Theorem stoweidlem22 38915

Description: If a set of real functions from a common domain is closed under addition, multiplication and it contains constants, then it is closed under subtraction. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Hypotheses

Ref	Expression
stoweidlem22.8	⊢ Ⅎ𝑡𝜑
stoweidlem22.9	⊢ Ⅎ𝑡𝐹
stoweidlem22.10	⊢ Ⅎ𝑡𝐺
stoweidlem22.1	⊢ 𝐻 = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) − (𝐺‘𝑡)))
stoweidlem22.2	⊢ 𝐼 = (𝑡 ∈ 𝑇 ↦ -1)
stoweidlem22.3	⊢ 𝐿 = (𝑡 ∈ 𝑇 ↦ ((𝐼‘𝑡) · (𝐺‘𝑡)))
stoweidlem22.4	⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ)
stoweidlem22.5	⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)
stoweidlem22.6	⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
stoweidlem22.7	⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴)

Assertion

Ref	Expression
stoweidlem22	⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) − (𝐺‘𝑡))) ∈ 𝐴)

Distinct variable groups:   𝑓,𝑔,𝑡,𝐴   𝑓,𝐹,𝑔   𝑓,𝐺,𝑔   𝑓,𝐼,𝑔   𝑇,𝑓,𝑔,𝑡   𝜑,𝑓,𝑔   𝑔,𝐿   𝑥,𝑡,𝐴   𝑥,𝑇   𝜑,𝑥

Allowed substitution hints:   𝜑(𝑡)   𝐹(𝑥,𝑡)   𝐺(𝑥,𝑡)   𝐻(𝑥,𝑡,𝑓,𝑔)   𝐼(𝑥,𝑡)   𝐿(𝑥,𝑡,𝑓)

Proof of Theorem stoweidlem22

Step	Hyp	Ref	Expression
1		stoweidlem22.8	. . . 4 ⊢ Ⅎ𝑡𝜑
2		stoweidlem22.9	. . . . 5 ⊢ Ⅎ𝑡𝐹
3	2	nfel1 2765	. . . 4 ⊢ Ⅎ𝑡 𝐹 ∈ 𝐴
4		stoweidlem22.10	. . . . 5 ⊢ Ⅎ𝑡𝐺
5	4	nfel1 2765	. . . 4 ⊢ Ⅎ𝑡 𝐺 ∈ 𝐴
6	1, 3, 5	nf3an 1819	. . 3 ⊢ Ⅎ𝑡(𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴)
7		simpr 476	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇)
8		simpl1 1057	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → 𝜑)
9		stoweidlem22.2	. . . . . . . . . . . 12 ⊢ 𝐼 = (𝑡 ∈ 𝑇 ↦ -1)
10		neg1rr 11002	. . . . . . . . . . . . 13 ⊢ -1 ∈ ℝ
11		stoweidlem22.7	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴)
12	11	stoweidlem4 38897	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ -1 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ -1) ∈ 𝐴)
13	10, 12	mpan2 703	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ -1) ∈ 𝐴)
14	9, 13	syl5eqel 2692	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐼 ∈ 𝐴)
15		eleq1 2676	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝐼 → (𝑓 ∈ 𝐴 ↔ 𝐼 ∈ 𝐴))
16	15	anbi2d 736	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝐼 → ((𝜑 ∧ 𝑓 ∈ 𝐴) ↔ (𝜑 ∧ 𝐼 ∈ 𝐴)))
17		feq1 5939	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝐼 → (𝑓:𝑇⟶ℝ ↔ 𝐼:𝑇⟶ℝ))
18	16, 17	imbi12d 333	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝐼 → (((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ 𝐼 ∈ 𝐴) → 𝐼:𝑇⟶ℝ)))
19		stoweidlem22.4	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ)
20	18, 19	vtoclg 3239	. . . . . . . . . . . 12 ⊢ (𝐼 ∈ 𝐴 → ((𝜑 ∧ 𝐼 ∈ 𝐴) → 𝐼:𝑇⟶ℝ))
21	20	anabsi7 856	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐼 ∈ 𝐴) → 𝐼:𝑇⟶ℝ)
22	14, 21	mpdan 699	. . . . . . . . . 10 ⊢ (𝜑 → 𝐼:𝑇⟶ℝ)
23	8, 22	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → 𝐼:𝑇⟶ℝ)
24	23, 7	ffvelrnd 6268	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → (𝐼‘𝑡) ∈ ℝ)
25		simpl3 1059	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → 𝐺 ∈ 𝐴)
26		eleq1 2676	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝐺 → (𝑓 ∈ 𝐴 ↔ 𝐺 ∈ 𝐴))
27	26	anbi2d 736	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝐺 → ((𝜑 ∧ 𝑓 ∈ 𝐴) ↔ (𝜑 ∧ 𝐺 ∈ 𝐴)))
28		feq1 5939	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝐺 → (𝑓:𝑇⟶ℝ ↔ 𝐺:𝑇⟶ℝ))
29	27, 28	imbi12d 333	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝐺 → (((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ 𝐺 ∈ 𝐴) → 𝐺:𝑇⟶ℝ)))
30	29, 19	vtoclg 3239	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ 𝐴 → ((𝜑 ∧ 𝐺 ∈ 𝐴) → 𝐺:𝑇⟶ℝ))
31	30	anabsi7 856	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐺 ∈ 𝐴) → 𝐺:𝑇⟶ℝ)
32	31	3adant3 1074	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐺 ∈ 𝐴 ∧ 𝑡 ∈ 𝑇) → 𝐺:𝑇⟶ℝ)
33		simp3 1056	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐺 ∈ 𝐴 ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇)
34	32, 33	ffvelrnd 6268	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐺 ∈ 𝐴 ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑡) ∈ ℝ)
35	8, 25, 7, 34	syl3anc 1318	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑡) ∈ ℝ)
36	24, 35	remulcld 9949	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → ((𝐼‘𝑡) · (𝐺‘𝑡)) ∈ ℝ)
37		stoweidlem22.3	. . . . . . . 8 ⊢ 𝐿 = (𝑡 ∈ 𝑇 ↦ ((𝐼‘𝑡) · (𝐺‘𝑡)))
38	37	fvmpt2 6200	. . . . . . 7 ⊢ ((𝑡 ∈ 𝑇 ∧ ((𝐼‘𝑡) · (𝐺‘𝑡)) ∈ ℝ) → (𝐿‘𝑡) = ((𝐼‘𝑡) · (𝐺‘𝑡)))
39	7, 36, 38	syl2anc 691	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → (𝐿‘𝑡) = ((𝐼‘𝑡) · (𝐺‘𝑡)))
40	9	fvmpt2 6200	. . . . . . . . 9 ⊢ ((𝑡 ∈ 𝑇 ∧ -1 ∈ ℝ) → (𝐼‘𝑡) = -1)
41	10, 40	mpan2 703	. . . . . . . 8 ⊢ (𝑡 ∈ 𝑇 → (𝐼‘𝑡) = -1)
42	41	adantl 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → (𝐼‘𝑡) = -1)
43	42	oveq1d 6564	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → ((𝐼‘𝑡) · (𝐺‘𝑡)) = (-1 · (𝐺‘𝑡)))
44	35	recnd 9947	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑡) ∈ ℂ)
45	44	mulm1d 10361	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → (-1 · (𝐺‘𝑡)) = -(𝐺‘𝑡))
46	39, 43, 45	3eqtrd 2648	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → (𝐿‘𝑡) = -(𝐺‘𝑡))
47	46	oveq2d 6565	. . . 4 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → ((𝐹‘𝑡) + (𝐿‘𝑡)) = ((𝐹‘𝑡) + -(𝐺‘𝑡)))
48		simpl2 1058	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → 𝐹 ∈ 𝐴)
49		eleq1 2676	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝐹 → (𝑓 ∈ 𝐴 ↔ 𝐹 ∈ 𝐴))
50	49	anbi2d 736	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → ((𝜑 ∧ 𝑓 ∈ 𝐴) ↔ (𝜑 ∧ 𝐹 ∈ 𝐴)))
51		feq1 5939	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → (𝑓:𝑇⟶ℝ ↔ 𝐹:𝑇⟶ℝ))
52	50, 51	imbi12d 333	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → (((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ 𝐹 ∈ 𝐴) → 𝐹:𝑇⟶ℝ)))
53	52, 19	vtoclg 3239	. . . . . . . . 9 ⊢ (𝐹 ∈ 𝐴 → ((𝜑 ∧ 𝐹 ∈ 𝐴) → 𝐹:𝑇⟶ℝ))
54	53	anabsi7 856	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴) → 𝐹:𝑇⟶ℝ)
55	8, 48, 54	syl2anc 691	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → 𝐹:𝑇⟶ℝ)
56	55, 7	ffvelrnd 6268	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℝ)
57	56	recnd 9947	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℂ)
58	57, 44	negsubd 10277	. . . 4 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → ((𝐹‘𝑡) + -(𝐺‘𝑡)) = ((𝐹‘𝑡) − (𝐺‘𝑡)))
59	47, 58	eqtr2d 2645	. . 3 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → ((𝐹‘𝑡) − (𝐺‘𝑡)) = ((𝐹‘𝑡) + (𝐿‘𝑡)))
60	6, 59	mpteq2da 4671	. 2 ⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) − (𝐺‘𝑡))) = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) + (𝐿‘𝑡))))
61	14	3ad2ant1 1075	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → 𝐼 ∈ 𝐴)
62		nfmpt1 4675	. . . . . . . 8 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ -1)
63	9, 62	nfcxfr 2749	. . . . . . 7 ⊢ Ⅎ𝑡𝐼
64	63	nfeq2 2766	. . . . . 6 ⊢ Ⅎ𝑡 𝑓 = 𝐼
65	4	nfeq2 2766	. . . . . 6 ⊢ Ⅎ𝑡 𝑔 = 𝐺
66		stoweidlem22.6	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
67	64, 65, 66	stoweidlem6 38899	. . . . 5 ⊢ ((𝜑 ∧ 𝐼 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐼‘𝑡) · (𝐺‘𝑡))) ∈ 𝐴)
68	61, 67	syld3an2 1365	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐼‘𝑡) · (𝐺‘𝑡))) ∈ 𝐴)
69	37, 68	syl5eqel 2692	. . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → 𝐿 ∈ 𝐴)
70		stoweidlem22.5	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)
71		nfmpt1 4675	. . . . 5 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ ((𝐼‘𝑡) · (𝐺‘𝑡)))
72	37, 71	nfcxfr 2749	. . . 4 ⊢ Ⅎ𝑡𝐿
73	70, 2, 72	stoweidlem8 38901	. . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐿 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) + (𝐿‘𝑡))) ∈ 𝐴)
74	69, 73	syld3an3 1363	. 2 ⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) + (𝐿‘𝑡))) ∈ 𝐴)
75	60, 74	eqeltrd 2688	1 ⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) − (𝐺‘𝑡))) ∈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475  Ⅎwnf 1699   ∈ wcel 1977  Ⅎwnfc 2738   ↦ cmpt 4643  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  ℝcr 9814  1c1 9816   + caddc 9818   · cmul 9820   − cmin 10145  -cneg 10146

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-ltxr 9958  df-sub 10147  df-neg 10148

This theorem is referenced by:  stoweidlem33  38926