Theorem fmptco 7128

Description: Composition of two functions expressed as ordered-pair class abstractions. If 𝐹 has the equation (𝑥 + 2) and 𝐺 the equation (3∗𝑧) then (𝐺 ∘ 𝐹) has the equation (3∗(𝑥 + 2)). (Contributed by FL, 21-Jun-2012.) (Revised by Mario Carneiro, 24-Jul-2014.)

Hypotheses

Ref	Expression
fmptco.1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑅 ∈ 𝐵)
fmptco.2	⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝑅))
fmptco.3	⊢ (𝜑 → 𝐺 = (𝑦 ∈ 𝐵 ↦ 𝑆))
fmptco.4	⊢ (𝑦 = 𝑅 → 𝑆 = 𝑇)

Assertion

Ref	Expression
fmptco	⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ 𝑇))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑦,𝐵   𝑦,𝑅   𝜑,𝑥   𝑥,𝑆   𝑦,𝑇

Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑦)   𝑅(𝑥)   𝑆(𝑦)   𝑇(𝑥)   𝐹(𝑥, 𝑦)   𝐺(𝑥, 𝑦)

Proof of Theorem fmptco

Dummy variables 𝑣 𝑢 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relco 6104	. 2 ⊢ Rel (𝐺 ∘ 𝐹)
2		mptrel 5803	. 2 ⊢ Rel (𝑥 ∈ 𝐴 ↦ 𝑇)
3		fmptco.2	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝑅))
4		fmptco.1	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑅 ∈ 𝐵)
5	3, 4	fmpt3d 7114	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
6	5	ffund 6712	. . . . . . . . . 10 ⊢ (𝜑 → Fun 𝐹)
7		funbrfv 6931	. . . . . . . . . . 11 ⊢ (Fun 𝐹 → (𝑧𝐹𝑢 → (𝐹‘𝑧) = 𝑢))
8	7	imp 412	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑧𝐹𝑢) → (𝐹‘𝑧) = 𝑢)
9	6, 8	sylan 592	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧𝐹𝑢) → (𝐹‘𝑧) = 𝑢)
10	9	eqcomd 2767	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧𝐹𝑢) → 𝑢 = (𝐹‘𝑧))
11	10	a1d 26	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧𝐹𝑢) → (𝑢𝐺𝑤 → 𝑢 = (𝐹‘𝑧)))
12	11	expimpd 459	. . . . . 6 ⊢ (𝜑 → ((𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤) → 𝑢 = (𝐹‘𝑧)))
13	12	pm4.71rd 572	. . . . 5 ⊢ (𝜑 → ((𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤) ↔ (𝑢 = (𝐹‘𝑧) ∧ (𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤))))
14	13	exbidv 1954	. . . 4 ⊢ (𝜑 → (∃𝑢(𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤) ↔ ∃𝑢(𝑢 = (𝐹‘𝑧) ∧ (𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤))))
15		fvex 6896	. . . . . 6 ⊢ (𝐹‘𝑧) ∈ V
16		breq2 5107	. . . . . . 7 ⊢ (𝑢 = (𝐹‘𝑧) → (𝑧𝐹𝑢 ↔ 𝑧𝐹(𝐹‘𝑧)))
17		breq1 5106	. . . . . . 7 ⊢ (𝑢 = (𝐹‘𝑧) → (𝑢𝐺𝑤 ↔ (𝐹‘𝑧)𝐺𝑤))
18	16, 17	anbi12d 644	. . . . . 6 ⊢ (𝑢 = (𝐹‘𝑧) → ((𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤) ↔ (𝑧𝐹(𝐹‘𝑧) ∧ (𝐹‘𝑧)𝐺𝑤)))
19	15, 18	ceqsexv 3499	. . . . 5 ⊢ (∃𝑢(𝑢 = (𝐹‘𝑧) ∧ (𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤)) ↔ (𝑧𝐹(𝐹‘𝑧) ∧ (𝐹‘𝑧)𝐺𝑤))
20		funfvbrb 7048	. . . . . . . . 9 ⊢ (Fun 𝐹 → (𝑧 ∈ dom 𝐹 ↔ 𝑧𝐹(𝐹‘𝑧)))
21	6, 20	syl 18	. . . . . . . 8 ⊢ (𝜑 → (𝑧 ∈ dom 𝐹 ↔ 𝑧𝐹(𝐹‘𝑧)))
22	5	fdmd 6718	. . . . . . . . 9 ⊢ (𝜑 → dom 𝐹 = 𝐴)
23	22	eleq2d 2847	. . . . . . . 8 ⊢ (𝜑 → (𝑧 ∈ dom 𝐹 ↔ 𝑧 ∈ 𝐴))
24	21, 23	bitr3d 284	. . . . . . 7 ⊢ (𝜑 → (𝑧𝐹(𝐹‘𝑧) ↔ 𝑧 ∈ 𝐴))
25	3	fveq1d 6885	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝑧) = ((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧))
26		fmptco.3	. . . . . . . 8 ⊢ (𝜑 → 𝐺 = (𝑦 ∈ 𝐵 ↦ 𝑆))
27		eqidd 2762	. . . . . . . 8 ⊢ (𝜑 → 𝑤 = 𝑤)
28	25, 26, 27	breq123d 5117	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝑧)𝐺𝑤 ↔ ((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤))
29	24, 28	anbi12d 644	. . . . . 6 ⊢ (𝜑 → ((𝑧𝐹(𝐹‘𝑧) ∧ (𝐹‘𝑧)𝐺𝑤) ↔ (𝑧 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤)))
30		nfcv 2923	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑧
31		nfv 1947	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝜑
32		nffvmpt1 6894	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)
33		nfcv 2923	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐵 ↦ 𝑆)
34		nfcv 2923	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝑤
35	32, 33, 34	nfbr 5152	. . . . . . . . . . 11 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤
36		nfcsb1v 3871	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝑇
37	36	nfeq2 2940	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 𝑤 = ⦋𝑧 / 𝑥⦌𝑇
38	35, 37	nfbi 1936	. . . . . . . . . 10 ⊢ Ⅎ𝑥(((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇)
39	31, 38	nfim 1929	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝜑 → (((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇))
40		fveq2 6883	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧))
41	40	breq1d 5113	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑥)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ ((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤))
42		csbeq1a 3861	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → 𝑇 = ⦋𝑧 / 𝑥⦌𝑇)
43	42	eqeq2d 2772	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (𝑤 = 𝑇 ↔ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇))
44	41, 43	bibi12d 348	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → ((((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑥)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑤 = 𝑇) ↔ (((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇)))
45	44	imbi2d 343	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → ((𝜑 → (((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑥)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑤 = 𝑇)) ↔ (𝜑 → (((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇))))
46		vex 3455	. . . . . . . . . . . 12 ⊢ 𝑤 ∈ V
47		simpl 488	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 = 𝑅 ∧ 𝑢 = 𝑤) → 𝑦 = 𝑅)
48	47	eleq1d 2846	. . . . . . . . . . . . . 14 ⊢ ((𝑦 = 𝑅 ∧ 𝑢 = 𝑤) → (𝑦 ∈ 𝐵 ↔ 𝑅 ∈ 𝐵))
49		id 23	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑤 → 𝑢 = 𝑤)
50		fmptco.4	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑅 → 𝑆 = 𝑇)
51	49, 50	eqeqan12rd 2776	. . . . . . . . . . . . . 14 ⊢ ((𝑦 = 𝑅 ∧ 𝑢 = 𝑤) → (𝑢 = 𝑆 ↔ 𝑤 = 𝑇))
52	48, 51	anbi12d 644	. . . . . . . . . . . . 13 ⊢ ((𝑦 = 𝑅 ∧ 𝑢 = 𝑤) → ((𝑦 ∈ 𝐵 ∧ 𝑢 = 𝑆) ↔ (𝑅 ∈ 𝐵 ∧ 𝑤 = 𝑇)))
53		df-mpt 5187	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝐵 ↦ 𝑆) = {⟨𝑦, 𝑢⟩ ∣ (𝑦 ∈ 𝐵 ∧ 𝑢 = 𝑆)}
54	52, 53	brabga 5508	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ 𝐵 ∧ 𝑤 ∈ V) → (𝑅(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ (𝑅 ∈ 𝐵 ∧ 𝑤 = 𝑇)))
55	4, 46, 54	sylancl 598	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑅(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ (𝑅 ∈ 𝐵 ∧ 𝑤 = 𝑇)))
56		id 23	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐴)
57		eqid 2761	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑅) = (𝑥 ∈ 𝐴 ↦ 𝑅)
58	57	fvmpt2 7003	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑅 ∈ 𝐵) → ((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑥) = 𝑅)
59	56, 4, 58	syl2an2 699	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑥) = 𝑅)
60	59	breq1d 5113	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑥)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑅(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤))
61	4	biantrurd 542	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑤 = 𝑇 ↔ (𝑅 ∈ 𝐵 ∧ 𝑤 = 𝑇)))
62	55, 60, 61	3bitr4d 314	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑥)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑤 = 𝑇))
63	62	expcom 419	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → (((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑥)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑤 = 𝑇)))
64	30, 39, 45, 63	vtoclgaf 3536	. . . . . . . 8 ⊢ (𝑧 ∈ 𝐴 → (𝜑 → (((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇)))
65	64	impcom 413	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇))
66	65	pm5.32da 590	. . . . . 6 ⊢ (𝜑 → ((𝑧 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤) ↔ (𝑧 ∈ 𝐴 ∧ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇)))
67	29, 66	bitrd 282	. . . . 5 ⊢ (𝜑 → ((𝑧𝐹(𝐹‘𝑧) ∧ (𝐹‘𝑧)𝐺𝑤) ↔ (𝑧 ∈ 𝐴 ∧ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇)))
68	19, 67	bitrid 286	. . . 4 ⊢ (𝜑 → (∃𝑢(𝑢 = (𝐹‘𝑧) ∧ (𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤)) ↔ (𝑧 ∈ 𝐴 ∧ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇)))
69	14, 68	bitrd 282	. . 3 ⊢ (𝜑 → (∃𝑢(𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤) ↔ (𝑧 ∈ 𝐴 ∧ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇)))
70		vex 3455	. . . 4 ⊢ 𝑧 ∈ V
71	70, 46	opelco 5849	. . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ (𝐺 ∘ 𝐹) ↔ ∃𝑢(𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤))
72		df-mpt 5187	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑇) = {⟨𝑥, 𝑣⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑣 = 𝑇)}
73	72	eleq2i 2853	. . . 4 ⊢ (⟨𝑧, 𝑤⟩ ∈ (𝑥 ∈ 𝐴 ↦ 𝑇) ↔ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑣⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑣 = 𝑇)})
74		nfv 1947	. . . . . 6 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴
75	36	nfeq2 2940	. . . . . 6 ⊢ Ⅎ𝑥 𝑣 = ⦋𝑧 / 𝑥⦌𝑇
76	74, 75	nfan 1932	. . . . 5 ⊢ Ⅎ𝑥(𝑧 ∈ 𝐴 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝑇)
77		nfv 1947	. . . . 5 ⊢ Ⅎ𝑣(𝑧 ∈ 𝐴 ∧ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇)
78		eleq1w 2844	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
79	42	eqeq2d 2772	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑣 = 𝑇 ↔ 𝑣 = ⦋𝑧 / 𝑥⦌𝑇))
80	78, 79	anbi12d 644	. . . . 5 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ∧ 𝑣 = 𝑇) ↔ (𝑧 ∈ 𝐴 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝑇)))
81		eqeq1 2765	. . . . . 6 ⊢ (𝑣 = 𝑤 → (𝑣 = ⦋𝑧 / 𝑥⦌𝑇 ↔ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇))
82	81	anbi2d 642	. . . . 5 ⊢ (𝑣 = 𝑤 → ((𝑧 ∈ 𝐴 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝑇) ↔ (𝑧 ∈ 𝐴 ∧ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇)))
83	76, 77, 70, 46, 80, 82	opelopabf 5520	. . . 4 ⊢ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑣⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑣 = 𝑇)} ↔ (𝑧 ∈ 𝐴 ∧ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇))
84	73, 83	bitri 278	. . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ (𝑥 ∈ 𝐴 ↦ 𝑇) ↔ (𝑧 ∈ 𝐴 ∧ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇))
85	69, 71, 84	3bitr4g 317	. 2 ⊢ (𝜑 → (⟨𝑧, 𝑤⟩ ∈ (𝐺 ∘ 𝐹) ↔ ⟨𝑧, 𝑤⟩ ∈ (𝑥 ∈ 𝐴 ↦ 𝑇)))
86	1, 2, 85	eqrelrdv 5768	1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ 𝑇))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  Vcvv 3451  ⦋csb 3847  ⟨cop 4590   class class class wbr 5103  {copab 5167   ↦ cmpt 5186  dom cdm 5651   ∘ ccom 5655  Fun wfun 6531  ‘cfv 6537

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545

This theorem is used by:  fmptcof  7129  cofmpt  7131  fcompt  7132  fcoconst  7133  ofco  7716  ccatco  14979  rlimcn1  15748  rlimdiv  15806  ackbijnn  15990  setcepi  18256  prf1st  18371  prf2nd  18372  hofcllem  18425  prdsidlem  18956  pws0g  18960  mhmvlin  18989  pwsco1mhm  19021  pwsco2mhm  19022  smndex1iidm  19090  smndex2dlinvh  19109  pwsinvg  19256  pwssub  19257  ghmquskerco  19491  galactghm  19611  efginvrel1  19935  frgpup3lem  19984  gsumzf1o  20119  gsumconst  20141  gsummptshft  20143  gsumzmhm  20144  gsummhm2  20146  gsummptmhm  20147  gsumsub  20155  gsum2dlem2  20178  dprdfsub  20230  lmhmvsca  21313  frgpcyg  21872  evpmodpmf1o  21895  psrass1lem  22234  psrlinv  22256  psrcom  22268  evlslem2  22381  selvvvval  22444  psdmplcl  22476  psdmul  22480  coe1fval3  22519  psropprmul  22548  coe1z  22575  coe1mul2  22581  coe1tm  22585  ply1coe  22609  evls1sca  22634  ofco2  22759  mdetleib2  22896  mdetralt  22916  smadiadetlem3  22976  ptrescn  23951  lmcn2  23961  qtopeu  24028  flfcnp2  24319  tgpconncomp  24425  tsmssub  24461  tsmsxplem1  24465  negfcncf  25237  pcopt  25336  pcopt2  25337  pi1xfrcnvlem  25370  ovolctb  25804  ovolfs2  25885  uniioombllem2  25897  ismbf  25942  mbfconst  25947  limccnp2  26205  limcco  26206  dvcof  26261  dvcj  26263  dvfre  26264  dvmptcj  26281  dvmptco  26285  dvcnvlem  26289  dvlip  26306  dvlipcn  26307  itgsubstlem  26361  plyco  26553  dgrcolem1  26585  dgrcolem2  26586  dgrco  26587  plycjlem  26588  taylply2  26688  logcn  26968  leibpi  27263  efrlim  27290  jensenlem2  27308  amgmlem  27310  ftalem7  27399  dchrisum0  27840  gsumwrd2dccat  33632  mplvrpmfgalem  34169  psrmonprod  34177  esplyfval0  34189  esplyfvaln  34199  ofcfval4  34730  eulerpartgbij  34997  dstfrvclim1  35103  cvmliftlem6  36034  cvmliftphtlem  36061  cvmlift3lem5  36067  elmsubrn  36272  msubco  36275  circum  36418  mblfinlem2  38556  volsupnfl  38563  itgaddnc  38578  itgmulc2nc  38586  ftc1anclem1  38591  ftc1anclem2  38592  ftc1anclem3  38593  ftc1anclem4  38594  ftc1anclem5  38595  ftc1anclem7  38597  ftc1anclem8  38598  fnopabco  38637  upixp  38643  aks6d1c6lem4  43203  evlselv  43597  mendassa  44176  fsovrfovd  44994  fsovcnvlem  44998  cncfcompt  46862  dvcosax  46905  dirkercncflem4  47085  fourierdlem111  47196  meadjiunlem  47444  meadjiun  47445  fundcmpsurbijinjpreimafv  48458  itcovalpclem2  49752  itcovalt2lem2  49757  amgmwlem  50956  amgmlemALT  50957