Theorem bopcnlem3 7933

Description: Lemma for bopcn 7935.

Hypotheses

Ref	Expression
bopcn.1	|- C = (abs o. - )
bopcn.2	|- D = {<.<.w, v>., u>. | ((w e. (CC X. CC) /\ v e. (CC X. CC)) /\ u = sup({((1st` w)C(1st` v)), ((2nd` w)C(2nd` v))}, RR, < ))}
bopcn.j	|- J = (Open` C)
bopcn.k	|- K = (Open` D)
bopcn.5	|- O:(CC X. CC)-->CC
bopcn.6	|- F = {<.k, r>. | (k e. NN /\ r = (1st` (h` k)))}
bopcn.7	|- G = {<.k, r>. | (k e. NN /\ r = (2nd` (h` k)))}
bopcn.9	|- (((1st` (h` k)) e. CC /\ (2nd` (h` k)) e. CC) -> ((1st` (h` k))O(2nd` (h` k))) e. CC)
bopcn.10	|- (((F ~~> (1st` q) /\ G ~~> (2nd` q)) /\ (1 e. ZZ /\ A.m e. (ZZ>` 1)((F` m) e. CC /\ (G` m) e. CC /\ (H` m) = ((F` m)O(G` m))))) -> H ~~> ((1st` q)O(2nd` q)))
bopcn.8	|- H = {<.k, r>. | (k e. NN /\ r = (O` (h` k)))}

Assertion

Ref	Expression
bopcnlem3	|- ((h:NN-->(CC X. CC) /\ h(~~>m` D)q) -> H ~~> (O` q))

Distinct variable groups:   h,q,u,v,w,C   h,m,D,q   u,m,v,w,F   m,G,u,v,w   H,q   J,q   K,q   h,k,r,O,q   k,m,u,v,w,r

Proof of Theorem bopcnlem3

Step	Hyp	Ref	Expression
1		bopcn.1	. . 3 |- C = (abs o. - )
2		bopcn.2	. . 3 |- D = {<.<.w, v>., u>. | ((w e. (CC X. CC) /\ v e. (CC X. CC)) /\ u = sup({((1st` w)C(1st` v)), ((2nd` w)C(2nd` v))}, RR, < ))}
3		bopcn.j	. . 3 |- J = (Open` C)
4		bopcn.k	. . 3 |- K = (Open` D)
5		bopcn.5	. . 3 |- O:(CC X. CC)-->CC
6		bopcn.6	. . 3 |- F = {<.k, r>. | (k e. NN /\ r = (1st` (h` k)))}
7		bopcn.7	. . 3 |- G = {<.k, r>. | (k e. NN /\ r = (2nd` (h` k)))}
8		bopcn.9	. . 3 |- (((1st` (h` k)) e. CC /\ (2nd` (h` k)) e. CC) -> ((1st` (h` k))O(2nd` (h` k))) e. CC)
9		bopcn.10	. . 3 |- (((F ~~> (1st` q) /\ G ~~> (2nd` q)) /\ (1 e. ZZ /\ A.m e. (ZZ>` 1)((F` m) e. CC /\ (G` m) e. CC /\ (H` m) = ((F` m)O(G` m))))) -> H ~~> ((1st` q)O(2nd` q)))
10		bopcn.8	. . 3 |- H = {<.k, r>. | (k e. NN /\ r = (O` (h` k)))}
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	bopcnlem2 7932	. 2 |- ((h:NN-->(CC X. CC) /\ h(~~>m` D)q) -> H ~~> ((1st` q)O(2nd` q)))
12	1, 2	cn2met 7859	. . . . . . . 8 |- D e. Met
13		visset 1809	. . . . . . . 8 |- q e. V
14		ltso 5492	. . . . . . . . . . . . 13 |- < Or RR
15	14	supex 4557	. . . . . . . . . . . 12 |- sup({((1st` w)C(1st` v)), ((2nd` w)C(2nd` v))}, RR, < ) e. V
16	15, 2	dmoprab2 4113	. . . . . . . . . . 11 |- dom D = ((CC X. CC) X. (CC X. CC))
17	16	dmeqi 3307	. . . . . . . . . 10 |- dom dom D = dom ((CC X. CC) X. (CC X. CC))
18		dmxpid 3328	. . . . . . . . . 10 |- dom ((CC X. CC) X. (CC X. CC)) = (CC X. CC)
19	17, 18	eqtr2 1493	. . . . . . . . 9 |- (CC X. CC) = dom dom D
20	19	lmcl 7900	. . . . . . . 8 |- ((D e. Met /\ q e. V /\ h(~~>m` D)q) -> q e. (CC X. CC))
21	12, 13, 20	mp3an12 904	. . . . . . 7 |- (h(~~>m` D)q -> q e. (CC X. CC))
22		elxp6 4092	. . . . . . . 8 |- (q e. (CC X. CC) <-> (q = <.(1st` q), (2nd` q)>. /\ ((1st` q) e. CC /\ (2nd` q) e. CC)))
23	22	pm3.26bi 322	. . . . . . 7 |- (q e. (CC X. CC) -> q = <.(1st` q), (2nd` q)>.)
24	21, 23	syl 10	. . . . . 6 |- (h(~~>m` D)q -> q = <.(1st` q), (2nd` q)>.)
25	24	fveq2d 3719	. . . . 5 |- (h(~~>m` D)q -> (O` q) = (O` <.(1st` q), (2nd` q)>.))
26		df-opr 3956	. . . . 5 |- ((1st` q)O(2nd` q)) = (O` <.(1st` q), (2nd` q)>.)
27	25, 26	syl6reqr 1523	. . . 4 |- (h(~~>m` D)q -> ((1st` q)O(2nd` q)) = (O` q))
28	27	breq2d 2625	. . 3 |- (h(~~>m` D)q -> (H ~~> ((1st` q)O(2nd` q)) <-> H ~~> (O` q)))
29	28	adantl 388	. 2 |- ((h:NN-->(CC X. CC) /\ h(~~>m` D)q) -> (H ~~> ((1st` q)O(2nd` q)) <-> H ~~> (O` q)))
30	11, 29	mpbid 195	1 |- ((h:NN-->(CC X. CC) /\ h(~~>m` D)q) -> H ~~> (O` q))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 774   = wceq 954   e. wcel 956  A.wral 1642  Vcvv 1807  {cpr 2406  <.cop 2407   class class class wbr 2614  {copab 2661   X. cxp 3163  dom cdm 3165   o. ccom 3169  -->wf 3173  ` cfv 3177  (class class class)co 3954  {copab2 3955  1stc1st 4067  2ndc2nd 4068  supcsup 4553  CCcc 5212  RRcr 5213  1c1 5215   - cmin 5272  NNcn 5276  ZZcz 5278   < clt 5466  ZZ>cuz 6357  abscabs 6689   ~~> cli 6920  Metcme 7739  Opencopn 7742  ~~>mclm 7871

This theorem is referenced by:  bopcnlem4 7934

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-inf2 4605

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-nel 1585  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-pss 2051  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-f1 3190  df-fo 3191  df-f1o 3192  df-fv 3193  df-rdg 3923  df-opr 3956  df-oprab 3957  df-1st 4069  df-2nd 4070  df-1o 4123  df-oadd 4125  df-omul 4126  df-er 4251  df-ec 4253  df-qs 4256  df-en 4357  df-dom 4358  df-sdom 4359  df-sup 4554  df-ni 4980  df-pli 4981  df-mi 4982  df-lti 4983  df-plpq 5015  df-mpq 5016  df-enq 5017  df-nq 5018  df-plq 5019  df-mq 5020  df-rq 5021  df-ltq 5022  df-1q 5023  df-np 5066  df-1p 5067  df-plp 5068  df-mp 5069  df-ltp 5070  df-plpr 5144  df-mpr 5145  df-enr 5146  df-nr 5147  df-plr 5148  df-mr 5149  df-ltr 5150  df-0r 5151  df-1r 5152  df-m1r 5153  df-c 5220  df-0 5221  df-1 5222  df-i 5223  df-r 5224  df-plus 5225  df-mul 5226  df-lt 5227  df-sub 5336  df-neg 5338  df-pnf 5467  df-mnf 5468  df-xr 5469  df-ltxr 5470  df-le 5471  df-div 5680  df-n 5881  df-2 5925  df-n0 6055  df-z 6091  df-seq1 6253  df-uz 6358  df-exp 6509  df-sqr 6608  df-re 6690  df-im 6691  df-cj 6692  df-abs 6693  df-clim 6921  df-met 7743  df-lm 7874

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