Theorem xmettxlem 12678

Description: Lemma for xmettx 12679. (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 12676	. . . . . . . 8 |- ( ph -> P e. ( *Met ` ( X X. Y ) ) )
5		blrn 12581	. . . . . . . 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 294	. . . . . 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 12613	. . . . . . . . . . . . . 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 12613	. . . . . . . . . . . . . 14 |- ( N e. ( *Met ` Y ) -> K e. Top )
13	3, 12	syl 14	. . . . . . . . . . . . 13 |- ( ph -> K e. Top )
14		mpoexga 6110	. . . . . . . . . . . . 13 |- ( ( J e. Top /\ K e. Top ) -> ( r e. J , s e. K |-> ( r X. s ) ) e. _V )
15	10, 13, 14	syl2anc 408	. . . . . . . . . . . 12 |- ( ph -> ( r e. J , s e. K |-> ( r X. s ) ) e. _V )
16		rnexg 4804	. . . . . . . . . . . 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 483	. . . . . . . . . 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 12230	. . . . . . . . . 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 483	. . . . . . . . . . . 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 524	. . . . . . . . . . . . 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 6063	. . . . . . . . . . . . 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 525	. . . . . . . . . . . 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 12659	. . . . . . . . . . . 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 1216	. . . . . . . . . . 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 483	. . . . . . . . . . . 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 6064	. . . . . . . . . . . . 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 12659	. . . . . . . . . . . 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 1216	. . . . . . . . . . 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 109	. . . . . . . . . . . 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 12677	. . . . . . . . . . . 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 2172	. . . . . . . . . . 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 4553	. . . . . . . . . . . . 13 |- ( r = ( ( 1st ` z ) ( ball ` M ) p ) -> ( r X. s ) = ( ( ( 1st ` z ) ( ball ` M ) p ) X. s ) )
37	36	eqeq2d 2151	. . . . . . . . . . . 12 |- ( r = ( ( 1st ` z ) ( ball ` M ) p ) -> ( w = ( r X. s ) <-> w = ( ( ( 1st ` z ) ( ball ` M ) p ) X. s ) ) )
38		xpeq2 4554	. . . . . . . . . . . . 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 2151	. . . . . . . . . . . 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 2804	. . . . . . . . . . 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 1216	. . . . . . . . . 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 2139	. . . . . . . . . . . 12 |- ( r e. J , s e. K |-> ( r X. s ) ) = ( r e. J , s e. K |-> ( r X. s ) )
43	42	elrnmpog 5883	. . . . . . . . . . 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 2690	. . . . . . . . . 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 133	. . . . . . . . 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 3098	. . . . . . . 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 114	. . . . . . 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 2557	. . . . . 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 114	. . . 4 |- ( ph -> ( w e. ran ( ball ` P ) -> w e. ( topGen ` ran ( r e. J , s e. K |-> ( r X. s ) ) ) ) )
51	50	ssrdv 3103	. . 3 |- ( ph -> ran ( ball ` P ) C_ ( topGen ` ran ( r e. J , s e. K |-> ( r X. s ) ) ) )
52		blex 12556	. . . . 5 |- ( P e. ( *Met ` ( X X. Y ) ) -> ( ball ` P ) e. _V )
53		rnexg 4804	. . . . 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 12247	. . . 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 408	. . 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 166	. 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 12611	. . 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 2139	. . . 4 |- ran ( r e. J , s e. K |-> ( r X. s ) ) = ran ( r e. J , s e. K |-> ( r X. s ) )
62	61	txval 12424	. . 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 408	. 2 |- ( ph -> ( J tX K ) = ( topGen ` ran ( r e. J , s e. K |-> ( r X. s ) ) ) )
64	57, 60, 63	3sstr4d 3142	1 |- ( ph -> L C_ ( J tX K ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   E.wrex 2417   _Vcvv 2686   C_ wss 3071   {cpr 3528   X. cxp 4537   ran crn 4540   ` cfv 5123  (class class class)co 5774   e. cmpo 5776   1stc1st 6036   2ndc2nd 6037   supcsup 6869   RR*cxr 7799   < clt 7800   topGenctg 12135   *Metcxmet 12149   ballcbl 12151   MetOpencmopn 12154   Topctop 12164   tX ctx 12421

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738  ax-arch 7739  ax-caucvg 7740

This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-map 6544  df-sup 6871  df-inf 6872  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-2 8779  df-3 8780  df-4 8781  df-n0 8978  df-z 9055  df-uz 9327  df-q 9412  df-rp 9442  df-xneg 9559  df-xadd 9560  df-seqfrec 10219  df-exp 10293  df-cj 10614  df-re 10615  df-im 10616  df-rsqrt 10770  df-abs 10771  df-topgen 12141  df-psmet 12156  df-xmet 12157  df-bl 12159  df-mopn 12160  df-top 12165  df-topon 12178  df-bases 12210  df-tx 12422

This theorem is referenced by:  xmettx  12679