ulmval - Metamath Proof Explorer

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Theorem ulmval 24134

Description: Express the predicate: The sequence of functions

converges uniformly to

. (Contributed by Mario Carneiro, 26-Feb-2015.)

**Assertion**
Ref	Expression
ulmval	$/\$ $/\$

Distinct variable groups:

Allowed substitution hints:

(

)

Proof of Theorem ulmval Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ulmrel 24132	. . . 4
2		brrelex12 5155	. . . 4 $/\$ $/\$
3	1, 2	mpan 706	. . 3 $/\$
4	3	a1i 11	. 2 $/\$
5		3simpa 1058	. . . 4 $/\$ $/\$ $/\$
6		fvex 6201	. . . . . . 7
7		fex 6490	. . . . . . 7 $/\$
8	6, 7	mpan2 707	. . . . . 6
9	8	a1i 11	. . . . 5
10		fex 6490	. . . . . 6 $/\$
11	10	expcom 451	. . . . 5
12	9, 11	anim12d 586	. . . 4 $/\$ $/\$
13	5, 12	syl5 34	. . 3 $/\$ $/\$ $/\$
14	13	rexlimdvw 3034	. 2 $/\$ $/\$ $/\$
15		elex 3212	. . . . . 6
16		simpr1 1067	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
17		uzssz 11707	. . . . . . . . . . . . 13
18		ovex 6678	. . . . . . . . . . . . . 14
19		zex 11386	. . . . . . . . . . . . . 14
20		elpm2r 7875	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
21	18, 19, 20	mpanl12 718	. . . . . . . . . . . . 13 $/\$
22	16, 17, 21	sylancl 694	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
23		simpr2 1068	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
24		cnex 10017	. . . . . . . . . . . . . 14
25		simpl 473	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
26		elmapg 7870	. . . . . . . . . . . . . 14 $/\$
27	24, 25, 26	sylancr 695	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
28	23, 27	mpbird 247	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
29	22, 28	jca 554	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
30	29	ex 450	. . . . . . . . . 10 $/\$ $/\$ $/\$
31	30	rexlimdvw 3034	. . . . . . . . 9 $/\$ $/\$ $/\$
32	31	ssopab2dv 5004	. . . . . . . 8 $/\$ $/\$ $/\$
33		df-xp 5120	. . . . . . . 8 $/\$
34	32, 33	syl6sseqr 3652	. . . . . . 7 $/\$ $/\$
35		ovex 6678	. . . . . . . . 9
36	35, 18	xpex 6962	. . . . . . . 8
37	36	ssex 4802	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
38	34, 37	syl 17	. . . . . 6 $/\$ $/\$
39		oveq2 6658	. . . . . . . . . . 11
40	39	feq3d 6032	. . . . . . . . . 10
41		feq2 6027	. . . . . . . . . 10
42		raleq 3138	. . . . . . . . . . . 12
43	42	rexralbidv 3058	. . . . . . . . . . 11
44	43	ralbidv 2986	. . . . . . . . . 10
45	40, 41, 44	3anbi123d 1399	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
46	45	rexbidv 3052	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
47	46	opabbidv 4716	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
48		df-ulm 24131	. . . . . . 7 $/\$ $/\$
49	47, 48	fvmptg 6280	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
50	15, 38, 49	syl2anc 693	. . . . 5 $/\$ $/\$
51	50	breqd 4664	. . . 4 $/\$ $/\$
52		simpl 473	. . . . . . . 8 $/\$
53	52	feq1d 6030	. . . . . . 7 $/\$
54		simpr 477	. . . . . . . 8 $/\$
55	54	feq1d 6030	. . . . . . 7 $/\$
56	52	fveq1d 6193	. . . . . . . . . . . . . 14 $/\$
57	56	fveq1d 6193	. . . . . . . . . . . . 13 $/\$
58	54	fveq1d 6193	. . . . . . . . . . . . 13 $/\$
59	57, 58	oveq12d 6668	. . . . . . . . . . . 12 $/\$
60	59	fveq2d 6195	. . . . . . . . . . 11 $/\$
61	60	breq1d 4663	. . . . . . . . . 10 $/\$
62	61	ralbidv 2986	. . . . . . . . 9 $/\$
63	62	rexralbidv 3058	. . . . . . . 8 $/\$
64	63	ralbidv 2986	. . . . . . 7 $/\$
65	53, 55, 64	3anbi123d 1399	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
66	65	rexbidv 3052	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
67		eqid 2622	. . . . 5 $/\$ $/\$ $/\$ $/\$
68	66, 67	brabga 4989	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
69	51, 68	sylan9bb 736	. . 3 $/\$ $/\$ $/\$ $/\$
70	69	ex 450	. 2 $/\$ $/\$ $/\$
71	4, 14, 70	pm5.21ndd 369	1 $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

wral 2912

wrex 2913

cvv 3200

wss 3574 class class class wbr 4653

copab 4712

cxp 5112

wrel 5119

wf 5884

cfv 5888 (class class class)co 6650

cmap 7857

cpm 7858

cc 9934

clt 10074

cmin 10266

cz 11377

cuz 11687

crp 11832

cabs 13974

culm 24130

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-map 7859 df-pm 7860 df-neg 10269 df-z 11378 df-uz 11688 df-ulm 24131

This theorem is referenced by: ulmcl 24135 ulmf 24136 ulm2 24139

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