Theorem xmettxlem 14688

Description: Lemma for xmettx 14689. (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 14686	. . . . . . . 8 |- ( ph -> P e. ( *Met ` ( X X. Y ) ) )
5		blrn 14591	. . . . . . . 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 14623	. . . . . . . . . . . . . 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 14623	. . . . . . . . . . . . . 14 |- ( N e. ( *Met ` Y ) -> K e. Top )
13	3, 12	syl 14	. . . . . . . . . . . . 13 |- ( ph -> K e. Top )
14		mpoexga 6267	. . . . . . . . . . . . 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 4928	. . . . . . . . . . . 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 14240	. . . . . . . . . 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 6220	. . . . . . . . . . . . 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 14669	. . . . . . . . . . . 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 1249	. . . . . . . . . . 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 6221	. . . . . . . . . . . . 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 14669	. . . . . . . . . . . 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 1249	. . . . . . . . . . 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 14687	. . . . . . . . . . . 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 2226	. . . . . . . . . . 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 4674	. . . . . . . . . . . . 13 |- ( r = ( ( 1st ` z ) ( ball ` M ) p ) -> ( r X. s ) = ( ( ( 1st ` z ) ( ball ` M ) p ) X. s ) )
37	36	eqeq2d 2205	. . . . . . . . . . . 12 |- ( r = ( ( 1st ` z ) ( ball ` M ) p ) -> ( w = ( r X. s ) <-> w = ( ( ( 1st ` z ) ( ball ` M ) p ) X. s ) ) )
38		xpeq2 4675	. . . . . . . . . . . . 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 2205	. . . . . . . . . . . 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 2880	. . . . . . . . . . 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 1249	. . . . . . . . . 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 2193	. . . . . . . . . . . 12 |- ( r e. J , s e. K |-> ( r X. s ) ) = ( r e. J , s e. K |-> ( r X. s ) )
43	42	elrnmpog 6032	. . . . . . . . . . 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 2764	. . . . . . . . . 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 3181	. . . . . . . 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 2619	. . . . . 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 3186	. . 3 |- ( ph -> ran ( ball ` P ) C_ ( topGen ` ran ( r e. J , s e. K |-> ( r X. s ) ) ) )
52		blex 14566	. . . . 5 |- ( P e. ( *Met ` ( X X. Y ) ) -> ( ball ` P ) e. _V )
53		rnexg 4928	. . . . 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 14257	. . . 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 14621	. . 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 2193	. . . 4 |- ran ( r e. J , s e. K |-> ( r X. s ) ) = ran ( r e. J , s e. K |-> ( r X. s ) )
62	61	txval 14434	. . 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 3225	1 |- ( ph -> L C_ ( J tX K ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   E.wrex 2473   _Vcvv 2760   C_ wss 3154   {cpr 3620   X. cxp 4658   ran crn 4661   ` cfv 5255  (class class class)co 5919   e. cmpo 5921   1stc1st 6193   2ndc2nd 6194   supcsup 7043   RR*cxr 8055   < clt 8056   topGenctg 12868   *Metcxmet 14035   ballcbl 14037   MetOpencmopn 14040   Topctop 14176   tX ctx 14431

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-mulrcl 7973  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-precex 7984  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990  ax-pre-mulgt0 7991  ax-pre-mulext 7992  ax-arch 7993  ax-caucvg 7994

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-ilim 4401  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-frec 6446  df-map 6706  df-sup 7045  df-inf 7046  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-reap 8596  df-ap 8603  df-div 8694  df-inn 8985  df-2 9043  df-3 9044  df-4 9045  df-n0 9244  df-z 9321  df-uz 9596  df-q 9688  df-rp 9723  df-xneg 9841  df-xadd 9842  df-seqfrec 10522  df-exp 10613  df-cj 10989  df-re 10990  df-im 10991  df-rsqrt 11145  df-abs 11146  df-topgen 12874  df-psmet 14042  df-xmet 14043  df-bl 14045  df-mopn 14046  df-top 14177  df-topon 14190  df-bases 14222  df-tx 14432

This theorem is referenced by:  xmettx  14689