Theorem xmettxlem 15223

Description: Lemma for xmettx 15224. (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 15221	. . . . . . . 8 |- ( ph -> P e. ( *Met ` ( X X. Y ) ) )
5		blrn 15126	. . . . . . . 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 15158	. . . . . . . . . . . . . 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 15158	. . . . . . . . . . . . . 14 |- ( N e. ( *Met ` Y ) -> K e. Top )
13	3, 12	syl 14	. . . . . . . . . . . . 13 |- ( ph -> K e. Top )
14		mpoexga 6372	. . . . . . . . . . . . 13 |- ( ( J e. Top /\ K e. Top ) -> ( r e. J , s e. K |-> ( r X. s ) ) e. _V )
15	10, 13, 14	syl2anc 411	. . . . . . . . . . . 12 |- ( ph -> ( r e. J , s e. K |-> ( r X. s ) ) e. _V )
16		rnexg 4995	. . . . . . . . . . . 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 492	. . . . . . . . . 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 14775	. . . . . . . . . 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 492	. . . . . . . . . . . 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 535	. . . . . . . . . . . . 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 6323	. . . . . . . . . . . . 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 536	. . . . . . . . . . . 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 15204	. . . . . . . . . . . 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 1271	. . . . . . . . . . 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 492	. . . . . . . . . . . 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 6324	. . . . . . . . . . . . 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 15204	. . . . . . . . . . . 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 1271	. . . . . . . . . . 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 15222	. . . . . . . . . . . 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 2262	. . . . . . . . . . 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 4737	. . . . . . . . . . . . 13 |- ( r = ( ( 1st ` z ) ( ball ` M ) p ) -> ( r X. s ) = ( ( ( 1st ` z ) ( ball ` M ) p ) X. s ) )
37	36	eqeq2d 2241	. . . . . . . . . . . 12 |- ( r = ( ( 1st ` z ) ( ball ` M ) p ) -> ( w = ( r X. s ) <-> w = ( ( ( 1st ` z ) ( ball ` M ) p ) X. s ) ) )
38		xpeq2 4738	. . . . . . . . . . . . 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 2241	. . . . . . . . . . . 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 2923	. . . . . . . . . . 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 1271	. . . . . . . . . 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 2229	. . . . . . . . . . . 12 |- ( r e. J , s e. K |-> ( r X. s ) ) = ( r e. J , s e. K |-> ( r X. s ) )
43	42	elrnmpog 6129	. . . . . . . . . . 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 2804	. . . . . . . . . 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 3226	. . . . . . . 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 2656	. . . . . 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 3231	. . 3 |- ( ph -> ran ( ball ` P ) C_ ( topGen ` ran ( r e. J , s e. K |-> ( r X. s ) ) ) )
52		blex 15101	. . . . 5 |- ( P e. ( *Met ` ( X X. Y ) ) -> ( ball ` P ) e. _V )
53		rnexg 4995	. . . . 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 14792	. . . 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 411	. . 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 15156	. . 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 2229	. . . 4 |- ran ( r e. J , s e. K |-> ( r X. s ) ) = ran ( r e. J , s e. K |-> ( r X. s ) )
62	61	txval 14969	. . 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 411	. 2 |- ( ph -> ( J tX K ) = ( topGen ` ran ( r e. J , s e. K |-> ( r X. s ) ) ) )
64	57, 60, 63	3sstr4d 3270	1 |- ( ph -> L C_ ( J tX K ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   E.wrex 2509   _Vcvv 2800   C_ wss 3198   {cpr 3668   X. cxp 4721   ran crn 4724   ` cfv 5324  (class class class)co 6013   e. cmpo 6015   1stc1st 6296   2ndc2nd 6297   supcsup 7172   RR*cxr 8203   < clt 8204   topGenctg 13327   *Metcxmet 14540   ballcbl 14542   MetOpencmopn 14545   Topctop 14711   tX ctx 14966

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-mulrcl 8121  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-precex 8132  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138  ax-pre-mulgt0 8139  ax-pre-mulext 8140  ax-arch 8141  ax-caucvg 8142

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-isom 5333  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-frec 6552  df-map 6814  df-sup 7174  df-inf 7175  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-reap 8745  df-ap 8752  df-div 8843  df-inn 9134  df-2 9192  df-3 9193  df-4 9194  df-n0 9393  df-z 9470  df-uz 9746  df-q 9844  df-rp 9879  df-xneg 9997  df-xadd 9998  df-seqfrec 10700  df-exp 10791  df-cj 11393  df-re 11394  df-im 11395  df-rsqrt 11549  df-abs 11550  df-topgen 13333  df-psmet 14547  df-xmet 14548  df-bl 14550  df-mopn 14551  df-top 14712  df-topon 14725  df-bases 14757  df-tx 14967

This theorem is referenced by:  xmettx  15224