Theorem dvidsslem 15689

Description: Lemma for dvconstss 15694. Analogue of dvidlemap 15687 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 3263	. . . . . . 7 |- ( ph -> CC C_ CC )
3		dvidsslem.j	. . . . . . . . . 10 |- J = ( K ↾t S )
4		restsspw 13551	. . . . . . . . . 10 |- ( K ↾t S ) C_ ~P S
5	3, 4	eqsstri 3274	. . . . . . . . 9 |- J C_ ~P S
6		dvidsslem.x	. . . . . . . . 9 |- ( ph -> X e. J )
7	5, 6	sselid 3240	. . . . . . . 8 |- ( ph -> X e. ~P S )
8	7	elpwid 3686	. . . . . . 7 |- ( ph -> X C_ S )
9		cnex 8268	. . . . . . . 8 |- CC e. _V
10	9	a1i 9	. . . . . . 7 |- ( ph -> CC e. _V )
11		pmss12g 6923	. . . . . . 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 1275	. . . . . 6 |- ( ph -> ( CC ^pm X ) C_ ( CC ^pm S ) )
13		dvidsslem.1	. . . . . . 7 |- ( ph -> F : X --> CC )
14		fpmg 6922	. . . . . . 7 |- ( ( X e. J /\ CC e. _V /\ F : X --> CC ) -> F e. ( CC ^pm X ) )
15	6, 10, 13, 14	syl3anc 1274	. . . . . 6 |- ( ph -> F e. ( CC ^pm X ) )
16	12, 15	sseldd 3243	. . . . 5 |- ( ph -> F e. ( CC ^pm S ) )
17		dvfgg 15684	. . . . 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 15683	. . . . . . . 8 |- ( S e. { RR , CC } -> S C_ CC )
20	1, 19	syl 14	. . . . . . 7 |- ( ph -> S C_ CC )
21	20, 13, 8	dvbss 15681	. . . . . 6 |- ( ph -> dom ( S _D F ) C_ X )
22		reldvg 15675	. . . . . . . . 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 15529	. . . . . . . . . . . . . . 15 |- K e. Top
27	26	a1i 9	. . . . . . . . . . . . . 14 |- ( ph -> K e. Top )
28		resttop 15166	. . . . . . . . . . . . . 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 2321	. . . . . . . . . . . 12 |- ( ph -> J e. Top )
31		isopn3i 15131	. . . . . . . . . . . 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 2240	. . . . . . . . . 10 |- ( ph -> X = ( ( int ` J ) ` X ) )
34	33	eleq2d 2304	. . . . . . . . 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 15662	. . . . . . . . . 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 3252	. . . . . . . . . . . . 13 |- ( ph -> X C_ CC )
40		cncfmptc 15592	. . . . . . . . . . . . 13 |- ( ( B e. CC /\ X C_ CC /\ CC C_ CC ) -> ( z e. X |-> B ) e. ( X -cn-> CC ) )
41	38, 39, 2, 40	syl3anc 1274	. . . . . . . . . . . 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 2235	. . . . . . . . . . 11 |- ( z = x -> B = B )
45	42, 43, 44	cnmptlimc 15670	. . . . . . . . . 10 |- ( ( ph /\ x e. X ) -> B e. ( ( z e. X |-> B ) lim CC x ) )
46	36, 45	sselid 3240	. . . . . . . . 9 |- ( ( ph /\ x e. X ) -> B e. ( ( ( z e. X |-> B ) |` { w e. X | w # x } ) lim CC x ) )
47		breq1 4118	. . . . . . . . . . . . . 14 |- ( w = z -> ( w # x <-> z # x ) )
48	47	elrab 2976	. . . . . . . . . . . . 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 1252	. . . . . . . . . . . . . 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 4206	. . . . . . . . . . 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 3327	. . . . . . . . . . . 12 |- { w e. X | w # x } C_ X
55		resmpt 5092	. . . . . . . . . . . 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 2285	. . . . . . . . . 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 6074	. . . . . . . . 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 2314	. . . . . . . 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 2234	. . . . . . . . . 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 15678	. . . . . . . . 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 953	. . . . . . 7 |- ( ( ph /\ x e. X ) -> x ( S _D F ) B )
64		releldm 4998	. . . . . . 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 3261	. . . . 5 |- ( ph -> dom ( S _D F ) = X )
67	66	feq2d 5502	. . . 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 5515	. 2 |- ( ph -> ( S _D F ) Fn X )
70		fnconstg 5571	. . 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 5518	. . . . 5 |- ( ( ph /\ x e. X ) -> Fun ( S _D F ) )
74		funbrfvb 5723	. . . . 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 5904	. . . 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 2270	. 2 |- ( ( ph /\ x e. X ) -> ( ( S _D F ) ` x ) = ( ( X X. { B } ) ` x ) )
80	69, 71, 79	eqfnfvd 5784	1 |- ( ph -> ( S _D F ) = ( X X. { B } ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2205   {crab 2526   _Vcvv 2815   C_ wss 3214   ~Pcpw 3675   {csn 3695   {cpr 3696   class class class wbr 4115   |-> cmpt 4177   X. cxp 4753   dom cdm 4755   |` cres 4757   o. ccom 4759   Rel wrel 4760   Fun wfun 5352   Fn wfn 5353   -->wf 5354   ` cfv 5358  (class class class)co 6059   ^pm cpm 6897   CCcc 8142   RRcr 8143   - cmin 8462   # cap 8874   / cdiv 8967   abscabs 11712   ↾_t crest 13541   MetOpencmopn 14820   Topctop 14993   intcnt 15089   -cn->ccncf 15566   lim CC climc 15650   _D cdv 15651

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4231  ax-sep 4234  ax-nul 4242  ax-pow 4293  ax-pr 4328  ax-un 4560  ax-setind 4665  ax-iinf 4716  ax-cnex 8235  ax-resscn 8236  ax-1cn 8237  ax-1re 8238  ax-icn 8239  ax-addcl 8240  ax-addrcl 8241  ax-mulcl 8242  ax-mulrcl 8243  ax-addcom 8244  ax-mulcom 8245  ax-addass 8246  ax-mulass 8247  ax-distr 8248  ax-i2m1 8249  ax-0lt1 8250  ax-1rid 8251  ax-0id 8252  ax-rnegex 8253  ax-precex 8254  ax-cnre 8255  ax-pre-ltirr 8256  ax-pre-ltwlin 8257  ax-pre-lttrn 8258  ax-pre-apti 8259  ax-pre-ltadd 8260  ax-pre-mulgt0 8261  ax-pre-mulext 8262  ax-arch 8263  ax-caucvg 8264

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3626  df-pw 3677  df-sn 3701  df-pr 3702  df-op 3704  df-uni 3921  df-int 3956  df-iun 3999  df-br 4116  df-opab 4178  df-mpt 4179  df-tr 4215  df-id 4420  df-po 4423  df-iso 4424  df-iord 4493  df-on 4495  df-ilim 4496  df-suc 4498  df-iom 4719  df-xp 4761  df-rel 4762  df-cnv 4763  df-co 4764  df-dm 4765  df-rn 4766  df-res 4767  df-ima 4768  df-iota 5318  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-isom 5367  df-riota 6012  df-ov 6062  df-oprab 6063  df-mpo 6064  df-1st 6348  df-2nd 6349  df-recs 6550  df-frec 6636  df-map 6898  df-pm 6899  df-sup 7289  df-inf 7290  df-pnf 8327  df-mnf 8328  df-xr 8329  df-ltxr 8330  df-le 8331  df-sub 8464  df-neg 8465  df-reap 8868  df-ap 8875  df-div 8968  df-inn 9259  df-2 9317  df-3 9318  df-4 9319  df-n0 9518  df-z 9599  df-uz 9876  df-q 9974  df-rp 10009  df-xneg 10128  df-xadd 10129  df-seqfrec 10838  df-exp 10929  df-cj 11556  df-re 11557  df-im 11558  df-rsqrt 11713  df-abs 11714  df-rest 13543  df-topgen 13562  df-psmet 14822  df-xmet 14823  df-met 14824  df-bl 14825  df-mopn 14826  df-top 14994  df-topon 15007  df-bases 15039  df-ntr 15092  df-cn 15184  df-cnp 15185  df-cncf 15567  df-limced 15652  df-dvap 15653

This theorem is referenced by:  dvconstss  15694