Theorem fmptco 7083

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 6068	. 2 ⊢ Rel (𝐺 ∘ 𝐹)
2		mptrel 5779	. 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 7070	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
6	5	ffund 6674	. . . . . . . . . 10 ⊢ (𝜑 → Fun 𝐹)
7		funbrfv 6891	. . . . . . . . . . 11 ⊢ (Fun 𝐹 → (𝑧𝐹𝑢 → (𝐹‘𝑧) = 𝑢))
8	7	imp 406	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑧𝐹𝑢) → (𝐹‘𝑧) = 𝑢)
9	6, 8	sylan 580	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧𝐹𝑢) → (𝐹‘𝑧) = 𝑢)
10	9	eqcomd 2735	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧𝐹𝑢) → 𝑢 = (𝐹‘𝑧))
11	10	a1d 25	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧𝐹𝑢) → (𝑢𝐺𝑤 → 𝑢 = (𝐹‘𝑧)))
12	11	expimpd 453	. . . . . 6 ⊢ (𝜑 → ((𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤) → 𝑢 = (𝐹‘𝑧)))
13	12	pm4.71rd 562	. . . . 5 ⊢ (𝜑 → ((𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤) ↔ (𝑢 = (𝐹‘𝑧) ∧ (𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤))))
14	13	exbidv 1921	. . . 4 ⊢ (𝜑 → (∃𝑢(𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤) ↔ ∃𝑢(𝑢 = (𝐹‘𝑧) ∧ (𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤))))
15		fvex 6853	. . . . . 6 ⊢ (𝐹‘𝑧) ∈ V
16		breq2 5106	. . . . . . 7 ⊢ (𝑢 = (𝐹‘𝑧) → (𝑧𝐹𝑢 ↔ 𝑧𝐹(𝐹‘𝑧)))
17		breq1 5105	. . . . . . 7 ⊢ (𝑢 = (𝐹‘𝑧) → (𝑢𝐺𝑤 ↔ (𝐹‘𝑧)𝐺𝑤))
18	16, 17	anbi12d 632	. . . . . 6 ⊢ (𝑢 = (𝐹‘𝑧) → ((𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤) ↔ (𝑧𝐹(𝐹‘𝑧) ∧ (𝐹‘𝑧)𝐺𝑤)))
19	15, 18	ceqsexv 3495	. . . . 5 ⊢ (∃𝑢(𝑢 = (𝐹‘𝑧) ∧ (𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤)) ↔ (𝑧𝐹(𝐹‘𝑧) ∧ (𝐹‘𝑧)𝐺𝑤))
20		funfvbrb 7005	. . . . . . . . 9 ⊢ (Fun 𝐹 → (𝑧 ∈ dom 𝐹 ↔ 𝑧𝐹(𝐹‘𝑧)))
21	6, 20	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑧 ∈ dom 𝐹 ↔ 𝑧𝐹(𝐹‘𝑧)))
22	5	fdmd 6680	. . . . . . . . 9 ⊢ (𝜑 → dom 𝐹 = 𝐴)
23	22	eleq2d 2814	. . . . . . . 8 ⊢ (𝜑 → (𝑧 ∈ dom 𝐹 ↔ 𝑧 ∈ 𝐴))
24	21, 23	bitr3d 281	. . . . . . 7 ⊢ (𝜑 → (𝑧𝐹(𝐹‘𝑧) ↔ 𝑧 ∈ 𝐴))
25	3	fveq1d 6842	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝑧) = ((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧))
26		fmptco.3	. . . . . . . 8 ⊢ (𝜑 → 𝐺 = (𝑦 ∈ 𝐵 ↦ 𝑆))
27		eqidd 2730	. . . . . . . 8 ⊢ (𝜑 → 𝑤 = 𝑤)
28	25, 26, 27	breq123d 5116	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝑧)𝐺𝑤 ↔ ((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤))
29	24, 28	anbi12d 632	. . . . . 6 ⊢ (𝜑 → ((𝑧𝐹(𝐹‘𝑧) ∧ (𝐹‘𝑧)𝐺𝑤) ↔ (𝑧 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤)))
30		nfcv 2891	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑧
31		nfv 1914	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝜑
32		nffvmpt1 6851	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)
33		nfcv 2891	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐵 ↦ 𝑆)
34		nfcv 2891	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝑤
35	32, 33, 34	nfbr 5149	. . . . . . . . . . 11 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤
36		nfcsb1v 3883	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝑇
37	36	nfeq2 2909	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 𝑤 = ⦋𝑧 / 𝑥⦌𝑇
38	35, 37	nfbi 1903	. . . . . . . . . 10 ⊢ Ⅎ𝑥(((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇)
39	31, 38	nfim 1896	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝜑 → (((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇))
40		fveq2 6840	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧))
41	40	breq1d 5112	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑥)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ ((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤))
42		csbeq1a 3873	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → 𝑇 = ⦋𝑧 / 𝑥⦌𝑇)
43	42	eqeq2d 2740	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (𝑤 = 𝑇 ↔ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇))
44	41, 43	bibi12d 345	. . . . . . . . . 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 340	. . . . . . . . 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 3448	. . . . . . . . . . . 12 ⊢ 𝑤 ∈ V
47		simpl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 = 𝑅 ∧ 𝑢 = 𝑤) → 𝑦 = 𝑅)
48	47	eleq1d 2813	. . . . . . . . . . . . . 14 ⊢ ((𝑦 = 𝑅 ∧ 𝑢 = 𝑤) → (𝑦 ∈ 𝐵 ↔ 𝑅 ∈ 𝐵))
49		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑤 → 𝑢 = 𝑤)
50		fmptco.4	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑅 → 𝑆 = 𝑇)
51	49, 50	eqeqan12rd 2744	. . . . . . . . . . . . . 14 ⊢ ((𝑦 = 𝑅 ∧ 𝑢 = 𝑤) → (𝑢 = 𝑆 ↔ 𝑤 = 𝑇))
52	48, 51	anbi12d 632	. . . . . . . . . . . . 13 ⊢ ((𝑦 = 𝑅 ∧ 𝑢 = 𝑤) → ((𝑦 ∈ 𝐵 ∧ 𝑢 = 𝑆) ↔ (𝑅 ∈ 𝐵 ∧ 𝑤 = 𝑇)))
53		df-mpt 5184	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝐵 ↦ 𝑆) = {⟨𝑦, 𝑢⟩ ∣ (𝑦 ∈ 𝐵 ∧ 𝑢 = 𝑆)}
54	52, 53	brabga 5489	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ 𝐵 ∧ 𝑤 ∈ V) → (𝑅(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ (𝑅 ∈ 𝐵 ∧ 𝑤 = 𝑇)))
55	4, 46, 54	sylancl 586	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑅(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ (𝑅 ∈ 𝐵 ∧ 𝑤 = 𝑇)))
56		id 22	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐴)
57		eqid 2729	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑅) = (𝑥 ∈ 𝐴 ↦ 𝑅)
58	57	fvmpt2 6961	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑅 ∈ 𝐵) → ((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑥) = 𝑅)
59	56, 4, 58	syl2an2 686	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑥) = 𝑅)
60	59	breq1d 5112	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑥)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑅(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤))
61	4	biantrurd 532	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑤 = 𝑇 ↔ (𝑅 ∈ 𝐵 ∧ 𝑤 = 𝑇)))
62	55, 60, 61	3bitr4d 311	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑥)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑤 = 𝑇))
63	62	expcom 413	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → (((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑥)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑤 = 𝑇)))
64	30, 39, 45, 63	vtoclgaf 3539	. . . . . . . 8 ⊢ (𝑧 ∈ 𝐴 → (𝜑 → (((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇)))
65	64	impcom 407	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇))
66	65	pm5.32da 579	. . . . . 6 ⊢ (𝜑 → ((𝑧 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤) ↔ (𝑧 ∈ 𝐴 ∧ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇)))
67	29, 66	bitrd 279	. . . . 5 ⊢ (𝜑 → ((𝑧𝐹(𝐹‘𝑧) ∧ (𝐹‘𝑧)𝐺𝑤) ↔ (𝑧 ∈ 𝐴 ∧ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇)))
68	19, 67	bitrid 283	. . . 4 ⊢ (𝜑 → (∃𝑢(𝑢 = (𝐹‘𝑧) ∧ (𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤)) ↔ (𝑧 ∈ 𝐴 ∧ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇)))
69	14, 68	bitrd 279	. . 3 ⊢ (𝜑 → (∃𝑢(𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤) ↔ (𝑧 ∈ 𝐴 ∧ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇)))
70		vex 3448	. . . 4 ⊢ 𝑧 ∈ V
71	70, 46	opelco 5825	. . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ (𝐺 ∘ 𝐹) ↔ ∃𝑢(𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤))
72		df-mpt 5184	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑇) = {⟨𝑥, 𝑣⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑣 = 𝑇)}
73	72	eleq2i 2820	. . . 4 ⊢ (⟨𝑧, 𝑤⟩ ∈ (𝑥 ∈ 𝐴 ↦ 𝑇) ↔ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑣⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑣 = 𝑇)})
74		nfv 1914	. . . . . 6 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴
75	36	nfeq2 2909	. . . . . 6 ⊢ Ⅎ𝑥 𝑣 = ⦋𝑧 / 𝑥⦌𝑇
76	74, 75	nfan 1899	. . . . 5 ⊢ Ⅎ𝑥(𝑧 ∈ 𝐴 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝑇)
77		nfv 1914	. . . . 5 ⊢ Ⅎ𝑣(𝑧 ∈ 𝐴 ∧ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇)
78		eleq1w 2811	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
79	42	eqeq2d 2740	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑣 = 𝑇 ↔ 𝑣 = ⦋𝑧 / 𝑥⦌𝑇))
80	78, 79	anbi12d 632	. . . . 5 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ∧ 𝑣 = 𝑇) ↔ (𝑧 ∈ 𝐴 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝑇)))
81		eqeq1 2733	. . . . . 6 ⊢ (𝑣 = 𝑤 → (𝑣 = ⦋𝑧 / 𝑥⦌𝑇 ↔ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇))
82	81	anbi2d 630	. . . . 5 ⊢ (𝑣 = 𝑤 → ((𝑧 ∈ 𝐴 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝑇) ↔ (𝑧 ∈ 𝐴 ∧ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇)))
83	76, 77, 70, 46, 80, 82	opelopabf 5500	. . . 4 ⊢ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑣⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑣 = 𝑇)} ↔ (𝑧 ∈ 𝐴 ∧ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇))
84	73, 83	bitri 275	. . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ (𝑥 ∈ 𝐴 ↦ 𝑇) ↔ (𝑧 ∈ 𝐴 ∧ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇))
85	69, 71, 84	3bitr4g 314	. 2 ⊢ (𝜑 → (⟨𝑧, 𝑤⟩ ∈ (𝐺 ∘ 𝐹) ↔ ⟨𝑧, 𝑤⟩ ∈ (𝑥 ∈ 𝐴 ↦ 𝑇)))
86	1, 2, 85	eqrelrdv 5746	1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ 𝑇))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  Vcvv 3444  ⦋csb 3859  ⟨cop 4591   class class class wbr 5102  {copab 5164   ↦ cmpt 5183  dom cdm 5631   ∘ ccom 5635  Fun wfun 6493  ‘cfv 6499

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 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  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-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-fv 6507

This theorem is referenced by:  fmptcof  7084  cofmpt  7086  fcompt  7087  fcoconst  7088  ofco  7658  ccatco  14777  rlimcn1  15530  rlimdiv  15588  ackbijnn  15770  setcepi  18026  prf1st  18141  prf2nd  18142  hofcllem  18195  prdsidlem  18672  pws0g  18676  mhmvlin  18704  pwsco1mhm  18735  pwsco2mhm  18736  smndex1iidm  18804  smndex2dlinvh  18820  pwsinvg  18961  pwssub  18962  ghmquskerco  19192  galactghm  19310  efginvrel1  19634  frgpup3lem  19683  gsumzf1o  19818  gsumconst  19840  gsummptshft  19842  gsumzmhm  19843  gsummhm2  19845  gsummptmhm  19846  gsumsub  19854  gsum2dlem2  19877  dprdfsub  19929  lmhmvsca  20928  frgpcyg  21459  evpmodpmf1o  21481  psrass1lem  21817  psrlinv  21840  psrcom  21853  evlslem2  21962  psdmplcl  22025  psdmul  22029  coe1fval3  22069  psropprmul  22098  coe1z  22125  coe1mul2  22131  coe1tm  22135  ply1coe  22161  evls1sca  22186  ofco2  22314  mdetleib2  22451  mdetralt  22471  smadiadetlem3  22531  ptrescn  23502  lmcn2  23512  qtopeu  23579  flfcnp2  23870  tgpconncomp  23976  tsmssub  24012  tsmsxplem1  24016  negfcncf  24793  pcopt  24898  pcopt2  24899  pi1xfrcnvlem  24932  ovolctb  25367  ovolfs2  25448  uniioombllem2  25460  ismbf  25505  mbfconst  25510  limccnp2  25769  limcco  25770  dvcof  25828  dvcj  25830  dvfre  25831  dvmptcj  25848  dvmptco  25852  dvcnvlem  25856  dvlip  25874  dvlipcn  25875  itgsubstlem  25931  plyco  26122  dgrcolem1  26155  dgrcolem2  26156  dgrco  26157  plycjlem  26158  taylply2  26251  taylply2OLD  26252  logcn  26532  leibpi  26828  efrlim  26855  efrlimOLD  26856  jensenlem2  26874  amgmlem  26876  ftalem7  26965  dchrisum0  27407  gsumwrd2dccat  32980  ofcfval4  34068  eulerpartgbij  34336  dstfrvclim1  34442  cvmliftlem6  35250  cvmliftphtlem  35277  cvmlift3lem5  35283  elmsubrn  35488  msubco  35491  circum  35634  mblfinlem2  37625  volsupnfl  37632  itgaddnc  37647  itgmulc2nc  37655  ftc1anclem1  37660  ftc1anclem2  37661  ftc1anclem3  37662  ftc1anclem4  37663  ftc1anclem5  37664  ftc1anclem7  37666  ftc1anclem8  37667  fnopabco  37690  upixp  37696  aks6d1c6lem4  42134  selvvvval  42546  evlselv  42548  mendassa  43152  fsovrfovd  43971  fsovcnvlem  43975  cncfcompt  45854  dvcosax  45897  dirkercncflem4  46077  fourierdlem111  46188  meadjiunlem  46436  meadjiun  46437  fundcmpsurbijinjpreimafv  47381  itcovalpclem2  48633  itcovalt2lem2  48638  amgmwlem  49764  amgmlemALT  49765