Theorem xmettxlem 15536

Description: Lemma for xmettx 15537. (Contributed by Jim Kingdon, 15-Oct-2023.)

Hypotheses

Ref	Expression
xmetxp.p	|- P = ( u e. ( X X. Y ) , v e. ( X X. Y ) |-> sup ( { ( ( 1st ` u ) M ( 1st ` v ) ) , ( ( 2nd ` u ) N ( 2nd ` v ) ) } , RR* , < ) )
xmetxp.1	|- ( ph -> M e. ( *Met ` X ) )
xmetxp.2	|- ( ph -> N e. ( *Met ` Y ) )
xmettx.j	|- J = ( MetOpen ` M )
xmettx.k	|- K = ( MetOpen ` N )
xmettx.l	|- L = ( MetOpen ` P )

Assertion

Ref	Expression
xmettxlem	|- ( ph -> L C_ ( J tX K ) )

Distinct variable groups:   u, M, v   u, N, v   u, X, v   u, Y, v

Allowed substitution hints:   ph( v, u)   P( v, u)   J( v, u)   K( v, u)   L( v, u)

Proof of Theorem xmettxlem

Dummy variables p r s w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xmetxp.p	. . . . . . . . 9 |- P = ( u e. ( X X. Y ) , v e. ( X X. Y ) |-> sup ( { ( ( 1st ` u ) M ( 1st ` v ) ) , ( ( 2nd ` u ) N ( 2nd ` v ) ) } , RR* , < ) )
2		xmetxp.1	. . . . . . . . 9 |- ( ph -> M e. ( *Met ` X ) )
3		xmetxp.2	. . . . . . . . 9 |- ( ph -> N e. ( *Met ` Y ) )
4	1, 2, 3	xmetxp 15534	. . . . . . . 8 |- ( ph -> P e. ( *Met ` ( X X. Y ) ) )
5		blrn 15439	. . . . . . . 8 |- ( P e. ( *Met ` ( X X. Y ) ) -> ( w e. ran ( ball ` P ) <-> E. z e. ( X X. Y ) E. p e. RR* w = ( z ( ball ` P ) p ) ) )
6	4, 5	syl 14	. . . . . . 7 |- ( ph -> ( w e. ran ( ball ` P ) <-> E. z e. ( X X. Y ) E. p e. RR* w = ( z ( ball ` P ) p ) ) )
7	6	biimpa 296	. . . . . 6 |- ( ( ph /\ w e. ran ( ball ` P ) ) -> E. z e. ( X X. Y ) E. p e. RR* w = ( z ( ball ` P ) p ) )
8		xmettx.j	. . . . . . . . . . . . . . 15 |- J = ( MetOpen ` M )
9	8	mopntop 15471	. . . . . . . . . . . . . 14 |- ( M e. ( *Met ` X ) -> J e. Top )
10	2, 9	syl 14	. . . . . . . . . . . . 13 |- ( ph -> J e. Top )
11		xmettx.k	. . . . . . . . . . . . . . 15 |- K = ( MetOpen ` N )
12	11	mopntop 15471	. . . . . . . . . . . . . 14 |- ( N e. ( *Met ` Y ) -> K e. Top )
13	3, 12	syl 14	. . . . . . . . . . . . 13 |- ( ph -> K e. Top )
14		mpoexga 6441	. . . . . . . . . . . . 13 |- ( ( J e. Top /\ K e. Top ) -> ( r e. J , s e. K |-> ( r X. s ) ) e. _V )
15	10, 13, 14	syl2anc 415	. . . . . . . . . . . 12 |- ( ph -> ( r e. J , s e. K |-> ( r X. s ) ) e. _V )
16		rnexg 5045	. . . . . . . . . . . 12 |- ( ( r e. J , s e. K |-> ( r X. s ) ) e. _V -> ran ( r e. J , s e. K |-> ( r X. s ) ) e. _V )
17	15, 16	syl 14	. . . . . . . . . . 11 |- ( ph -> ran ( r e. J , s e. K |-> ( r X. s ) ) e. _V )
18	17	ad3antrrr 496	. . . . . . . . . 10 |- ( ( ( ( ph /\ w e. ran ( ball ` P ) ) /\ ( z e. ( X X. Y ) /\ p e. RR* ) ) /\ w = ( z ( ball ` P ) p ) ) -> ran ( r e. J , s e. K |-> ( r X. s ) ) e. _V )
19		bastg 15088	. . . . . . . . . 10 |- ( ran ( r e. J , s e. K |-> ( r X. s ) ) e. _V -> ran ( r e. J , s e. K |-> ( r X. s ) ) C_ ( topGen ` ran ( r e. J , s e. K |-> ( r X. s ) ) ) )
20	18, 19	syl 14	. . . . . . . . 9 |- ( ( ( ( ph /\ w e. ran ( ball ` P ) ) /\ ( z e. ( X X. Y ) /\ p e. RR* ) ) /\ w = ( z ( ball ` P ) p ) ) -> ran ( r e. J , s e. K |-> ( r X. s ) ) C_ ( topGen ` ran ( r e. J , s e. K |-> ( r X. s ) ) ) )
21	2	ad3antrrr 496	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ w e. ran ( ball ` P ) ) /\ ( z e. ( X X. Y ) /\ p e. RR* ) ) /\ w = ( z ( ball ` P ) p ) ) -> M e. ( *Met ` X ) )
22		simplrl 541	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ w e. ran ( ball ` P ) ) /\ ( z e. ( X X. Y ) /\ p e. RR* ) ) /\ w = ( z ( ball ` P ) p ) ) -> z e. ( X X. Y ) )
23		xp1st 6392	. . . . . . . . . . . . 13 |- ( z e. ( X X. Y ) -> ( 1st ` z ) e. X )
24	22, 23	syl 14	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ w e. ran ( ball ` P ) ) /\ ( z e. ( X X. Y ) /\ p e. RR* ) ) /\ w = ( z ( ball ` P ) p ) ) -> ( 1st ` z ) e. X )
25		simplrr 542	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ w e. ran ( ball ` P ) ) /\ ( z e. ( X X. Y ) /\ p e. RR* ) ) /\ w = ( z ( ball ` P ) p ) ) -> p e. RR* )
26	8	blopn 15517	. . . . . . . . . . . 12 |- ( ( M e. ( *Met ` X ) /\ ( 1st ` z ) e. X /\ p e. RR* ) -> ( ( 1st ` z ) ( ball ` M ) p ) e. J )
27	21, 24, 25, 26	syl3anc 1278	. . . . . . . . . . 11 |- ( ( ( ( ph /\ w e. ran ( ball ` P ) ) /\ ( z e. ( X X. Y ) /\ p e. RR* ) ) /\ w = ( z ( ball ` P ) p ) ) -> ( ( 1st ` z ) ( ball ` M ) p ) e. J )
28	3	ad3antrrr 496	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ w e. ran ( ball ` P ) ) /\ ( z e. ( X X. Y ) /\ p e. RR* ) ) /\ w = ( z ( ball ` P ) p ) ) -> N e. ( *Met ` Y ) )
29		xp2nd 6393	. . . . . . . . . . . . 13 |- ( z e. ( X X. Y ) -> ( 2nd ` z ) e. Y )
30	22, 29	syl 14	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ w e. ran ( ball ` P ) ) /\ ( z e. ( X X. Y ) /\ p e. RR* ) ) /\ w = ( z ( ball ` P ) p ) ) -> ( 2nd ` z ) e. Y )
31	11	blopn 15517	. . . . . . . . . . . 12 |- ( ( N e. ( *Met ` Y ) /\ ( 2nd ` z ) e. Y /\ p e. RR* ) -> ( ( 2nd ` z ) ( ball ` N ) p ) e. K )
32	28, 30, 25, 31	syl3anc 1278	. . . . . . . . . . 11 |- ( ( ( ( ph /\ w e. ran ( ball ` P ) ) /\ ( z e. ( X X. Y ) /\ p e. RR* ) ) /\ w = ( z ( ball ` P ) p ) ) -> ( ( 2nd ` z ) ( ball ` N ) p ) e. K )
33		simpr 110	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ w e. ran ( ball ` P ) ) /\ ( z e. ( X X. Y ) /\ p e. RR* ) ) /\ w = ( z ( ball ` P ) p ) ) -> w = ( z ( ball ` P ) p ) )
34	1, 21, 28, 25, 22	xmetxpbl 15535	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ w e. ran ( ball ` P ) ) /\ ( z e. ( X X. Y ) /\ p e. RR* ) ) /\ w = ( z ( ball ` P ) p ) ) -> ( z ( ball ` P ) p ) = ( ( ( 1st ` z ) ( ball ` M ) p ) X. ( ( 2nd ` z ) ( ball ` N ) p ) ) )
35	33, 34	eqtrd 2271	. . . . . . . . . . 11 |- ( ( ( ( ph /\ w e. ran ( ball ` P ) ) /\ ( z e. ( X X. Y ) /\ p e. RR* ) ) /\ w = ( z ( ball ` P ) p ) ) -> w = ( ( ( 1st ` z ) ( ball ` M ) p ) X. ( ( 2nd ` z ) ( ball ` N ) p ) ) )
36		xpeq1 4786	. . . . . . . . . . . . 13 |- ( r = ( ( 1st ` z ) ( ball ` M ) p ) -> ( r X. s ) = ( ( ( 1st ` z ) ( ball ` M ) p ) X. s ) )
37	36	eqeq2d 2250	. . . . . . . . . . . 12 |- ( r = ( ( 1st ` z ) ( ball ` M ) p ) -> ( w = ( r X. s ) <-> w = ( ( ( 1st ` z ) ( ball ` M ) p ) X. s ) ) )
38		xpeq2 4787	. . . . . . . . . . . . 13 |- ( s = ( ( 2nd ` z ) ( ball ` N ) p ) -> ( ( ( 1st ` z ) ( ball ` M ) p ) X. s ) = ( ( ( 1st ` z ) ( ball ` M ) p ) X. ( ( 2nd ` z ) ( ball ` N ) p ) ) )
39	38	eqeq2d 2250	. . . . . . . . . . . 12 |- ( s = ( ( 2nd ` z ) ( ball ` N ) p ) -> ( w = ( ( ( 1st ` z ) ( ball ` M ) p ) X. s ) <-> w = ( ( ( 1st ` z ) ( ball ` M ) p ) X. ( ( 2nd ` z ) ( ball ` N ) p ) ) ) )
40	37, 39	rspc2ev 2945	. . . . . . . . . . 11 |- ( ( ( ( 1st ` z ) ( ball ` M ) p ) e. J /\ ( ( 2nd ` z ) ( ball ` N ) p ) e. K /\ w = ( ( ( 1st ` z ) ( ball ` M ) p ) X. ( ( 2nd ` z ) ( ball ` N ) p ) ) ) -> E. r e. J E. s e. K w = ( r X. s ) )
41	27, 32, 35, 40	syl3anc 1278	. . . . . . . . . 10 |- ( ( ( ( ph /\ w e. ran ( ball ` P ) ) /\ ( z e. ( X X. Y ) /\ p e. RR* ) ) /\ w = ( z ( ball ` P ) p ) ) -> E. r e. J E. s e. K w = ( r X. s ) )
42		eqid 2238	. . . . . . . . . . . 12 |- ( r e. J , s e. K |-> ( r X. s ) ) = ( r e. J , s e. K |-> ( r X. s ) )
43	42	elrnmpog 6194	. . . . . . . . . . 11 |- ( w e. _V -> ( w e. ran ( r e. J , s e. K |-> ( r X. s ) ) <-> E. r e. J E. s e. K w = ( r X. s ) ) )
44	43	elv 2825	. . . . . . . . . 10 |- ( w e. ran ( r e. J , s e. K |-> ( r X. s ) ) <-> E. r e. J E. s e. K w = ( r X. s ) )
45	41, 44	sylibr 134	. . . . . . . . 9 |- ( ( ( ( ph /\ w e. ran ( ball ` P ) ) /\ ( z e. ( X X. Y ) /\ p e. RR* ) ) /\ w = ( z ( ball ` P ) p ) ) -> w e. ran ( r e. J , s e. K |-> ( r X. s ) ) )
46	20, 45	sseldd 3249	. . . . . . . 8 |- ( ( ( ( ph /\ w e. ran ( ball ` P ) ) /\ ( z e. ( X X. Y ) /\ p e. RR* ) ) /\ w = ( z ( ball ` P ) p ) ) -> w e. ( topGen ` ran ( r e. J , s e. K |-> ( r X. s ) ) ) )
47	46	ex 115	. . . . . . 7 |- ( ( ( ph /\ w e. ran ( ball ` P ) ) /\ ( z e. ( X X. Y ) /\ p e. RR* ) ) -> ( w = ( z ( ball ` P ) p ) -> w e. ( topGen ` ran ( r e. J , s e. K |-> ( r X. s ) ) ) ) )
48	47	rexlimdvva 2676	. . . . . 6 |- ( ( ph /\ w e. ran ( ball ` P ) ) -> ( E. z e. ( X X. Y ) E. p e. RR* w = ( z ( ball ` P ) p ) -> w e. ( topGen ` ran ( r e. J , s e. K |-> ( r X. s ) ) ) ) )
49	7, 48	mpd 13	. . . . 5 |- ( ( ph /\ w e. ran ( ball ` P ) ) -> w e. ( topGen ` ran ( r e. J , s e. K |-> ( r X. s ) ) ) )
50	49	ex 115	. . . 4 |- ( ph -> ( w e. ran ( ball ` P ) -> w e. ( topGen ` ran ( r e. J , s e. K |-> ( r X. s ) ) ) ) )
51	50	ssrdv 3254	. . 3 |- ( ph -> ran ( ball ` P ) C_ ( topGen ` ran ( r e. J , s e. K |-> ( r X. s ) ) ) )
52		blex 15414	. . . . 5 |- ( P e. ( *Met ` ( X X. Y ) ) -> ( ball ` P ) e. _V )
53		rnexg 5045	. . . . 5 |- ( ( ball ` P ) e. _V -> ran ( ball ` P ) e. _V )
54	4, 52, 53	3syl 17	. . . 4 |- ( ph -> ran ( ball ` P ) e. _V )
55		tgss3 15105	. . . 4 |- ( ( ran ( ball ` P ) e. _V /\ ran ( r e. J , s e. K |-> ( r X. s ) ) e. _V ) -> ( ( topGen ` ran ( ball ` P ) ) C_ ( topGen ` ran ( r e. J , s e. K |-> ( r X. s ) ) ) <-> ran ( ball ` P ) C_ ( topGen ` ran ( r e. J , s e. K |-> ( r X. s ) ) ) ) )
56	54, 17, 55	syl2anc 415	. . 3 |- ( ph -> ( ( topGen ` ran ( ball ` P ) ) C_ ( topGen ` ran ( r e. J , s e. K |-> ( r X. s ) ) ) <-> ran ( ball ` P ) C_ ( topGen ` ran ( r e. J , s e. K |-> ( r X. s ) ) ) ) )
57	51, 56	mpbird 167	. 2 |- ( ph -> ( topGen ` ran ( ball ` P ) ) C_ ( topGen ` ran ( r e. J , s e. K |-> ( r X. s ) ) ) )
58		xmettx.l	. . . 4 |- L = ( MetOpen ` P )
59	58	mopnval 15469	. . 3 |- ( P e. ( *Met ` ( X X. Y ) ) -> L = ( topGen ` ran ( ball ` P ) ) )
60	4, 59	syl 14	. 2 |- ( ph -> L = ( topGen ` ran ( ball ` P ) ) )
61		eqid 2238	. . . 4 |- ran ( r e. J , s e. K |-> ( r X. s ) ) = ran ( r e. J , s e. K |-> ( r X. s ) )
62	61	txval 15282	. . 3 |- ( ( J e. Top /\ K e. Top ) -> ( J tX K ) = ( topGen ` ran ( r e. J , s e. K |-> ( r X. s ) ) ) )
63	10, 13, 62	syl2anc 415	. 2 |- ( ph -> ( J tX K ) = ( topGen ` ran ( r e. J , s e. K |-> ( r X. s ) ) ) )
64	57, 60, 63	3sstr4d 3293	1 |- ( ph -> L C_ ( J tX K ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   E.wrex 2529   _Vcvv 2821   C_ wss 3220   {cpr 3709   X. cxp 4770   ran crn 4773   ` cfv 5375  (class class class)co 6078   e. cmpo 6080   1stc1st 6365   2ndc2nd 6366   supcsup 7315   RR*cxr 8352   < clt 8353   topGenctg 13588   *Metcxmet 14848   ballcbl 14850   MetOpencmopn 14853   Topctop 15024   tX ctx 15279

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8263  ax-resscn 8264  ax-1cn 8265  ax-1re 8266  ax-icn 8267  ax-addcl 8268  ax-addrcl 8269  ax-mulcl 8270  ax-mulrcl 8271  ax-addcom 8272  ax-mulcom 8273  ax-addass 8274  ax-mulass 8275  ax-distr 8276  ax-i2m1 8277  ax-0lt1 8278  ax-1rid 8279  ax-0id 8280  ax-rnegex 8281  ax-precex 8282  ax-cnre 8283  ax-pre-ltirr 8284  ax-pre-ltwlin 8285  ax-pre-lttrn 8286  ax-pre-apti 8287  ax-pre-ltadd 8288  ax-pre-mulgt0 8289  ax-pre-mulext 8290  ax-arch 8291  ax-caucvg 8292

This theorem depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-isom 5384  df-riota 6031  df-ov 6081  df-oprab 6082  df-mpo 6083  df-1st 6367  df-2nd 6368  df-recs 6569  df-frec 6655  df-map 6917  df-sup 7317  df-inf 7318  df-pnf 8355  df-mnf 8356  df-xr 8357  df-ltxr 8358  df-le 8359  df-sub 8492  df-neg 8493  df-reap 8896  df-ap 8903  df-div 8996  df-inn 9287  df-2 9345  df-3 9346  df-4 9347  df-n0 9546  df-z 9627  df-uz 9904  df-q 10002  df-rp 10037  df-xneg 10156  df-xadd 10157  df-seqfrec 10866  df-exp 10957  df-cj 11588  df-re 11589  df-im 11590  df-rsqrt 11745  df-abs 11746  df-topgen 13594  df-psmet 14855  df-xmet 14856  df-bl 14858  df-mopn 14859  df-top 15025  df-topon 15038  df-bases 15070  df-tx 15280

This theorem is referenced by:  xmettx  15537