Theorem xplmi2 9252

Description: Two sequences converge if the sequence of their ordered pairs converges. Part of Proposition 14-2.6 of [Gleason] p. 230. Note: The hypothesis S e. _V is redundant but is kept for convenience.

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
xplmi2	|- ((H:NN-->(X X. Y) /\ H(~~>m` D)R) -> ((F:NN-->X /\ F(~~>m` B)(1st` R)) /\ (G:NN-->Y /\ G(~~>m` C)(2nd` R))))

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 xplmi2

Step	Hyp	Ref	Expression
1		fvex 4689	. . 3 |- (1st` R) e. _V
2		fvex 4689	. . 3 |- (2nd` R) e. _V
3		xplm.1	. . 3 |- X = dom dom B
4		xplm.3	. . 3 |- Y = dom dom C
5		xplm.5	. . 3 |- B e. Met
6		xplm.6	. . 3 |- C e. Met
7		xplm.7	. . 3 |- 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, < ))}
8		xplmi.9	. . 3 |- F = {<.k, w>. | (k e. NN /\ w = (1st` (H` k)))}
9		xplmi.10	. . 3 |- G = {<.k, w>. | (k e. NN /\ w = (2nd` (H` k)))}
10	1, 2, 3, 4, 5, 6, 7, 8, 9	xplmi 9251	. 2 |- ((H:NN-->(X X. Y) /\ H(~~>m` D)<.(1st` R), (2nd` R)>.) -> ((F:NN-->X /\ F(~~>m` B)(1st` R)) /\ (G:NN-->Y /\ G(~~>m` C)(2nd` R))))
11	3, 4, 5, 6, 7	metxp 9111	. . . . . 6 |- D e. Met
12		xplm.a	. . . . . 6 |- R e. _V
13		ltso 6681	. . . . . . . . . . 11 |- < Or RR
14	13	supex 5667	. . . . . . . . . 10 |- sup({((1st` x)B(1st` y)), ((2nd` x)C(2nd` y))}, RR, < ) e. _V
15	14, 7	dmoprab2 5065	. . . . . . . . 9 |- dom D = ((X X. Y) X. (X X. Y))
16	15	dmeqi 4158	. . . . . . . 8 |- dom dom D = dom ((X X. Y) X. (X X. Y))
17		dmxpid 4179	. . . . . . . 8 |- dom ((X X. Y) X. (X X. Y)) = (X X. Y)
18	16, 17	eqtr2i 1909	. . . . . . 7 |- (X X. Y) = dom dom D
19	18	lmcl 9227	. . . . . 6 |- ((D e. Met /\ R e. _V /\ H(~~>m` D)R) -> R e. (X X. Y))
20	11, 12, 19	mp3an12 1181	. . . . 5 |- (H(~~>m` D)R -> R e. (X X. Y))
21		elxp6 5041	. . . . . 6 |- (R e. (X X. Y) <-> (R = <.(1st` R), (2nd` R)>. /\ ((1st` R) e. X /\ (2nd` R) e. Y)))
22	21	simplbi 349	. . . . 5 |- (R e. (X X. Y) -> R = <.(1st` R), (2nd` R)>.)
23	20, 22	syl 12	. . . 4 |- (H(~~>m` D)R -> R = <.(1st` R), (2nd` R)>.)
24	23	breq2d 3350	. . 3 |- (H(~~>m` D)R -> (H(~~>m` D)R <-> H(~~>m` D)<.(1st` R), (2nd` R)>.))
25	24	ibi 652	. 2 |- (H(~~>m` D)R -> H(~~>m` D)<.(1st` R), (2nd` R)>.)
26	10, 25	sylan2 500	1 |- ((H:NN-->(X X. Y) /\ H(~~>m` D)R) -> ((F:NN-->X /\ F(~~>m` B)(1st` R)) /\ (G:NN-->Y /\ G(~~>m` C)(2nd` R))))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  _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  NNcn 6449   < clt 6653  Metcme 9066  ~~>mclm 9197

This theorem is referenced by:  bopcnlem1 9259  vacnlem5 9671

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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