	15	ralbidv 3156	. . . . . . . 8 ⊢ (ℎ = 𝑔 → (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑔‘𝑡) ∧ (𝑔‘𝑡) ≤ 1)))
17	16, 8	elrab2 3662	. . . . . . 7 ⊢ (𝑔 ∈ 𝑌 ↔ (𝑔 ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑔‘𝑡) ∧ (𝑔‘𝑡) ≤ 1)))
18	17	simplbi 497	. . . . . 6 ⊢ (𝑔 ∈ 𝑌 → 𝑔 ∈ 𝐴)
19	18	3ad2ant3 1135	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → 𝑔 ∈ 𝐴)
20		stoweidlem16.5	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
21	2, 11, 19, 20	syl3anc 1373	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
22	1, 21	eqeltrid 2832	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → 𝐻 ∈ 𝐴)
23		stoweidlem16.1	. . . . 5 ⊢ Ⅎ𝑡𝜑
24		nfra1 3261	. . . . . . . 8 ⊢ Ⅎ𝑡∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)
25		nfcv 2891	. . . . . . . 8 ⊢ Ⅎ𝑡𝐴
26	24, 25	nfrabw 3443	. . . . . . 7 ⊢ Ⅎ𝑡{ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)}
27	8, 26	nfcxfr 2889	. . . . . 6 ⊢ Ⅎ𝑡𝑌
28	27	nfcri 2883	. . . . 5 ⊢ Ⅎ𝑡 𝑓 ∈ 𝑌
29	27	nfcri 2883	. . . . 5 ⊢ Ⅎ𝑡 𝑔 ∈ 𝑌
30	23, 28, 29	nf3an 1901	. . . 4 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌)
31	2, 11	jca 511	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → (𝜑 ∧ 𝑓 ∈ 𝐴))
32	31	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → (𝜑 ∧ 𝑓 ∈ 𝐴))
33		stoweidlem16.4	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ)
34	32, 33	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → 𝑓:𝑇⟶ℝ)
35		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇)
36	34, 35	ffvelcdmd 7057	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → (𝑓‘𝑡) ∈ ℝ)
37	2, 19	jca 511	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → (𝜑 ∧ 𝑔 ∈ 𝐴))
38		eleq1w 2811	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑔 → (𝑓 ∈ 𝐴 ↔ 𝑔 ∈ 𝐴))
39	38	anbi2d 630	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → ((𝜑 ∧ 𝑓 ∈ 𝐴) ↔ (𝜑 ∧ 𝑔 ∈ 𝐴)))
40		feq1 6666	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → (𝑓:𝑇⟶ℝ ↔ 𝑔:𝑇⟶ℝ))
41	39, 40	imbi12d 344	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → (((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ 𝑔 ∈ 𝐴) → 𝑔:𝑇⟶ℝ)))
42	41, 33	vtoclg 3520	. . . . . . . . . 10 ⊢ (𝑔 ∈ 𝐴 → ((𝜑 ∧ 𝑔 ∈ 𝐴) → 𝑔:𝑇⟶ℝ))
43	19, 37, 42	sylc 65	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → 𝑔:𝑇⟶ℝ)
44	43	ffvelcdmda 7056	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → (𝑔‘𝑡) ∈ ℝ)
45	9	simprbi 496	. . . . . . . . . . 11 ⊢ (𝑓 ∈ 𝑌 → ∀𝑡 ∈ 𝑇 (0 ≤ (𝑓‘𝑡) ∧ (𝑓‘𝑡) ≤ 1))
46	45	3ad2ant2 1134	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → ∀𝑡 ∈ 𝑇 (0 ≤ (𝑓‘𝑡) ∧ (𝑓‘𝑡) ≤ 1))
47	46	r19.21bi 3229	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → (0 ≤ (𝑓‘𝑡) ∧ (𝑓‘𝑡) ≤ 1))
48	47	simpld 494	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → 0 ≤ (𝑓‘𝑡))
49	17	simprbi 496	. . . . . . . . . . 11 ⊢ (𝑔 ∈ 𝑌 → ∀𝑡 ∈ 𝑇 (0 ≤ (𝑔‘𝑡) ∧ (𝑔‘𝑡) ≤ 1))
50	49	3ad2ant3 1135	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → ∀𝑡 ∈ 𝑇 (0 ≤ (𝑔‘𝑡) ∧ (𝑔‘𝑡) ≤ 1))
51	50	r19.21bi 3229	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → (0 ≤ (𝑔‘𝑡) ∧ (𝑔‘𝑡) ≤ 1))
52	51	simpld 494	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → 0 ≤ (𝑔‘𝑡))
53	36, 44, 48, 52	mulge0d 11755	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → 0 ≤ ((𝑓‘𝑡) · (𝑔‘𝑡)))
54	36, 44	remulcld 11204	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → ((𝑓‘𝑡) · (𝑔‘𝑡)) ∈ ℝ)
55	1	fvmpt2 6979	. . . . . . . 8 ⊢ ((𝑡 ∈ 𝑇 ∧ ((𝑓‘𝑡) · (𝑔‘𝑡)) ∈ ℝ) → (𝐻‘𝑡) = ((𝑓‘𝑡) · (𝑔‘𝑡)))
56	35, 54, 55	syl2anc 584	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → (𝐻‘𝑡) = ((𝑓‘𝑡) · (𝑔‘𝑡)))
57	53, 56	breqtrrd 5135	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → 0 ≤ (𝐻‘𝑡))
58		1red 11175	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → 1 ∈ ℝ)
59	47	simprd 495	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → (𝑓‘𝑡) ≤ 1)
60	51	simprd 495	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → (𝑔‘𝑡) ≤ 1)
61	36, 58, 44, 58, 48, 52, 59, 60	lemul12ad 12125	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → ((𝑓‘𝑡) · (𝑔‘𝑡)) ≤ (1 · 1))
62		1t1e1 12343	. . . . . . . 8 ⊢ (1 · 1) = 1
63	61, 62	breqtrdi 5148	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → ((𝑓‘𝑡) · (𝑔‘𝑡)) ≤ 1)
64	56, 63	eqbrtrd 5129	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → (𝐻‘𝑡) ≤ 1)
65	57, 64	jca 511	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → (0 ≤ (𝐻‘𝑡) ∧ (𝐻‘𝑡) ≤ 1))
66	65	ex 412	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → (𝑡 ∈ 𝑇 → (0 ≤ (𝐻‘𝑡) ∧ (𝐻‘𝑡) ≤ 1)))
67	30, 66	ralrimi 3235	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → ∀𝑡 ∈ 𝑇 (0 ≤ (𝐻‘𝑡) ∧ (𝐻‘𝑡) ≤ 1))
68		nfmpt1 5206	. . . . . . 7 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡)))
69	1, 68	nfcxfr 2889	. . . . . 6 ⊢ Ⅎ𝑡𝐻
70	69	nfeq2 2909	. . . . 5 ⊢ Ⅎ𝑡 ℎ = 𝐻
71		fveq1 6857	. . . . . . 7 ⊢ (ℎ = 𝐻 → (ℎ‘𝑡) = (𝐻‘𝑡))
72	71	breq2d 5119	. . . . . 6 ⊢ (ℎ = 𝐻 → (0 ≤ (ℎ‘𝑡) ↔ 0 ≤ (𝐻‘𝑡)))
73	71	breq1d 5117	. . . . . 6 ⊢ (ℎ = 𝐻 → ((ℎ‘𝑡) ≤ 1 ↔ (𝐻‘𝑡) ≤ 1))
74	72, 73	anbi12d 632	. . . . 5 ⊢ (ℎ = 𝐻 → ((0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ (0 ≤ (𝐻‘𝑡) ∧ (𝐻‘𝑡) ≤ 1)))
75	70, 74	ralbid 3250	. . . 4 ⊢ (ℎ = 𝐻 → (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ ∀𝑡 ∈ 𝑇 (0 ≤ (𝐻‘𝑡) ∧ (𝐻‘𝑡) ≤ 1)))
76	75	elrab 3659	. . 3 ⊢ (𝐻 ∈ {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)} ↔ (𝐻 ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (𝐻‘𝑡) ∧ (𝐻‘𝑡) ≤ 1)))
77	22, 67, 76	sylanbrc 583	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → 𝐻 ∈ {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)})
78	77, 8	eleqtrrdi 2839	1 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → 𝐻 ∈ 𝑌)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540  Ⅎwnf 1783   ∈ wcel 2109  ∀wral 3044  {crab 3405   class class class wbr 5107   ↦ cmpt 5188  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387  ℝcr 11067  0cc0 11068  1c1 11069   · cmul 11073   ≤ cle 11209

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-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

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-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-po 5546  df-so 5547  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-er 8671  df-en 8919  df-dom 8920  df-sdom 8921  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408

This theorem is referenced by:  stoweidlem48  46046  stoweidlem51  46049