Theorem stoweidlem16 47168

Description: Lemma for stoweid 47215. 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 6895	. . . . . . . . . . 11 ⊢ (ℎ = 𝑓 → (ℎ‘𝑡) = (𝑓‘𝑡))
4	3	breq2d 5118	. . . . . . . . . 10 ⊢ (ℎ = 𝑓 → (0 ≤ (ℎ‘𝑡) ↔ 0 ≤ (𝑓‘𝑡)))
5	3	breq1d 5116	. . . . . . . . . 10 ⊢ (ℎ = 𝑓 → ((ℎ‘𝑡) ≤ 1 ↔ (𝑓‘𝑡) ≤ 1))
6	4, 5	anbi12d 644	. . . . . . . . 9 ⊢ (ℎ = 𝑓 → ((0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ (0 ≤ (𝑓‘𝑡) ∧ (𝑓‘𝑡) ≤ 1)))
7	6	ralbidv 3189	. . . . . . . 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 3652	. . . . . . 7 ⊢ (𝑓 ∈ 𝑌 ↔ (𝑓 ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑓‘𝑡) ∧ (𝑓‘𝑡) ≤ 1)))
10	9	simplbi 502	. . . . . 6 ⊢ (𝑓 ∈ 𝑌 → 𝑓 ∈ 𝐴)
11	10	3ad2ant2 1152	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → 𝑓 ∈ 𝐴)
12		fveq1 6895	. . . . . . . . . . 11 ⊢ (ℎ = 𝑔 → (ℎ‘𝑡) = (𝑔‘𝑡))
13	12	breq2d 5118	. . . . . . . . . 10 ⊢ (ℎ = 𝑔 → (0 ≤ (ℎ‘𝑡) ↔ 0 ≤ (𝑔‘𝑡)))
14	12	breq1d 5116	. . . . . . . . . 10 ⊢ (ℎ = 𝑔 → ((ℎ‘𝑡) ≤ 1 ↔ (𝑔‘𝑡) ≤ 1))
15	13, 14	anbi12d 644	. . . . . . . . 9 ⊢ (ℎ = 𝑔 → ((0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ (0 ≤ (𝑔‘𝑡) ∧ (𝑔‘𝑡) ≤ 1)))
16	15	ralbidv 3189	. . . . . . . 8 ⊢ (ℎ = 𝑔 → (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑔‘𝑡) ∧ (𝑔‘𝑡) ≤ 1)))
17	16, 8	elrab2 3652	. . . . . . 7 ⊢ (𝑔 ∈ 𝑌 ↔ (𝑔 ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑔‘𝑡) ∧ (𝑔‘𝑡) ≤ 1)))
18	17	simplbi 502	. . . . . 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 2868	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → 𝐻 ∈ 𝐴)
23		stoweidlem16.1	. . . . 5 ⊢ Ⅎ𝑡𝜑
24		nfra1 3290	. . . . . . . 8 ⊢ Ⅎ𝑡∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)
25		nfcv 2926	. . . . . . . 8 ⊢ Ⅎ𝑡𝐴
26	24, 25	nfrabw 3451	. . . . . . 7 ⊢ Ⅎ𝑡{ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)}
27	8, 26	nfcxfr 2924	. . . . . 6 ⊢ Ⅎ𝑡𝑌
28	27	nfcri 2918	. . . . 5 ⊢ Ⅎ𝑡 𝑓 ∈ 𝑌
29	27	nfcri 2918	. . . . 5 ⊢ Ⅎ𝑡 𝑔 ∈ 𝑌
30	23, 28, 29	nf3an 1935	. . . 4 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌)
31	2, 11	jca 521	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → (𝜑 ∧ 𝑓 ∈ 𝐴))
32	31	adantr 486	. . . . . . . . . 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 490	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇)
36	34, 35	ffvelcdmd 7096	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → (𝑓‘𝑡) ∈ ℝ)
37	2, 19	jca 521	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → (𝜑 ∧ 𝑔 ∈ 𝐴))
38		eleq1w 2847	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑔 → (𝑓 ∈ 𝐴 ↔ 𝑔 ∈ 𝐴))
39	38	anbi2d 642	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → ((𝜑 ∧ 𝑓 ∈ 𝐴) ↔ (𝜑 ∧ 𝑔 ∈ 𝐴)))
40		feq1 6698	. . . . . . . . . . . 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 3521	. . . . . . . . . 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 7095	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → (𝑔‘𝑡) ∈ ℝ)
45	9	simprbi 503	. . . . . . . . . . 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 3258	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → (0 ≤ (𝑓‘𝑡) ∧ (𝑓‘𝑡) ≤ 1))
48	47	simpld 500	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → 0 ≤ (𝑓‘𝑡))
49	17	simprbi 503	. . . . . . . . . . 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 3258	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → (0 ≤ (𝑔‘𝑡) ∧ (𝑔‘𝑡) ≤ 1))
52	51	simpld 500	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → 0 ≤ (𝑔‘𝑡))
53	36, 44, 48, 52	mulge0d 11906	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → 0 ≤ ((𝑓‘𝑡) · (𝑔‘𝑡)))
54	36, 44	remulcld 11352	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → ((𝑓‘𝑡) · (𝑔‘𝑡)) ∈ ℝ)
55	1	fvmpt2 7016	. . . . . . . 8 ⊢ ((𝑡 ∈ 𝑇 ∧ ((𝑓‘𝑡) · (𝑔‘𝑡)) ∈ ℝ) → (𝐻‘𝑡) = ((𝑓‘𝑡) · (𝑔‘𝑡)))
56	35, 54, 55	syl2anc 596	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → (𝐻‘𝑡) = ((𝑓‘𝑡) · (𝑔‘𝑡)))
57	53, 56	breqtrrd 5136	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → 0 ≤ (𝐻‘𝑡))
58		1red 11322	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → 1 ∈ ℝ)
59	47	simprd 501	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → (𝑓‘𝑡) ≤ 1)
60	51	simprd 501	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → (𝑔‘𝑡) ≤ 1)
61	36, 58, 44, 58, 48, 52, 59, 60	lemul12ad 12272	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → ((𝑓‘𝑡) · (𝑔‘𝑡)) ≤ (1 · 1))
62		1t1e1 12517	. . . . . . . 8 ⊢ (1 · 1) = 1
63	61, 62	breqtrdi 5149	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → ((𝑓‘𝑡) · (𝑔‘𝑡)) ≤ 1)
64	56, 63	eqbrtrd 5130	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → (𝐻‘𝑡) ≤ 1)
65	57, 64	jca 521	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → (0 ≤ (𝐻‘𝑡) ∧ (𝐻‘𝑡) ≤ 1))
66	65	ex 418	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → (𝑡 ∈ 𝑇 → (0 ≤ (𝐻‘𝑡) ∧ (𝐻‘𝑡) ≤ 1)))
67	30, 66	ralrimi 3264	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → ∀𝑡 ∈ 𝑇 (0 ≤ (𝐻‘𝑡) ∧ (𝐻‘𝑡) ≤ 1))
68		nfmpt1 5207	. . . . . . 7 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡)))
69	1, 68	nfcxfr 2924	. . . . . 6 ⊢ Ⅎ𝑡𝐻
70	69	nfeq2 2943	. . . . 5 ⊢ Ⅎ𝑡 ℎ = 𝐻
71		fveq1 6895	. . . . . . 7 ⊢ (ℎ = 𝐻 → (ℎ‘𝑡) = (𝐻‘𝑡))
72	71	breq2d 5118	. . . . . 6 ⊢ (ℎ = 𝐻 → (0 ≤ (ℎ‘𝑡) ↔ 0 ≤ (𝐻‘𝑡)))
73	71	breq1d 5116	. . . . . 6 ⊢ (ℎ = 𝐻 → ((ℎ‘𝑡) ≤ 1 ↔ (𝐻‘𝑡) ≤ 1))
74	72, 73	anbi12d 644	. . . . 5 ⊢ (ℎ = 𝐻 → ((0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ (0 ≤ (𝐻‘𝑡) ∧ (𝐻‘𝑡) ≤ 1)))
75	70, 74	ralbid 3279	. . . 4 ⊢ (ℎ = 𝐻 → (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ ∀𝑡 ∈ 𝑇 (0 ≤ (𝐻‘𝑡) ∧ (𝐻‘𝑡) ≤ 1)))
76	75	elrab 3648	. . 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 595	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → 𝐻 ∈ {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)})
78	77, 8	eleqtrrdi 2875	1 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → 𝐻 ∈ 𝑌)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569  Ⅎwnf 1815   ∈ wcel 2146  ∀wral 3080  {crab 3416   class class class wbr 5106   ↦ cmpt 5189  ⟶wf 6543  ‘cfv 6547  (class class class)co 7430  ℝcr 11212  0cc0 11213  1c1 11214   · cmul 11218   ≤ cle 11357

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7761  ax-resscn 11270  ax-1cn 11271  ax-icn 11272  ax-addcl 11273  ax-addrcl 11274  ax-mulcl 11275  ax-mulrcl 11276  ax-mulcom 11277  ax-addass 11278  ax-mulass 11279  ax-distr 11280  ax-i2m1 11281  ax-1ne0 11282  ax-1rid 11283  ax-rnegex 11284  ax-rrecex 11285  ax-cnre 11286  ax-pre-lttri 11287  ax-pre-lttrn 11288  ax-pre-ltadd 11289  ax-pre-mulgt0 11290

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-po 5563  df-so 5564  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-er 8731  df-en 8988  df-dom 8989  df-sdom 8990  df-pnf 11358  df-mnf 11359  df-xr 11360  df-ltxr 11361  df-le 11362  df-sub 11556  df-neg 11557

This theorem is used by:  stoweidlem48  47200  stoweidlem51  47203