Theorem dvidsslem 15410

Description: Lemma for dvconstss 15415. Analogue of dvidlemap 15408 where F is defined on an open subset of the real or complex numbers. (Contributed by Jim Kingdon, 3-Oct-2025.)

Hypotheses

Ref	Expression
dvidsslem.s	|- ( ph -> S e. { RR , CC } )
dvidsslem.j	|- J = ( K ↾t S )
dvidsslem.k	|- K = ( MetOpen ` ( abs o. - ) )
dvidsslem.1	|- ( ph -> F : X --> CC )
dvidsslem.x	|- ( ph -> X e. J )
dvidsslem.2	|- ( ( ph /\ ( x e. X /\ z e. X /\ z # x ) ) -> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) = B )
dvidsslem.3	|- B e. CC

Assertion

Ref	Expression
dvidsslem	|- ( ph -> ( S _D F ) = ( X X. { B } ) )

Distinct variable groups:   x, z, B   x, F, z   ph, x, z   x, S, z   x, X, z

Allowed substitution hints:   J( x, z)   K( x, z)

Proof of Theorem dvidsslem

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvidsslem.s	. . . . 5 |- ( ph -> S e. { RR , CC } )
2		ssidd 3246	. . . . . . 7 |- ( ph -> CC C_ CC )
3		dvidsslem.j	. . . . . . . . . 10 |- J = ( K ↾t S )
4		restsspw 13325	. . . . . . . . . 10 |- ( K ↾t S ) C_ ~P S
5	3, 4	eqsstri 3257	. . . . . . . . 9 |- J C_ ~P S
6		dvidsslem.x	. . . . . . . . 9 |- ( ph -> X e. J )
7	5, 6	sselid 3223	. . . . . . . 8 |- ( ph -> X e. ~P S )
8	7	elpwid 3661	. . . . . . 7 |- ( ph -> X C_ S )
9		cnex 8149	. . . . . . . 8 |- CC e. _V
10	9	a1i 9	. . . . . . 7 |- ( ph -> CC e. _V )
11		pmss12g 6839	. . . . . . 7 |- ( ( ( CC C_ CC /\ X C_ S ) /\ ( CC e. _V /\ S e. { RR , CC } ) ) -> ( CC ^pm X ) C_ ( CC ^pm S ) )
12	2, 8, 10, 1, 11	syl22anc 1272	. . . . . 6 |- ( ph -> ( CC ^pm X ) C_ ( CC ^pm S ) )
13		dvidsslem.1	. . . . . . 7 |- ( ph -> F : X --> CC )
14		fpmg 6838	. . . . . . 7 |- ( ( X e. J /\ CC e. _V /\ F : X --> CC ) -> F e. ( CC ^pm X ) )
15	6, 10, 13, 14	syl3anc 1271	. . . . . 6 |- ( ph -> F e. ( CC ^pm X ) )
16	12, 15	sseldd 3226	. . . . 5 |- ( ph -> F e. ( CC ^pm S ) )
17		dvfgg 15405	. . . . 5 |- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) -> ( S _D F ) : dom ( S _D F ) --> CC )
18	1, 16, 17	syl2anc 411	. . . 4 |- ( ph -> ( S _D F ) : dom ( S _D F ) --> CC )
19		recnprss 15404	. . . . . . . 8 |- ( S e. { RR , CC } -> S C_ CC )
20	1, 19	syl 14	. . . . . . 7 |- ( ph -> S C_ CC )
21	20, 13, 8	dvbss 15402	. . . . . 6 |- ( ph -> dom ( S _D F ) C_ X )
22		reldvg 15396	. . . . . . . . 9 |- ( ( S C_ CC /\ F e. ( CC ^pm S ) ) -> Rel ( S _D F ) )
23	20, 16, 22	syl2anc 411	. . . . . . . 8 |- ( ph -> Rel ( S _D F ) )
24	23	adantr 276	. . . . . . 7 |- ( ( ph /\ x e. X ) -> Rel ( S _D F ) )
25		dvidsslem.k	. . . . . . . . . . . . . . . 16 |- K = ( MetOpen ` ( abs o. - ) )
26	25	cntoptop 15250	. . . . . . . . . . . . . . 15 |- K e. Top
27	26	a1i 9	. . . . . . . . . . . . . 14 |- ( ph -> K e. Top )
28		resttop 14887	. . . . . . . . . . . . . 14 |- ( ( K e. Top /\ S e. { RR , CC } ) -> ( K ↾t S ) e. Top )
29	27, 1, 28	syl2anc 411	. . . . . . . . . . . . 13 |- ( ph -> ( K ↾t S ) e. Top )
30	3, 29	eqeltrid 2316	. . . . . . . . . . . 12 |- ( ph -> J e. Top )
31		isopn3i 14852	. . . . . . . . . . . 12 |- ( ( J e. Top /\ X e. J ) -> ( ( int ` J ) ` X ) = X )
32	30, 6, 31	syl2anc 411	. . . . . . . . . . 11 |- ( ph -> ( ( int ` J ) ` X ) = X )
33	32	eqcomd 2235	. . . . . . . . . 10 |- ( ph -> X = ( ( int ` J ) ` X ) )
34	33	eleq2d 2299	. . . . . . . . 9 |- ( ph -> ( x e. X <-> x e. ( ( int ` J ) ` X ) ) )
35	34	biimpa 296	. . . . . . . 8 |- ( ( ph /\ x e. X ) -> x e. ( ( int ` J ) ` X ) )
36		limcresi 15383	. . . . . . . . . 10 |- ( ( z e. X |-> B ) lim CC x ) C_ ( ( ( z e. X |-> B ) |` { w e. X | w # x } ) lim CC x )
37		dvidsslem.3	. . . . . . . . . . . . . 14 |- B e. CC
38	37	a1i 9	. . . . . . . . . . . . 13 |- ( ph -> B e. CC )
39	8, 20	sstrd 3235	. . . . . . . . . . . . 13 |- ( ph -> X C_ CC )
40		cncfmptc 15313	. . . . . . . . . . . . 13 |- ( ( B e. CC /\ X C_ CC /\ CC C_ CC ) -> ( z e. X |-> B ) e. ( X -cn-> CC ) )
41	38, 39, 2, 40	syl3anc 1271	. . . . . . . . . . . 12 |- ( ph -> ( z e. X |-> B ) e. ( X -cn-> CC ) )
42	41	adantr 276	. . . . . . . . . . 11 |- ( ( ph /\ x e. X ) -> ( z e. X |-> B ) e. ( X -cn-> CC ) )
43		simpr 110	. . . . . . . . . . 11 |- ( ( ph /\ x e. X ) -> x e. X )
44		eqidd 2230	. . . . . . . . . . 11 |- ( z = x -> B = B )
45	42, 43, 44	cnmptlimc 15391	. . . . . . . . . 10 |- ( ( ph /\ x e. X ) -> B e. ( ( z e. X |-> B ) lim CC x ) )
46	36, 45	sselid 3223	. . . . . . . . 9 |- ( ( ph /\ x e. X ) -> B e. ( ( ( z e. X |-> B ) |` { w e. X | w # x } ) lim CC x ) )
47		breq1 4089	. . . . . . . . . . . . . 14 |- ( w = z -> ( w # x <-> z # x ) )
48	47	elrab 2960	. . . . . . . . . . . . 13 |- ( z e. { w e. X | w # x } <-> ( z e. X /\ z # x ) )
49		dvidsslem.2	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( x e. X /\ z e. X /\ z # x ) ) -> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) = B )
50	49	3exp2 1249	. . . . . . . . . . . . . 14 |- ( ph -> ( x e. X -> ( z e. X -> ( z # x -> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) = B ) ) ) )
51	50	imp43 355	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. X ) /\ ( z e. X /\ z # x ) ) -> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) = B )
52	48, 51	sylan2b 287	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. X ) /\ z e. { w e. X | w # x } ) -> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) = B )
53	52	mpteq2dva 4177	. . . . . . . . . . 11 |- ( ( ph /\ x e. X ) -> ( z e. { w e. X | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) = ( z e. { w e. X | w # x } |-> B ) )
54		ssrab2 3310	. . . . . . . . . . . 12 |- { w e. X | w # x } C_ X
55		resmpt 5059	. . . . . . . . . . . 12 |- ( { w e. X | w # x } C_ X -> ( ( z e. X |-> B ) |` { w e. X | w # x } ) = ( z e. { w e. X | w # x } |-> B ) )
56	54, 55	ax-mp 5	. . . . . . . . . . 11 |- ( ( z e. X |-> B ) |` { w e. X | w # x } ) = ( z e. { w e. X | w # x } |-> B )
57	53, 56	eqtr4di 2280	. . . . . . . . . 10 |- ( ( ph /\ x e. X ) -> ( z e. { w e. X | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) = ( ( z e. X |-> B ) |` { w e. X | w # x } ) )
58	57	oveq1d 6028	. . . . . . . . 9 |- ( ( ph /\ x e. X ) -> ( ( z e. { w e. X | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) lim CC x ) = ( ( ( z e. X |-> B ) |` { w e. X | w # x } ) lim CC x ) )
59	46, 58	eleqtrrd 2309	. . . . . . . 8 |- ( ( ph /\ x e. X ) -> B e. ( ( z e. { w e. X | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) lim CC x ) )
60		eqid 2229	. . . . . . . . . 10 |- ( z e. { w e. X | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) = ( z e. { w e. X | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) )
61	3, 25, 60, 20, 13, 8	eldvap 15399	. . . . . . . . 9 |- ( ph -> ( x ( S _D F ) B <-> ( x e. ( ( int ` J ) ` X ) /\ B e. ( ( z e. { w e. X | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) lim CC x ) ) ) )
62	61	adantr 276	. . . . . . . 8 |- ( ( ph /\ x e. X ) -> ( x ( S _D F ) B <-> ( x e. ( ( int ` J ) ` X ) /\ B e. ( ( z e. { w e. X | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) lim CC x ) ) ) )
63	35, 59, 62	mpbir2and 950	. . . . . . 7 |- ( ( ph /\ x e. X ) -> x ( S _D F ) B )
64		releldm 4965	. . . . . . 7 |- ( ( Rel ( S _D F ) /\ x ( S _D F ) B ) -> x e. dom ( S _D F ) )
65	24, 63, 64	syl2anc 411	. . . . . 6 |- ( ( ph /\ x e. X ) -> x e. dom ( S _D F ) )
66	21, 65	eqelssd 3244	. . . . 5 |- ( ph -> dom ( S _D F ) = X )
67	66	feq2d 5467	. . . 4 |- ( ph -> ( ( S _D F ) : dom ( S _D F ) --> CC <-> ( S _D F ) : X --> CC ) )
68	18, 67	mpbid 147	. . 3 |- ( ph -> ( S _D F ) : X --> CC )
69	68	ffnd 5480	. 2 |- ( ph -> ( S _D F ) Fn X )
70		fnconstg 5531	. . 3 |- ( B e. CC -> ( X X. { B } ) Fn X )
71	37, 70	mp1i 10	. 2 |- ( ph -> ( X X. { B } ) Fn X )
72	18	adantr 276	. . . . . 6 |- ( ( ph /\ x e. X ) -> ( S _D F ) : dom ( S _D F ) --> CC )
73	72	ffund 5483	. . . . 5 |- ( ( ph /\ x e. X ) -> Fun ( S _D F ) )
74		funbrfvb 5682	. . . . 5 |- ( ( Fun ( S _D F ) /\ x e. dom ( S _D F ) ) -> ( ( ( S _D F ) ` x ) = B <-> x ( S _D F ) B ) )
75	73, 65, 74	syl2anc 411	. . . 4 |- ( ( ph /\ x e. X ) -> ( ( ( S _D F ) ` x ) = B <-> x ( S _D F ) B ) )
76	63, 75	mpbird 167	. . 3 |- ( ( ph /\ x e. X ) -> ( ( S _D F ) ` x ) = B )
77		fvconst2g 5863	. . . 4 |- ( ( B e. CC /\ x e. X ) -> ( ( X X. { B } ) ` x ) = B )
78	38, 77	sylan 283	. . 3 |- ( ( ph /\ x e. X ) -> ( ( X X. { B } ) ` x ) = B )
79	76, 78	eqtr4d 2265	. 2 |- ( ( ph /\ x e. X ) -> ( ( S _D F ) ` x ) = ( ( X X. { B } ) ` x ) )
80	69, 71, 79	eqfnfvd 5743	1 |- ( ph -> ( S _D F ) = ( X X. { B } ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   {crab 2512   _Vcvv 2800   C_ wss 3198   ~Pcpw 3650   {csn 3667   {cpr 3668   class class class wbr 4086   |-> cmpt 4148   X. cxp 4721   dom cdm 4723   |` cres 4725   o. ccom 4727   Rel wrel 4728   Fun wfun 5318   Fn wfn 5319   -->wf 5320   ` cfv 5324  (class class class)co 6013   ^pm cpm 6813   CCcc 8023   RRcr 8024   - cmin 8343   # cap 8754   / cdiv 8845   abscabs 11551   ↾_t crest 13315   MetOpencmopn 14548   Topctop 14714   intcnt 14810   -cn->ccncf 15287   lim CC climc 15371   _D cdv 15372

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 8116  ax-resscn 8117  ax-1cn 8118  ax-1re 8119  ax-icn 8120  ax-addcl 8121  ax-addrcl 8122  ax-mulcl 8123  ax-mulrcl 8124  ax-addcom 8125  ax-mulcom 8126  ax-addass 8127  ax-mulass 8128  ax-distr 8129  ax-i2m1 8130  ax-0lt1 8131  ax-1rid 8132  ax-0id 8133  ax-rnegex 8134  ax-precex 8135  ax-cnre 8136  ax-pre-ltirr 8137  ax-pre-ltwlin 8138  ax-pre-lttrn 8139  ax-pre-apti 8140  ax-pre-ltadd 8141  ax-pre-mulgt0 8142  ax-pre-mulext 8143  ax-arch 8144  ax-caucvg 8145

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-pm 6815  df-sup 7177  df-inf 7178  df-pnf 8209  df-mnf 8210  df-xr 8211  df-ltxr 8212  df-le 8213  df-sub 8345  df-neg 8346  df-reap 8748  df-ap 8755  df-div 8846  df-inn 9137  df-2 9195  df-3 9196  df-4 9197  df-n0 9396  df-z 9473  df-uz 9749  df-q 9847  df-rp 9882  df-xneg 10000  df-xadd 10001  df-seqfrec 10703  df-exp 10794  df-cj 11396  df-re 11397  df-im 11398  df-rsqrt 11552  df-abs 11553  df-rest 13317  df-topgen 13336  df-psmet 14550  df-xmet 14551  df-met 14552  df-bl 14553  df-mopn 14554  df-top 14715  df-topon 14728  df-bases 14760  df-ntr 14813  df-cn 14905  df-cnp 14906  df-cncf 15288  df-limced 15373  df-dvap 15374

This theorem is referenced by:  dvconstss  15415