Theorem fmptco 7077

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 5775	. 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 7063	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
6	5	ffund 6667	. . . . . . . . . 10 ⊢ (𝜑 → Fun 𝐹)
7		funbrfv 6883	. . . . . . . . . . 11 ⊢ (Fun 𝐹 → (𝑧𝐹𝑢 → (𝐹‘𝑧) = 𝑢))
8	7	imp 406	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑧𝐹𝑢) → (𝐹‘𝑧) = 𝑢)
9	6, 8	sylan 581	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧𝐹𝑢) → (𝐹‘𝑧) = 𝑢)
10	9	eqcomd 2743	. . . . . . . 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 1923	. . . 4 ⊢ (𝜑 → (∃𝑢(𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤) ↔ ∃𝑢(𝑢 = (𝐹‘𝑧) ∧ (𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤))))
15		fvex 6848	. . . . . 6 ⊢ (𝐹‘𝑧) ∈ V
16		breq2 5090	. . . . . . 7 ⊢ (𝑢 = (𝐹‘𝑧) → (𝑧𝐹𝑢 ↔ 𝑧𝐹(𝐹‘𝑧)))
17		breq1 5089	. . . . . . 7 ⊢ (𝑢 = (𝐹‘𝑧) → (𝑢𝐺𝑤 ↔ (𝐹‘𝑧)𝐺𝑤))
18	16, 17	anbi12d 633	. . . . . 6 ⊢ (𝑢 = (𝐹‘𝑧) → ((𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤) ↔ (𝑧𝐹(𝐹‘𝑧) ∧ (𝐹‘𝑧)𝐺𝑤)))
19	15, 18	ceqsexv 3479	. . . . 5 ⊢ (∃𝑢(𝑢 = (𝐹‘𝑧) ∧ (𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤)) ↔ (𝑧𝐹(𝐹‘𝑧) ∧ (𝐹‘𝑧)𝐺𝑤))
20		funfvbrb 6998	. . . . . . . . 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 6673	. . . . . . . . 9 ⊢ (𝜑 → dom 𝐹 = 𝐴)
23	22	eleq2d 2823	. . . . . . . 8 ⊢ (𝜑 → (𝑧 ∈ dom 𝐹 ↔ 𝑧 ∈ 𝐴))
24	21, 23	bitr3d 281	. . . . . . 7 ⊢ (𝜑 → (𝑧𝐹(𝐹‘𝑧) ↔ 𝑧 ∈ 𝐴))
25	3	fveq1d 6837	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝑧) = ((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧))
26		fmptco.3	. . . . . . . 8 ⊢ (𝜑 → 𝐺 = (𝑦 ∈ 𝐵 ↦ 𝑆))
27		eqidd 2738	. . . . . . . 8 ⊢ (𝜑 → 𝑤 = 𝑤)
28	25, 26, 27	breq123d 5100	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝑧)𝐺𝑤 ↔ ((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤))
29	24, 28	anbi12d 633	. . . . . 6 ⊢ (𝜑 → ((𝑧𝐹(𝐹‘𝑧) ∧ (𝐹‘𝑧)𝐺𝑤) ↔ (𝑧 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤)))
30		nfcv 2899	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑧
31		nfv 1916	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝜑
32		nffvmpt1 6846	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)
33		nfcv 2899	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐵 ↦ 𝑆)
34		nfcv 2899	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝑤
35	32, 33, 34	nfbr 5133	. . . . . . . . . . 11 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤
36		nfcsb1v 3862	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝑇
37	36	nfeq2 2917	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 𝑤 = ⦋𝑧 / 𝑥⦌𝑇
38	35, 37	nfbi 1905	. . . . . . . . . 10 ⊢ Ⅎ𝑥(((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇)
39	31, 38	nfim 1898	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝜑 → (((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇))
40		fveq2 6835	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧))
41	40	breq1d 5096	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑥)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ ((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤))
42		csbeq1a 3852	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → 𝑇 = ⦋𝑧 / 𝑥⦌𝑇)
43	42	eqeq2d 2748	. . . . . . . . . . 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 3434	. . . . . . . . . . . 12 ⊢ 𝑤 ∈ V
47		simpl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 = 𝑅 ∧ 𝑢 = 𝑤) → 𝑦 = 𝑅)
48	47	eleq1d 2822	. . . . . . . . . . . . . 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 2752	. . . . . . . . . . . . . 14 ⊢ ((𝑦 = 𝑅 ∧ 𝑢 = 𝑤) → (𝑢 = 𝑆 ↔ 𝑤 = 𝑇))
52	48, 51	anbi12d 633	. . . . . . . . . . . . 13 ⊢ ((𝑦 = 𝑅 ∧ 𝑢 = 𝑤) → ((𝑦 ∈ 𝐵 ∧ 𝑢 = 𝑆) ↔ (𝑅 ∈ 𝐵 ∧ 𝑤 = 𝑇)))
53		df-mpt 5168	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝐵 ↦ 𝑆) = {⟨𝑦, 𝑢⟩ ∣ (𝑦 ∈ 𝐵 ∧ 𝑢 = 𝑆)}
54	52, 53	brabga 5483	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ 𝐵 ∧ 𝑤 ∈ V) → (𝑅(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ (𝑅 ∈ 𝐵 ∧ 𝑤 = 𝑇)))
55	4, 46, 54	sylancl 587	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑅(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ (𝑅 ∈ 𝐵 ∧ 𝑤 = 𝑇)))
56		id 22	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐴)
57		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑅) = (𝑥 ∈ 𝐴 ↦ 𝑅)
58	57	fvmpt2 6954	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑅 ∈ 𝐵) → ((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑥) = 𝑅)
59	56, 4, 58	syl2an2 687	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑥) = 𝑅)
60	59	breq1d 5096	. . . . . . . . . . 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 3520	. . . . . . . 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 3434	. . . 4 ⊢ 𝑧 ∈ V
71	70, 46	opelco 5821	. . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ (𝐺 ∘ 𝐹) ↔ ∃𝑢(𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤))
72		df-mpt 5168	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑇) = {⟨𝑥, 𝑣⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑣 = 𝑇)}
73	72	eleq2i 2829	. . . 4 ⊢ (⟨𝑧, 𝑤⟩ ∈ (𝑥 ∈ 𝐴 ↦ 𝑇) ↔ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑣⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑣 = 𝑇)})
74		nfv 1916	. . . . . 6 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴
75	36	nfeq2 2917	. . . . . 6 ⊢ Ⅎ𝑥 𝑣 = ⦋𝑧 / 𝑥⦌𝑇
76	74, 75	nfan 1901	. . . . 5 ⊢ Ⅎ𝑥(𝑧 ∈ 𝐴 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝑇)
77		nfv 1916	. . . . 5 ⊢ Ⅎ𝑣(𝑧 ∈ 𝐴 ∧ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇)
78		eleq1w 2820	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
79	42	eqeq2d 2748	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑣 = 𝑇 ↔ 𝑣 = ⦋𝑧 / 𝑥⦌𝑇))
80	78, 79	anbi12d 633	. . . . 5 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ∧ 𝑣 = 𝑇) ↔ (𝑧 ∈ 𝐴 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝑇)))
81		eqeq1 2741	. . . . . 6 ⊢ (𝑣 = 𝑤 → (𝑣 = ⦋𝑧 / 𝑥⦌𝑇 ↔ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇))
82	81	anbi2d 631	. . . . 5 ⊢ (𝑣 = 𝑤 → ((𝑧 ∈ 𝐴 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝑇) ↔ (𝑧 ∈ 𝐴 ∧ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇)))
83	76, 77, 70, 46, 80, 82	opelopabf 5494	. . . 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 5742	1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ 𝑇))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  Vcvv 3430  ⦋csb 3838  ⟨cop 4574   class class class wbr 5086  {copab 5148   ↦ cmpt 5167  dom cdm 5625   ∘ ccom 5629  Fun wfun 6487  ‘cfv 6493

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-fv 6501

This theorem is referenced by:  fmptcof  7078  cofmpt  7080  fcompt  7081  fcoconst  7082  ofco  7650  ccatco  14791  rlimcn1  15544  rlimdiv  15602  ackbijnn  15787  setcepi  18049  prf1st  18164  prf2nd  18165  hofcllem  18218  prdsidlem  18731  pws0g  18735  mhmvlin  18763  pwsco1mhm  18794  pwsco2mhm  18795  smndex1iidm  18863  smndex2dlinvh  18882  pwsinvg  19023  pwssub  19024  ghmquskerco  19253  galactghm  19373  efginvrel1  19697  frgpup3lem  19746  gsumzf1o  19881  gsumconst  19903  gsummptshft  19905  gsumzmhm  19906  gsummhm2  19908  gsummptmhm  19909  gsumsub  19917  gsum2dlem2  19940  dprdfsub  19992  lmhmvsca  21035  frgpcyg  21566  evpmodpmf1o  21589  psrass1lem  21925  psrlinv  21947  psrcom  21959  evlslem2  22070  psdmplcl  22141  psdmul  22145  coe1fval3  22185  psropprmul  22214  coe1z  22241  coe1mul2  22247  coe1tm  22251  ply1coe  22276  evls1sca  22301  ofco2  22429  mdetleib2  22566  mdetralt  22586  smadiadetlem3  22646  ptrescn  23617  lmcn2  23627  qtopeu  23694  flfcnp2  23985  tgpconncomp  24091  tsmssub  24127  tsmsxplem1  24131  negfcncf  24903  pcopt  25002  pcopt2  25003  pi1xfrcnvlem  25036  ovolctb  25470  ovolfs2  25551  uniioombllem2  25563  ismbf  25608  mbfconst  25613  limccnp2  25872  limcco  25873  dvcof  25928  dvcj  25930  dvfre  25931  dvmptcj  25948  dvmptco  25952  dvcnvlem  25956  dvlip  25973  dvlipcn  25974  itgsubstlem  26028  plyco  26219  dgrcolem1  26251  dgrcolem2  26252  dgrco  26253  plycjlem  26254  taylply2  26347  taylply2OLD  26348  logcn  26627  leibpi  26922  efrlim  26949  efrlimOLD  26950  jensenlem2  26968  amgmlem  26970  ftalem7  27059  dchrisum0  27500  gsumwrd2dccat  33157  mplvrpmfgalem  33706  psrmonprod  33714  esplyfval0  33726  esplyfvaln  33736  ofcfval4  34268  eulerpartgbij  34535  dstfrvclim1  34641  cvmliftlem6  35491  cvmliftphtlem  35518  cvmlift3lem5  35524  elmsubrn  35729  msubco  35732  circum  35875  mblfinlem2  37996  volsupnfl  38003  itgaddnc  38018  itgmulc2nc  38026  ftc1anclem1  38031  ftc1anclem2  38032  ftc1anclem3  38033  ftc1anclem4  38034  ftc1anclem5  38035  ftc1anclem7  38037  ftc1anclem8  38038  fnopabco  38061  upixp  38067  aks6d1c6lem4  42629  selvvvval  43035  evlselv  43037  mendassa  43639  fsovrfovd  44457  fsovcnvlem  44461  cncfcompt  46332  dvcosax  46375  dirkercncflem4  46555  fourierdlem111  46666  meadjiunlem  46914  meadjiun  46915  fundcmpsurbijinjpreimafv  47882  itcovalpclem2  49162  itcovalt2lem2  49167  amgmwlem  50292  amgmlemALT  50293