Theorem xmettxlem 14048

Description: Lemma for xmettx 14049. (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 14046	. . . . . . . 8 |- ( ph -> P e. ( *Met ` ( X X. Y ) ) )
5		blrn 13951	. . . . . . . 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 13983	. . . . . . . . . . . . . 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 13983	. . . . . . . . . . . . . 14 |- ( N e. ( *Met ` Y ) -> K e. Top )
13	3, 12	syl 14	. . . . . . . . . . . . 13 |- ( ph -> K e. Top )
14		mpoexga 6215	. . . . . . . . . . . . 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 4894	. . . . . . . . . . . 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 13600	. . . . . . . . . 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 6168	. . . . . . . . . . . . 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 14029	. . . . . . . . . . . 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 1238	. . . . . . . . . . 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 6169	. . . . . . . . . . . . 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 14029	. . . . . . . . . . . 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 1238	. . . . . . . . . . 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 14047	. . . . . . . . . . . 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 2210	. . . . . . . . . . 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 4642	. . . . . . . . . . . . 13 |- ( r = ( ( 1st ` z ) ( ball ` M ) p ) -> ( r X. s ) = ( ( ( 1st ` z ) ( ball ` M ) p ) X. s ) )
37	36	eqeq2d 2189	. . . . . . . . . . . 12 |- ( r = ( ( 1st ` z ) ( ball ` M ) p ) -> ( w = ( r X. s ) <-> w = ( ( ( 1st ` z ) ( ball ` M ) p ) X. s ) ) )
38		xpeq2 4643	. . . . . . . . . . . . 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 2189	. . . . . . . . . . . 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 2858	. . . . . . . . . . 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 1238	. . . . . . . . . 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 2177	. . . . . . . . . . . 12 |- ( r e. J , s e. K |-> ( r X. s ) ) = ( r e. J , s e. K |-> ( r X. s ) )
43	42	elrnmpog 5989	. . . . . . . . . . 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 2743	. . . . . . . . . 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 3158	. . . . . . . 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 2602	. . . . . 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 3163	. . 3 |- ( ph -> ran ( ball ` P ) C_ ( topGen ` ran ( r e. J , s e. K |-> ( r X. s ) ) ) )
52		blex 13926	. . . . 5 |- ( P e. ( *Met ` ( X X. Y ) ) -> ( ball ` P ) e. _V )
53		rnexg 4894	. . . . 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 13617	. . . 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 13981	. . 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 2177	. . . 4 |- ran ( r e. J , s e. K |-> ( r X. s ) ) = ran ( r e. J , s e. K |-> ( r X. s ) )
62	61	txval 13794	. . 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 3202	1 |- ( ph -> L C_ ( J tX K ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   E.wrex 2456   _Vcvv 2739   C_ wss 3131   {cpr 3595   X. cxp 4626   ran crn 4629   ` cfv 5218  (class class class)co 5877   e. cmpo 5879   1stc1st 6141   2ndc2nd 6142   supcsup 6983   RR*cxr 7993   < clt 7994   topGenctg 12708   *Metcxmet 13479   ballcbl 13481   MetOpencmopn 13484   Topctop 13536   tX ctx 13791

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931  ax-arch 7932  ax-caucvg 7933

This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-isom 5227  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-frec 6394  df-map 6652  df-sup 6985  df-inf 6986  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632  df-inn 8922  df-2 8980  df-3 8981  df-4 8982  df-n0 9179  df-z 9256  df-uz 9531  df-q 9622  df-rp 9656  df-xneg 9774  df-xadd 9775  df-seqfrec 10448  df-exp 10522  df-cj 10853  df-re 10854  df-im 10855  df-rsqrt 11009  df-abs 11010  df-topgen 12714  df-psmet 13486  df-xmet 13487  df-bl 13489  df-mopn 13490  df-top 13537  df-topon 13550  df-bases 13582  df-tx 13792

This theorem is referenced by:  xmettx  14049