Theorem stoweidlem16 46758

Description: Lemma for stoweid 46805. 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 1154	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → 𝜑)
3		fveq1 6880	. . . . . . . . . . 11 ⊢ (ℎ = 𝑓 → (ℎ‘𝑡) = (𝑓‘𝑡))
4	3	breq2d 5121	. . . . . . . . . 10 ⊢ (ℎ = 𝑓 → (0 ≤ (ℎ‘𝑡) ↔ 0 ≤ (𝑓‘𝑡)))
5	3	breq1d 5119	. . . . . . . . . 10 ⊢ (ℎ = 𝑓 → ((ℎ‘𝑡) ≤ 1 ↔ (𝑓‘𝑡) ≤ 1))
6	4, 5	anbi12d 643	. . . . . . . . 9 ⊢ (ℎ = 𝑓 → ((0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ (0 ≤ (𝑓‘𝑡) ∧ (𝑓‘𝑡) ≤ 1)))
7	6	ralbidv 3188	. . . . . . . 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 3654	. . . . . . 7 ⊢ (𝑓 ∈ 𝑌 ↔ (𝑓 ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑓‘𝑡) ∧ (𝑓‘𝑡) ≤ 1)))
10	9	simplbi 501	. . . . . 6 ⊢ (𝑓 ∈ 𝑌 → 𝑓 ∈ 𝐴)
11	10	3ad2ant2 1152	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → 𝑓 ∈ 𝐴)
12		fveq1 6880	. . . . . . . . . . 11 ⊢ (ℎ = 𝑔 → (ℎ‘𝑡) = (𝑔‘𝑡))
13	12	breq2d 5121	. . . . . . . . . 10 ⊢ (ℎ = 𝑔 → (0 ≤ (ℎ‘𝑡) ↔ 0 ≤ (𝑔‘𝑡)))
14	12	breq1d 5119	. . . . . . . . . 10 ⊢ (ℎ = 𝑔 → ((ℎ‘𝑡) ≤ 1 ↔ (𝑔‘𝑡) ≤ 1))
15	13, 14	anbi12d 643	. . . . . . . . 9 ⊢ (ℎ = 𝑔 → ((0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ (0 ≤ (𝑔‘𝑡) ∧ (𝑔‘𝑡) ≤ 1)))
16	15	ralbidv 3188	. . . . . . . 8 ⊢ (ℎ = 𝑔 → (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑔‘𝑡) ∧ (𝑔‘𝑡) ≤ 1)))
17	16, 8	elrab2 3654	. . . . . . 7 ⊢ (𝑔 ∈ 𝑌 ↔ (𝑔 ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑔‘𝑡) ∧ (𝑔‘𝑡) ≤ 1)))
18	17	simplbi 501	. . . . . 6 ⊢ (𝑔 ∈ 𝑌 → 𝑔 ∈ 𝐴)
19	18	3ad2ant3 1153	. . . . 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 1398	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
22	1, 21	eqeltrid 2867	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → 𝐻 ∈ 𝐴)
23		stoweidlem16.1	. . . . 5 ⊢ Ⅎ𝑡𝜑
24		nfra1 3289	. . . . . . . 8 ⊢ Ⅎ𝑡∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)
25		nfcv 2925	. . . . . . . 8 ⊢ Ⅎ𝑡𝐴
26	24, 25	nfrabw 3452	. . . . . . 7 ⊢ Ⅎ𝑡{ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)}
27	8, 26	nfcxfr 2923	. . . . . 6 ⊢ Ⅎ𝑡𝑌
28	27	nfcri 2917	. . . . 5 ⊢ Ⅎ𝑡 𝑓 ∈ 𝑌
29	27	nfcri 2917	. . . . 5 ⊢ Ⅎ𝑡 𝑔 ∈ 𝑌
30	23, 28, 29	nf3an 1931	. . . 4 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌)
31	2, 11	jca 520	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → (𝜑 ∧ 𝑓 ∈ 𝐴))
32	31	adantr 485	. . . . . . . . . 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 18	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → 𝑓:𝑇⟶ℝ)
35		simpr 489	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇)
36	34, 35	ffvelcdmd 7080	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → (𝑓‘𝑡) ∈ ℝ)
37	2, 19	jca 520	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → (𝜑 ∧ 𝑔 ∈ 𝐴))
38		eleq1w 2846	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑔 → (𝑓 ∈ 𝐴 ↔ 𝑔 ∈ 𝐴))
39	38	anbi2d 641	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → ((𝜑 ∧ 𝑓 ∈ 𝐴) ↔ (𝜑 ∧ 𝑔 ∈ 𝐴)))
40		feq1 6683	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → (𝑓:𝑇⟶ℝ ↔ 𝑔:𝑇⟶ℝ))
41	39, 40	imbi12d 347	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → (((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ 𝑔 ∈ 𝐴) → 𝑔:𝑇⟶ℝ)))
42	41, 33	vtoclg 3522	. . . . . . . . . 10 ⊢ (𝑔 ∈ 𝐴 → ((𝜑 ∧ 𝑔 ∈ 𝐴) → 𝑔:𝑇⟶ℝ))
43	19, 37, 42	sylc 66	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → 𝑔:𝑇⟶ℝ)
44	43	ffvelcdmda 7079	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → (𝑔‘𝑡) ∈ ℝ)
45	9	simprbi 502	. . . . . . . . . . 11 ⊢ (𝑓 ∈ 𝑌 → ∀𝑡 ∈ 𝑇 (0 ≤ (𝑓‘𝑡) ∧ (𝑓‘𝑡) ≤ 1))
46	45	3ad2ant2 1152	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → ∀𝑡 ∈ 𝑇 (0 ≤ (𝑓‘𝑡) ∧ (𝑓‘𝑡) ≤ 1))
47	46	r19.21bi 3257	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → (0 ≤ (𝑓‘𝑡) ∧ (𝑓‘𝑡) ≤ 1))
48	47	simpld 499	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → 0 ≤ (𝑓‘𝑡))
49	17	simprbi 502	. . . . . . . . . . 11 ⊢ (𝑔 ∈ 𝑌 → ∀𝑡 ∈ 𝑇 (0 ≤ (𝑔‘𝑡) ∧ (𝑔‘𝑡) ≤ 1))
50	49	3ad2ant3 1153	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → ∀𝑡 ∈ 𝑇 (0 ≤ (𝑔‘𝑡) ∧ (𝑔‘𝑡) ≤ 1))
51	50	r19.21bi 3257	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → (0 ≤ (𝑔‘𝑡) ∧ (𝑔‘𝑡) ≤ 1))
52	51	simpld 499	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → 0 ≤ (𝑔‘𝑡))
53	36, 44, 48, 52	mulge0d 11795	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → 0 ≤ ((𝑓‘𝑡) · (𝑔‘𝑡)))
54	36, 44	remulcld 11243	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → ((𝑓‘𝑡) · (𝑔‘𝑡)) ∈ ℝ)
55	1	fvmpt2 7001	. . . . . . . 8 ⊢ ((𝑡 ∈ 𝑇 ∧ ((𝑓‘𝑡) · (𝑔‘𝑡)) ∈ ℝ) → (𝐻‘𝑡) = ((𝑓‘𝑡) · (𝑔‘𝑡)))
56	35, 54, 55	syl2anc 595	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → (𝐻‘𝑡) = ((𝑓‘𝑡) · (𝑔‘𝑡)))
57	53, 56	breqtrrd 5139	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → 0 ≤ (𝐻‘𝑡))
58		1red 11213	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → 1 ∈ ℝ)
59	47	simprd 500	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → (𝑓‘𝑡) ≤ 1)
60	51	simprd 500	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → (𝑔‘𝑡) ≤ 1)
61	36, 58, 44, 58, 48, 52, 59, 60	lemul12ad 12161	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → ((𝑓‘𝑡) · (𝑔‘𝑡)) ≤ (1 · 1))
62		1t1e1 12406	. . . . . . . 8 ⊢ (1 · 1) = 1
63	61, 62	breqtrdi 5152	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → ((𝑓‘𝑡) · (𝑔‘𝑡)) ≤ 1)
64	56, 63	eqbrtrd 5133	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → (𝐻‘𝑡) ≤ 1)
65	57, 64	jca 520	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → (0 ≤ (𝐻‘𝑡) ∧ (𝐻‘𝑡) ≤ 1))
66	65	ex 417	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → (𝑡 ∈ 𝑇 → (0 ≤ (𝐻‘𝑡) ∧ (𝐻‘𝑡) ≤ 1)))
67	30, 66	ralrimi 3263	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → ∀𝑡 ∈ 𝑇 (0 ≤ (𝐻‘𝑡) ∧ (𝐻‘𝑡) ≤ 1))
68		nfmpt1 5210	. . . . . . 7 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡)))
69	1, 68	nfcxfr 2923	. . . . . 6 ⊢ Ⅎ𝑡𝐻
70	69	nfeq2 2942	. . . . 5 ⊢ Ⅎ𝑡 ℎ = 𝐻
71		fveq1 6880	. . . . . . 7 ⊢ (ℎ = 𝐻 → (ℎ‘𝑡) = (𝐻‘𝑡))
72	71	breq2d 5121	. . . . . 6 ⊢ (ℎ = 𝐻 → (0 ≤ (ℎ‘𝑡) ↔ 0 ≤ (𝐻‘𝑡)))
73	71	breq1d 5119	. . . . . 6 ⊢ (ℎ = 𝐻 → ((ℎ‘𝑡) ≤ 1 ↔ (𝐻‘𝑡) ≤ 1))
74	72, 73	anbi12d 643	. . . . 5 ⊢ (ℎ = 𝐻 → ((0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ (0 ≤ (𝐻‘𝑡) ∧ (𝐻‘𝑡) ≤ 1)))
75	70, 74	ralbid 3278	. . . 4 ⊢ (ℎ = 𝐻 → (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ ∀𝑡 ∈ 𝑇 (0 ≤ (𝐻‘𝑡) ∧ (𝐻‘𝑡) ≤ 1)))
76	75	elrab 3650	. . 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 594	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → 𝐻 ∈ {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)})
78	77, 8	eleqtrrdi 2874	1 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → 𝐻 ∈ 𝑌)

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570  Ⅎwnf 1813   ∈ wcel 2143  ∀wral 3079  {crab 3416   class class class wbr 5109   ↦ cmpt 5192  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  ℝcr 11103  0cc0 11104  1c1 11105   · cmul 11109   ≤ cle 11248

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-resscn 11161  ax-1cn 11162  ax-icn 11163  ax-addcl 11164  ax-addrcl 11165  ax-mulcl 11166  ax-mulrcl 11167  ax-mulcom 11168  ax-addass 11169  ax-mulass 11170  ax-distr 11171  ax-i2m1 11172  ax-1ne0 11173  ax-1rid 11174  ax-rnegex 11175  ax-rrecex 11176  ax-cnre 11177  ax-pre-lttri 11178  ax-pre-lttrn 11179  ax-pre-ltadd 11180  ax-pre-mulgt0 11181

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-po 5569  df-so 5570  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-er 8690  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11447  df-neg 11448

This theorem is used by:  stoweidlem48  46790  stoweidlem51  46793