Theorem xplmi 9251

Description: Two sequences converge if the sequence of their ordered pairs converges. One direction of Proposition 14-2.6 of [Gleason] p. 230. Warning: The HTML proof page is 0.5MB in size.

Hypotheses

Ref	Expression
xplm.a	|- R e. _V
xplm.b	|- S e. _V
xplm.1	|- X = dom dom B
xplm.3	|- Y = dom dom C
xplm.5	|- B e. Met
xplm.6	|- C e. Met
xplm.7	|- D = {<.<.x, y>., z>. | ((x e. (X X. Y) /\ y e. (X X. Y)) /\ z = sup({((1st` x)B(1st` y)), ((2nd` x)C(2nd` y))}, RR, < ))}
xplmi.9	|- F = {<.k, w>. | (k e. NN /\ w = (1st` (H` k)))}
xplmi.10	|- G = {<.k, w>. | (k e. NN /\ w = (2nd` (H` k)))}

Assertion

Ref	Expression
xplmi	|- ((H:NN-->(X X. Y) /\ H(~~>m` D)<.R, S>.) -> ((F:NN-->X /\ F(~~>m` B)R) /\ (G:NN-->Y /\ G(~~>m` C)S)))

Distinct variable groups:   x,y,z,B   x,C,y,z   x,R,y,z   x,S,y,z   w,k,x,y,z,X   k,Y,w,x,y,z   x,F,y,z   x,G,y,z   k,H,w

Proof of Theorem xplmi

Step	Hyp	Ref	Expression
1		simprl 450	. . . . . . . 8 |- ((H(~~>m` D)<.R, S>. /\ (R e. X /\ S e. Y)) -> R e. X)
2	1	a1i 8	. . . . . . 7 |- (H:NN-->(X X. Y) -> ((H(~~>m` D)<.R, S>. /\ (R e. X /\ S e. Y)) -> R e. X))
3		opex 3527	. . . . . . . . . . . . 13 |- <.R, S>. e. _V
4		xplm.1	. . . . . . . . . . . . . . 15 |- X = dom dom B
5		xplm.3	. . . . . . . . . . . . . . 15 |- Y = dom dom C
6		xplm.5	. . . . . . . . . . . . . . 15 |- B e. Met
7		xplm.6	. . . . . . . . . . . . . . 15 |- C e. Met
8		xplm.7	. . . . . . . . . . . . . . 15 |- D = {<.<.x, y>., z>. | ((x e. (X X. Y) /\ y e. (X X. Y)) /\ z = sup({((1st` x)B(1st` y)), ((2nd` x)C(2nd` y))}, RR, < ))}
9	4, 5, 6, 7, 8	metxp 9111	. . . . . . . . . . . . . 14 |- D e. Met
10		ltso 6681	. . . . . . . . . . . . . . . . . . 19 |- < Or RR
11	10	supex 5667	. . . . . . . . . . . . . . . . . 18 |- sup({((1st` x)B(1st` y)), ((2nd` x)C(2nd` y))}, RR, < ) e. _V
12	11, 8	dmoprab2 5065	. . . . . . . . . . . . . . . . 17 |- dom D = ((X X. Y) X. (X X. Y))
13	12	dmeqi 4158	. . . . . . . . . . . . . . . 16 |- dom dom D = dom ((X X. Y) X. (X X. Y))
14		dmxpid 4179	. . . . . . . . . . . . . . . 16 |- dom ((X X. Y) X. (X X. Y)) = (X X. Y)
15	13, 14	eqtr2i 1909	. . . . . . . . . . . . . . 15 |- (X X. Y) = dom dom D
16		1z 7368	. . . . . . . . . . . . . . 15 |- 1 e. ZZ
17		nnuz 7608	. . . . . . . . . . . . . . 15 |- NN = (ZZ>=` 1)
18	15, 16, 17	lmcvg2 9211	. . . . . . . . . . . . . 14 |- (((D e. Met /\ <.R, S>. e. _V /\ H(~~>m` D)<.R, S>.) /\ (v e. RR /\ 0 < v)) -> E.j e. NN A.m e. NN (j <_ m -> ((H` m)D<.R, S>.) < v))
19	9, 18	mp3anl1 1185	. . . . . . . . . . . . 13 |- (((<.R, S>. e. _V /\ H(~~>m` D)<.R, S>.) /\ (v e. RR /\ 0 < v)) -> E.j e. NN A.m e. NN (j <_ m -> ((H` m)D<.R, S>.) < v))
20	3, 19	mpanl1 770	. . . . . . . . . . . 12 |- ((H(~~>m` D)<.R, S>. /\ (v e. RR /\ 0 < v)) -> E.j e. NN A.m e. NN (j <_ m -> ((H` m)D<.R, S>.) < v))
21	20	adantlr 429	. . . . . . . . . . 11 |- (((H(~~>m` D)<.R, S>. /\ (R e. X /\ S e. Y)) /\ (v e. RR /\ 0 < v)) -> E.j e. NN A.m e. NN (j <_ m -> ((H` m)D<.R, S>.) < v))
22	21	adantll 428	. . . . . . . . . 10 |- (((H:NN-->(X X. Y) /\ (H(~~>m` D)<.R, S>. /\ (R e. X /\ S e. Y))) /\ (v e. RR /\ 0 < v)) -> E.j e. NN A.m e. NN (j <_ m -> ((H` m)D<.R, S>.) < v))
23		ffvelrn 4787	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((H:NN-->(X X. Y) /\ m e. NN) -> (H` m) e. (X X. Y))
24		elxp6 5041	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ((H` m) e. (X X. Y) <-> ((H` m) = <.(1st` (H` m)), (2nd` (H` m))>. /\ ((1st` (H` m)) e. X /\ (2nd` (H` m)) e. Y)))
25	24	simplbi 349	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((H` m) e. (X X. Y) -> (H` m) = <.(1st` (H` m)), (2nd` (H` m))>.)
26	23, 25	syl 12	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((H:NN-->(X X. Y) /\ m e. NN) -> (H` m) = <.(1st` (H` m)), (2nd` (H` m))>.)
27		fveq2 4681	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- (k = m -> (H` k) = (H` m))
28	27	fveq2d 4685	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (k = m -> (1st` (H` k)) = (1st` (H` m)))
29		xplmi.9	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- F = {<.k, w>. | (k e. NN /\ w = (1st` (H` k)))}
30		fvex 4689	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (1st` (H` m)) e. _V
31	28, 29, 30	fvopab4 4743	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (m e. NN -> (F` m) = (1st` (H` m)))
32	27	fveq2d 4685	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (k = m -> (2nd` (H` k)) = (2nd` (H` m)))
33		xplmi.10	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- G = {<.k, w>. | (k e. NN /\ w = (2nd` (H` k)))}
34		fvex 4689	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (2nd` (H` m)) e. _V
35	32, 33, 34	fvopab4 4743	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (m e. NN -> (G` m) = (2nd` (H` m)))
36	31, 35	opeq12d 3166	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (m e. NN -> <.(F` m), (G` m)>. = <.(1st` (H` m)), (2nd` (H` m))>.)
37	36	adantl 424	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((H:NN-->(X X. Y) /\ m e. NN) -> <.(F` m), (G` m)>. = <.(1st` (H` m)), (2nd` (H` m))>.)
38	26, 37	eqtr4d 1928	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((H:NN-->(X X. Y) /\ m e. NN) -> (H` m) = <.(F` m), (G` m)>.)
39	38	opreq1d 4897	. . . . . . . . . . . . . . . . . . . . . 22 |- ((H:NN-->(X X. Y) /\ m e. NN) -> ((H` m)D<.R, S>.) = (<.(F` m), (G` m)>.D<.R, S>.))
40	39	breq1d 3348	. . . . . . . . . . . . . . . . . . . . 21 |- ((H:NN-->(X X. Y) /\ m e. NN) -> (((H` m)D<.R, S>.) < v <-> (<.(F` m), (G` m)>.D<.R, S>.) < v))
41	40	adantr 425	. . . . . . . . . . . . . . . . . . . 20 |- (((H:NN-->(X X. Y) /\ m e. NN) /\ ((R e. X /\ S e. Y) /\ v e. RR)) -> (((H` m)D<.R, S>.) < v <-> (<.(F` m), (G` m)>.D<.R, S>.) < v))
42	5	metcl 9088	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((C e. Met /\ (G` m) e. Y /\ S e. Y) -> ((G` m)CS) e. RR)
43	7, 42	mp3an1 1178	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (((G` m) e. Y /\ S e. Y) -> ((G` m)CS) e. RR)
44		ffvelrn 4787	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((G:NN-->Y /\ m e. NN) -> (G` m) e. Y)
45		ffvelrn 4787	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ((H:NN-->(X X. Y) /\ k e. NN) -> (H` k) e. (X X. Y))
46		elxp7 5042	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ((H` k) e. (X X. Y) <-> ((H` k) e. (_V X. _V) /\ ((1st` (H` k)) e. X /\ (2nd` (H` k)) e. Y)))
47	46	simprbi 353	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ((H` k) e. (X X. Y) -> ((1st` (H` k)) e. X /\ (2nd` (H` k)) e. Y))
48	47	simprd 352	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ((H` k) e. (X X. Y) -> (2nd` (H` k)) e. Y)
49	45, 48	syl 12	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ((H:NN-->(X X. Y) /\ k e. NN) -> (2nd` (H` k)) e. Y)
50	49	r19.21aiva 2176	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (H:NN-->(X X. Y) -> A.k e. NN (2nd` (H` k)) e. Y)
51	33	fopab2 4796	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (A.k e. NN (2nd` (H` k)) e. Y <-> G:NN-->Y)
52	50, 51	sylib 215	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (H:NN-->(X X. Y) -> G:NN-->Y)
53	44, 52	sylan 497	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((H:NN-->(X X. Y) /\ m e. NN) -> (G` m) e. Y)
54	43, 53	sylan 497	. . . . . . . . . . . . . . . . . . . . . . 23 |- (((H:NN-->(X X. Y) /\ m e. NN) /\ S e. Y) -> ((G` m)CS) e. RR)
55	54	adantrl 430	. . . . . . . . . . . . . . . . . . . . . 22 |- (((H:NN-->(X X. Y) /\ m e. NN) /\ (R e. X /\ S e. Y)) -> ((G` m)CS) e. RR)
56	55	adantrr 431	. . . . . . . . . . . . . . . . . . . . 21 |- (((H:NN-->(X X. Y) /\ m e. NN) /\ ((R e. X /\ S e. Y) /\ v e. RR)) -> ((G` m)CS) e. RR)
57	4	metcl 9088	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((B e. Met /\ (F` m) e. X /\ R e. X) -> ((F` m)BR) e. RR)
58	6, 57	mp3an1 1178	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (((F` m) e. X /\ R e. X) -> ((F` m)BR) e. RR)
59		ffvelrn 4787	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((F:NN-->X /\ m e. NN) -> (F` m) e. X)
60	47	simplld 348	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ((H` k) e. (X X. Y) -> (1st` (H` k)) e. X)
61	45, 60	syl 12	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ((H:NN-->(X X. Y) /\ k e. NN) -> (1st` (H` k)) e. X)
62	61	r19.21aiva 2176	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (H:NN-->(X X. Y) -> A.k e. NN (1st` (H` k)) e. X)
63	29	fopab2 4796	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (A.k e. NN (1st` (H` k)) e. X <-> F:NN-->X)
64	62, 63	sylib 215	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (H:NN-->(X X. Y) -> F:NN-->X)
65	59, 64	sylan 497	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((H:NN-->(X X. Y) /\ m e. NN) -> (F` m) e. X)
66	58, 65	sylan 497	. . . . . . . . . . . . . . . . . . . . . . 23 |- (((H:NN-->(X X. Y) /\ m e. NN) /\ R e. X) -> ((F` m)BR) e. RR)
67	66	adantrr 431	. . . . . . . . . . . . . . . . . . . . . 22 |- (((H:NN-->(X X. Y) /\ m e. NN) /\ (R e. X /\ S e. Y)) -> ((F` m)BR) e. RR)
68	67	adantrr 431	. . . . . . . . . . . . . . . . . . . . 21 |- (((H:NN-->(X X. Y) /\ m e. NN) /\ ((R e. X /\ S e. Y) /\ v e. RR)) -> ((F` m)BR) e. RR)
69		fvex 4689	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- (F` m) e. _V
70	69	op1st 5026	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (1st` <.(F` m), (G` m)>.) = (F` m)
71	70	eqcomi 1888	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (F` m) = (1st` <.(F` m), (G` m)>.)
72		fvex 4689	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- (G` m) e. _V
73	69, 72	op2nd 5027	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (2nd` <.(F` m), (G` m)>.) = (G` m)
74	73	eqcomi 1888	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (G` m) = (2nd` <.(F` m), (G` m)>.)
75		xplm.a	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- R e. _V
76	75	op1st 5026	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (1st` <.R, S>.) = R
77	76	eqcomi 1888	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- R = (1st` <.R, S>.)
78		xplm.b	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- S e. _V
79	75, 78	op2nd 5027	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (2nd` <.R, S>.) = S
80	79	eqcomi 1888	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- S = (2nd` <.R, S>.)
81	4, 5, 6, 7, 8, 71, 74, 77, 80	metxptval 9107	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (((<.(F` m), (G` m)>. e. (X X. Y) /\ <.R, S>. e. (X X. Y)) /\ ((G` m)CS) <_ ((F` m)BR)) -> (<.(F` m), (G` m)>.D<.R, S>.) = ((F` m)BR))
82		opelxpi 4040	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (((F` m) e. X /\ (G` m) e. Y) -> <.(F` m), (G` m)>. e. (X X. Y))
83	65, 53, 82	syl11anc 524	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ((H:NN-->(X X. Y) /\ m e. NN) -> <.(F` m), (G` m)>. e. (X X. Y))
84		opelxpi 4040	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ((R e. X /\ S e. Y) -> <.R, S>. e. (X X. Y))
85	83, 84	anim12i 360	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (((H:NN-->(X X. Y) /\ m e. NN) /\ (R e. X /\ S e. Y)) -> (<.(F` m), (G` m)>. e. (X X. Y) /\ <.R, S>. e. (X X. Y)))
86	81, 85	sylan 497	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((((H:NN-->(X X. Y) /\ m e. NN) /\ (R e. X /\ S e. Y)) /\ ((G` m)CS) <_ ((F` m)BR)) -> (<.(F` m), (G` m)>.D<.R, S>.) = ((F` m)BR))
87	86	breq1d 3348	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((((H:NN-->(X X. Y) /\ m e. NN) /\ (R e. X /\ S e. Y)) /\ ((G` m)CS) <_ ((F` m)BR)) -> ((<.(F` m), (G` m)>.D<.R, S>.) < v <-> ((F` m)BR) < v))
88	87	biimpd 170	. . . . . . . . . . . . . . . . . . . . . 22 |- ((((H:NN-->(X X. Y) /\ m e. NN) /\ (R e. X /\ S e. Y)) /\ ((G` m)CS) <_ ((F` m)BR)) -> ((<.(F` m), (G` m)>.D<.R, S>.) < v -> ((F` m)BR) < v))
89	88	adantlrr 435	. . . . . . . . . . . . . . . . . . . . 21 |- ((((H:NN-->(X X. Y) /\ m e. NN) /\ ((R e. X /\ S e. Y) /\ v e. RR)) /\ ((G` m)CS) <_ ((F` m)BR)) -> ((<.(F` m), (G` m)>.D<.R, S>.) < v -> ((F` m)BR) < v))
90	4, 5, 6, 7, 8, 71, 74, 77, 80	metxpfval 9108	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (((<.(F` m), (G` m)>. e. (X X. Y) /\ <.R, S>. e. (X X. Y)) /\ ((F` m)BR) <_ ((G` m)CS)) -> (<.(F` m), (G` m)>.D<.R, S>.) = ((G` m)CS))
91	90, 85	sylan 497	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((((H:NN-->(X X. Y) /\ m e. NN) /\ (R e. X /\ S e. Y)) /\ ((F` m)BR) <_ ((G` m)CS)) -> (<.(F` m), (G` m)>.D<.R, S>.) = ((G` m)CS))
92	91	adantlrr 435	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((((H:NN-->(X X. Y) /\ m e. NN) /\ ((R e. X /\ S e. Y) /\ v e. RR)) /\ ((F` m)BR) <_ ((G` m)CS)) -> (<.(F` m), (G` m)>.D<.R, S>.) = ((G` m)CS))
93	92	breq1d 3348	. . . . . . . . . . . . . . . . . . . . . 22 |- ((((H:NN-->(X X. Y) /\ m e. NN) /\ ((R e. X /\ S e. Y) /\ v e. RR)) /\ ((F` m)BR) <_ ((G` m)CS)) -> ((<.(F` m), (G` m)>.D<.R, S>.) < v <-> ((G` m)CS) < v))
94		simprr 451	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (((H:NN-->(X X. Y) /\ m e. NN) /\ ((R e. X /\ S e. Y) /\ v e. RR)) -> v e. RR)
95		lelttr 6693	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((((F` m)BR) e. RR /\ ((G` m)CS) e. RR /\ v e. RR) -> ((((F` m)BR) <_ ((G` m)CS) /\ ((G` m)CS) < v) -> ((F` m)BR) < v))
96	68, 56, 94, 95	syl111anc 1100	. . . . . . . . . . . . . . . . . . . . . . 23 |- (((H:NN-->(X X. Y) /\ m e. NN) /\ ((R e. X /\ S e. Y) /\ v e. RR)) -> ((((F` m)BR) <_ ((G` m)CS) /\ ((G` m)CS) < v) -> ((F` m)BR) < v))
97	96	expdimp 406	. . . . . . . . . . . . . . . . . . . . . 22 |- ((((H:NN-->(X X. Y) /\ m e. NN) /\ ((R e. X /\ S e. Y) /\ v e. RR)) /\ ((F` m)BR) <_ ((G` m)CS)) -> (((G` m)CS) < v -> ((F` m)BR) < v))
98	93, 97	sylbid 220	. . . . . . . . . . . . . . . . . . . . 21 |- ((((H:NN-->(X X. Y) /\ m e. NN) /\ ((R e. X /\ S e. Y) /\ v e. RR)) /\ ((F` m)BR) <_ ((G` m)CS)) -> ((<.(F` m), (G` m)>.D<.R, S>.) < v -> ((F` m)BR) < v))
99	56, 68, 89, 98	lecasei 6804	. . . . . . . . . . . . . . . . . . . 20 |- (((H:NN-->(X X. Y) /\ m e. NN) /\ ((R e. X /\ S e. Y) /\ v e. RR)) -> ((<.(F` m), (G` m)>.D<.R, S>.) < v -> ((F` m)BR) < v))
100	41, 99	sylbid 220	. . . . . . . . . . . . . . . . . . 19 |- (((H:NN-->(X X. Y) /\ m e. NN) /\ ((R e. X /\ S e. Y) /\ v e. RR)) -> (((H` m)D<.R, S>.) < v -> ((F` m)BR) < v))
101	100	exp43 415	. . . . . . . . . . . . . . . . . 18 |- (H:NN-->(X X. Y) -> (m e. NN -> ((R e. X /\ S e. Y) -> (v e. RR -> (((H` m)D<.R, S>.) < v -> ((F` m)BR) < v)))))
102	101	com12 14	. . . . . . . . . . . . . . . . 17 |- (m e. NN -> (H:NN-->(X X. Y) -> ((R e. X /\ S e. Y) -> (v e. RR -> (((H` m)D<.R, S>.) < v -> ((F` m)BR) < v)))))
103	102	com4l 43	. . . . . . . . . . . . . . . 16 |- (H:NN-->(X X. Y) -> ((R e. X /\ S e. Y) -> (v e. RR -> (m e. NN -> (((H` m)D<.R, S>.) < v -> ((F` m)BR) < v)))))
104	103	imp41 395	. . . . . . . . . . . . . . 15 |- ((((H:NN-->(X X. Y) /\ (R e. X /\ S e. Y)) /\ v e. RR) /\ m e. NN) -> (((H` m)D<.R, S>.) < v -> ((F` m)BR) < v))
105	104	imim2d 28	. . . . . . . . . . . . . 14 |- ((((H:NN-->(X X. Y) /\ (R e. X /\ S e. Y)) /\ v e. RR) /\ m e. NN) -> ((j <_ m -> ((H` m)D<.R, S>.) < v) -> (j <_ m -> ((F` m)BR) < v)))
106	105	ralimdvaa 2171	. . . . . . . . . . . . 13 |- (((H:NN-->(X X. Y) /\ (R e. X /\ S e. Y)) /\ v e. RR) -> (A.m e. NN (j <_ m -> ((H` m)D<.R, S>.) < v) -> A.m e. NN (j <_ m -> ((F` m)BR) < v)))
107	106	reximdv 2202	. . . . . . . . . . . 12 |- (((H:NN-->(X X. Y) /\ (R e. X /\ S e. Y)) /\ v e. RR) -> (E.j e. NN A.m e. NN (j <_ m -> ((H` m)D<.R, S>.) < v) -> E.j e. NN A.m e. NN (j <_ m -> ((F` m)BR) < v)))
108	107	adantrr 431	. . . . . . . . . . 11 |- (((H:NN-->(X X. Y) /\ (R e. X /\ S e. Y)) /\ (v e. RR /\ 0 < v)) -> (E.j e. NN A.m e. NN (j <_ m -> ((H` m)D<.R, S>.) < v) -> E.j e. NN A.m e. NN (j <_ m -> ((F` m)BR) < v)))
109	108	adantlrl 434	. . . . . . . . . 10 |- (((H:NN-->(X X. Y) /\ (H(~~>m` D)<.R, S>. /\ (R e. X /\ S e. Y))) /\ (v e. RR /\ 0 < v)) -> (E.j e. NN A.m e. NN (j <_ m -> ((H` m)D<.R, S>.) < v) -> E.j e. NN A.m e. NN (j <_ m -> ((F` m)BR) < v)))
110	22, 109	mpd 29	. . . . . . . . 9 |- (((H:NN-->(X X. Y) /\ (H(~~>m` D)<.R, S>. /\ (R e. X /\ S e. Y))) /\ (v e. RR /\ 0 < v)) -> E.j e. NN A.m e. NN (j <_ m -> ((F` m)BR) < v))
111	110	exp43 415	. . . . . . . 8 |- (H:NN-->(X X. Y) -> ((H(~~>m` D)<.R, S>. /\ (R e. X /\ S e. Y)) -> (v e. RR -> (0 < v -> E.j e. NN A.m e. NN (j <_ m -> ((F` m)BR) < v)))))
112	111	r19.21adv 2181	. . . . . . 7 |- (H:NN-->(X X. Y) -> ((H(~~>m` D)<.R, S>. /\ (R e. X /\ S e. Y)) -> A.v e. RR (0 < v -> E.j e. NN A.m e. NN (j <_ m -> ((F` m)BR) < v))))
113	2, 112	jcad 661	. . . . . 6 |- (H:NN-->(X X. Y) -> ((H(~~>m` D)<.R, S>. /\ (R e. X /\ S e. Y)) -> (R e. X /\ A.v e. RR (0 < v -> E.j e. NN A.m e. NN (j <_ m -> ((F` m)BR) < v)))))
114	4, 16, 17	lmbrf 9208	. . . . . . . 8 |- ((B e. Met /\ R e. _V /\ F:NN-->X) -> (F(~~>m` B)R <-> (R e. X /\ A.v e. RR (0 < v -> E.j e. NN A.m e. NN (j <_ m -> ((F` m)BR) < v)))))
115	6, 75, 114	mp3an12 1181	. . . . . . 7 |- (F:NN-->X -> (F(~~>m` B)R <-> (R e. X /\ A.v e. RR (0 < v -> E.j e. NN A.m e. NN (j <_ m -> ((F` m)BR) < v)))))
116	64, 115	syl 12	. . . . . 6 |- (H:NN-->(X X. Y) -> (F(~~>m` B)R <-> (R e. X /\ A.v e. RR (0 < v -> E.j e. NN A.m e. NN (j <_ m -> ((F` m)BR) < v)))))
117	113, 116	sylibrd 221	. . . . 5 |- (H:NN-->(X X. Y) -> ((H(~~>m` D)<.R, S>. /\ (R e. X /\ S e. Y)) -> F(~~>m` B)R))
118	117, 64	jctild 662	. . . 4 |- (H:NN-->(X X. Y) -> ((H(~~>m` D)<.R, S>. /\ (R e. X /\ S e. Y)) -> (F:NN-->X /\ F(~~>m` B)R)))
119		simprr 451	. . . . . . . 8 |- ((H(~~>m` D)<.R, S>. /\ (R e. X /\ S e. Y)) -> S e. Y)
120	119	a1i 8	. . . . . . 7 |- (H:NN-->(X X. Y) -> ((H(~~>m` D)<.R, S>. /\ (R e. X /\ S e. Y)) -> S e. Y))
121	86	adantlrr 435	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((((H:NN-->(X X. Y) /\ m e. NN) /\ ((R e. X /\ S e. Y) /\ v e. RR)) /\ ((G` m)CS) <_ ((F` m)BR)) -> (<.(F` m), (G` m)>.D<.R, S>.) = ((F` m)BR))
122	121	breq1d 3348	. . . . . . . . . . . . . . . . . . . . . 22 |- ((((H:NN-->(X X. Y) /\ m e. NN) /\ ((R e. X /\ S e. Y) /\ v e. RR)) /\ ((G` m)CS) <_ ((F` m)BR)) -> ((<.(F` m), (G` m)>.D<.R, S>.) < v <-> ((F` m)BR) < v))
123		lelttr 6693	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((((G` m)CS) e. RR /\ ((F` m)BR) e. RR /\ v e. RR) -> ((((G` m)CS) <_ ((F` m)BR) /\ ((F` m)BR) < v) -> ((G` m)CS) < v))
124	56, 68, 94, 123	syl111anc 1100	. . . . . . . . . . . . . . . . . . . . . . 23 |- (((H:NN-->(X X. Y) /\ m e. NN) /\ ((R e. X /\ S e. Y) /\ v e. RR)) -> ((((G` m)CS) <_ ((F` m)BR) /\ ((F` m)BR) < v) -> ((G` m)CS) < v))
125	124	expdimp 406	. . . . . . . . . . . . . . . . . . . . . 22 |- ((((H:NN-->(X X. Y) /\ m e. NN) /\ ((R e. X /\ S e. Y) /\ v e. RR)) /\ ((G` m)CS) <_ ((F` m)BR)) -> (((F` m)BR) < v -> ((G` m)CS) < v))
126	122, 125	sylbid 220	. . . . . . . . . . . . . . . . . . . . 21 |- ((((H:NN-->(X X. Y) /\ m e. NN) /\ ((R e. X /\ S e. Y) /\ v e. RR)) /\ ((G` m)CS) <_ ((F` m)BR)) -> ((<.(F` m), (G` m)>.D<.R, S>.) < v -> ((G` m)CS) < v))
127	91	breq1d 3348	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((((H:NN-->(X X. Y) /\ m e. NN) /\ (R e. X /\ S e. Y)) /\ ((F` m)BR) <_ ((G` m)CS)) -> ((<.(F` m), (G` m)>.D<.R, S>.) < v <-> ((G` m)CS) < v))
128	127	biimpd 170	. . . . . . . . . . . . . . . . . . . . . 22 |- ((((H:NN-->(X X. Y) /\ m e. NN) /\ (R e. X /\ S e. Y)) /\ ((F` m)BR) <_ ((G` m)CS)) -> ((<.(F` m), (G` m)>.D<.R, S>.) < v -> ((G` m)CS) < v))
129	128	adantlrr 435	. . . . . . . . . . . . . . . . . . . . 21 |- ((((H:NN-->(X X. Y) /\ m e. NN) /\ ((R e. X /\ S e. Y) /\ v e. RR)) /\ ((F` m)BR) <_ ((G` m)CS)) -> ((<.(F` m), (G` m)>.D<.R, S>.) < v -> ((G` m)CS) < v))
130	56, 68, 126, 129	lecasei 6804	. . . . . . . . . . . . . . . . . . . 20 |- (((H:NN-->(X X. Y) /\ m e. NN) /\ ((R e. X /\ S e. Y) /\ v e. RR)) -> ((<.(F` m), (G` m)>.D<.R, S>.) < v -> ((G` m)CS) < v))
131	41, 130	sylbid 220	. . . . . . . . . . . . . . . . . . 19 |- (((H:NN-->(X X. Y) /\ m e. NN) /\ ((R e. X /\ S e. Y) /\ v e. RR)) -> (((H` m)D<.R, S>.) < v -> ((G` m)CS) < v))
132	131	exp43 415	. . . . . . . . . . . . . . . . . 18 |- (H:NN-->(X X. Y) -> (m e. NN -> ((R e. X /\ S e. Y) -> (v e. RR -> (((H` m)D<.R, S>.) < v -> ((G` m)CS) < v)))))
133	132	com12 14	. . . . . . . . . . . . . . . . 17 |- (m e. NN -> (H:NN-->(X X. Y) -> ((R e. X /\ S e. Y) -> (v e. RR -> (((H` m)D<.R, S>.) < v -> ((G` m)CS) < v)))))
134	133	com4l 43	. . . . . . . . . . . . . . . 16 |- (H:NN-->(X X. Y) -> ((R e. X /\ S e. Y) -> (v e. RR -> (m e. NN -> (((H` m)D<.R, S>.) < v -> ((G` m)CS) < v)))))
135	134	imp41 395	. . . . . . . . . . . . . . 15 |- ((((H:NN-->(X X. Y) /\ (R e. X /\ S e. Y)) /\ v e. RR) /\ m e. NN) -> (((H` m)D<.R, S>.) < v -> ((G` m)CS) < v))
136	135	imim2d 28	. . . . . . . . . . . . . 14 |- ((((H:NN-->(X X. Y) /\ (R e. X /\ S e. Y)) /\ v e. RR) /\ m e. NN) -> ((j <_ m -> ((H` m)D<.R, S>.) < v) -> (j <_ m -> ((G` m)CS) < v)))
137	136	ralimdvaa 2171	. . . . . . . . . . . . 13 |- (((H:NN-->(X X. Y) /\ (R e. X /\ S e. Y)) /\ v e. RR) -> (A.m e. NN (j <_ m -> ((H` m)D<.R, S>.) < v) -> A.m e. NN (j <_ m -> ((G` m)CS) < v)))
138	137	reximdv 2202	. . . . . . . . . . . 12 |- (((H:NN-->(X X. Y) /\ (R e. X /\ S e. Y)) /\ v e. RR) -> (E.j e. NN A.m e. NN (j <_ m -> ((H` m)D<.R, S>.) < v) -> E.j e. NN A.m e. NN (j <_ m -> ((G` m)CS) < v)))
139	138	adantrr 431	. . . . . . . . . . 11 |- (((H:NN-->(X X. Y) /\ (R e. X /\ S e. Y)) /\ (v e. RR /\ 0 < v)) -> (E.j e. NN A.m e. NN (j <_ m -> ((H` m)D<.R, S>.) < v) -> E.j e. NN A.m e. NN (j <_ m -> ((G` m)CS) < v)))
140	139	adantlrl 434	. . . . . . . . . 10 |- (((H:NN-->(X X. Y) /\ (H(~~>m` D)<.R, S>. /\ (R e. X /\ S e. Y))) /\ (v e. RR /\ 0 < v)) -> (E.j e. NN A.m e. NN (j <_ m -> ((H` m)D<.R, S>.) < v) -> E.j e. NN A.m e. NN (j <_ m -> ((G` m)CS) < v)))
141	22, 140	mpd 29	. . . . . . . . 9 |- (((H:NN-->(X X. Y) /\ (H(~~>m` D)<.R, S>. /\ (R e. X /\ S e. Y))) /\ (v e. RR /\ 0 < v)) -> E.j e. NN A.m e. NN (j <_ m -> ((G` m)CS) < v))
142	141	exp43 415	. . . . . . . 8 |- (H:NN-->(X X. Y) -> ((H(~~>m` D)<.R, S>. /\ (R e. X /\ S e. Y)) -> (v e. RR -> (0 < v -> E.j e. NN A.m e. NN (j <_ m -> ((G` m)CS) < v)))))
143	142	r19.21adv 2181	. . . . . . 7 |- (H:NN-->(X X. Y) -> ((H(~~>m` D)<.R, S>. /\ (R e. X /\ S e. Y)) -> A.v e. RR (0 < v -> E.j e. NN A.m e. NN (j <_ m -> ((G` m)CS) < v))))
144	120, 143	jcad 661	. . . . . 6 |- (H:NN-->(X X. Y) -> ((H(~~>m` D)<.R, S>. /\ (R e. X /\ S e. Y)) -> (S e. Y /\ A.v e. RR (0 < v -> E.j e. NN A.m e. NN (j <_ m -> ((G` m)CS) < v)))))
145	5, 16, 17	lmbrf 9208	. . . . . . . 8 |- ((C e. Met /\ S e. _V /\ G:NN-->Y) -> (G(~~>m` C)S <-> (S e. Y /\ A.v e. RR (0 < v -> E.j e. NN A.m e. NN (j <_ m -> ((G` m)CS) < v)))))
146	7, 78, 145	mp3an12 1181	. . . . . . 7 |- (G:NN-->Y -> (G(~~>m` C)S <-> (S e. Y /\ A.v e. RR (0 < v -> E.j e. NN A.m e. NN (j <_ m -> ((G` m)CS) < v)))))
147	52, 146	syl 12	. . . . . 6 |- (H:NN-->(X X. Y) -> (G(~~>m` C)S <-> (S e. Y /\ A.v e. RR (0 < v -> E.j e. NN A.m e. NN (j <_ m -> ((G` m)CS) < v)))))
148	144, 147	sylibrd 221	. . . . 5 |- (H:NN-->(X X. Y) -> ((H(~~>m` D)<.R, S>. /\ (R e. X /\ S e. Y)) -> G(~~>m` C)S))
149	148, 52	jctild 662	. . . 4 |- (H:NN-->(X X. Y) -> ((H(~~>m` D)<.R, S>. /\ (R e. X /\ S e. Y)) -> (G:NN-->Y /\ G(~~>m` C)S)))
150	118, 149	jcad 661	. . 3 |- (H:NN-->(X X. Y) -> ((H(~~>m` D)<.R, S>. /\ (R e. X /\ S e. Y)) -> ((F:NN-->X /\ F(~~>m` B)R) /\ (G:NN-->Y /\ G(~~>m` C)S))))
151	15	lmcl 9227	. . . . . 6 |- ((D e. Met /\ <.R, S>. e. _V /\ H(~~>m` D)<.R, S>.) -> <.R, S>. e. (X X. Y))
152	9, 3, 151	mp3an12 1181	. . . . 5 |- (H(~~>m` D)<.R, S>. -> <.R, S>. e. (X X. Y))
153	78	opelxp 4036	. . . . 5 |- (<.R, S>. e. (X X. Y) <-> (R e. X /\ S e. Y))
154	152, 153	sylib 215	. . . 4 |- (H(~~>m` D)<.R, S>. -> (R e. X /\ S e. Y))
155	154	ancli 320	. . 3 |- (H(~~>m` D)<.R, S>. -> (H(~~>m` D)<.R, S>. /\ (R e. X /\ S e. Y)))
156	150, 155	syl5 20	. 2 |- (H:NN-->(X X. Y) -> (H(~~>m` D)<.R, S>. -> ((F:NN-->X /\ F(~~>m` B)R) /\ (G:NN-->Y /\ G(~~>m` C)S))))
157	156	imp 377	1 |- ((H:NN-->(X X. Y) /\ H(~~>m` D)<.R, S>.) -> ((F:NN-->X /\ F(~~>m` B)R) /\ (G:NN-->Y /\ G(~~>m` C)S)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106  _Vcvv 2292  {cpr 3045  <.cop 3046   class class class wbr 3338  {copab 3395   X. cxp 3984  dom cdm 3986  -->wf 3994  ` cfv 3998  (class class class)co 4884  {copab2 4885  1stc1st 5018  2ndc2nd 5019  supcsup 5663  RRcr 6385  0cc0 6386  1c1 6387   <_ cle 6448  NNcn 6449   < clt 6653  Metcme 9066  ~~>mclm 9197

This theorem is referenced by:  xplmi2 9252  xplm 9253

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-sup 5664  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-n 7108  df-2 7154  df-z 7345  df-uz 7587  df-met 9070  df-lm 9200

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