Theorem dvidsslem 14929

Description: Lemma for dvconstss 14934. Analogue of dvidlemap 14927 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 3204	. . . . . . 7 |- ( ph -> CC C_ CC )
3		dvidsslem.j	. . . . . . . . . 10 |- J = ( K ↾t S )
4		restsspw 12920	. . . . . . . . . 10 |- ( K ↾t S ) C_ ~P S
5	3, 4	eqsstri 3215	. . . . . . . . 9 |- J C_ ~P S
6		dvidsslem.x	. . . . . . . . 9 |- ( ph -> X e. J )
7	5, 6	sselid 3181	. . . . . . . 8 |- ( ph -> X e. ~P S )
8	7	elpwid 3616	. . . . . . 7 |- ( ph -> X C_ S )
9		cnex 8003	. . . . . . . 8 |- CC e. _V
10	9	a1i 9	. . . . . . 7 |- ( ph -> CC e. _V )
11		pmss12g 6734	. . . . . . 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 1250	. . . . . 6 |- ( ph -> ( CC ^pm X ) C_ ( CC ^pm S ) )
13		dvidsslem.1	. . . . . . 7 |- ( ph -> F : X --> CC )
14		fpmg 6733	. . . . . . 7 |- ( ( X e. J /\ CC e. _V /\ F : X --> CC ) -> F e. ( CC ^pm X ) )
15	6, 10, 13, 14	syl3anc 1249	. . . . . 6 |- ( ph -> F e. ( CC ^pm X ) )
16	12, 15	sseldd 3184	. . . . 5 |- ( ph -> F e. ( CC ^pm S ) )
17		dvfgg 14924	. . . . 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 14923	. . . . . . . 8 |- ( S e. { RR , CC } -> S C_ CC )
20	1, 19	syl 14	. . . . . . 7 |- ( ph -> S C_ CC )
21	20, 13, 8	dvbss 14921	. . . . . 6 |- ( ph -> dom ( S _D F ) C_ X )
22		reldvg 14915	. . . . . . . . 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 14769	. . . . . . . . . . . . . . 15 |- K e. Top
27	26	a1i 9	. . . . . . . . . . . . . 14 |- ( ph -> K e. Top )
28		resttop 14406	. . . . . . . . . . . . . 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 2283	. . . . . . . . . . . 12 |- ( ph -> J e. Top )
31		isopn3i 14371	. . . . . . . . . . . 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 2202	. . . . . . . . . 10 |- ( ph -> X = ( ( int ` J ) ` X ) )
34	33	eleq2d 2266	. . . . . . . . 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 14902	. . . . . . . . . 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 3193	. . . . . . . . . . . . 13 |- ( ph -> X C_ CC )
40		cncfmptc 14832	. . . . . . . . . . . . 13 |- ( ( B e. CC /\ X C_ CC /\ CC C_ CC ) -> ( z e. X |-> B ) e. ( X -cn-> CC ) )
41	38, 39, 2, 40	syl3anc 1249	. . . . . . . . . . . 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 2197	. . . . . . . . . . 11 |- ( z = x -> B = B )
45	42, 43, 44	cnmptlimc 14910	. . . . . . . . . 10 |- ( ( ph /\ x e. X ) -> B e. ( ( z e. X |-> B ) lim CC x ) )
46	36, 45	sselid 3181	. . . . . . . . 9 |- ( ( ph /\ x e. X ) -> B e. ( ( ( z e. X |-> B ) |` { w e. X | w # x } ) lim CC x ) )
47		breq1 4036	. . . . . . . . . . . . . 14 |- ( w = z -> ( w # x <-> z # x ) )
48	47	elrab 2920	. . . . . . . . . . . . 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 1227	. . . . . . . . . . . . . 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 4123	. . . . . . . . . . 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 3268	. . . . . . . . . . . 12 |- { w e. X | w # x } C_ X
55		resmpt 4994	. . . . . . . . . . . 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 2247	. . . . . . . . . 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 5937	. . . . . . . . 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 2276	. . . . . . . 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 2196	. . . . . . . . . 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 14918	. . . . . . . . 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 946	. . . . . . 7 |- ( ( ph /\ x e. X ) -> x ( S _D F ) B )
64		releldm 4901	. . . . . . 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 3202	. . . . 5 |- ( ph -> dom ( S _D F ) = X )
67	66	feq2d 5395	. . . 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 5408	. 2 |- ( ph -> ( S _D F ) Fn X )
70		fnconstg 5455	. . 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 5411	. . . . 5 |- ( ( ph /\ x e. X ) -> Fun ( S _D F ) )
74		funbrfvb 5603	. . . . 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 5776	. . . 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 2232	. 2 |- ( ( ph /\ x e. X ) -> ( ( S _D F ) ` x ) = ( ( X X. { B } ) ` x ) )
80	69, 71, 79	eqfnfvd 5662	1 |- ( ph -> ( S _D F ) = ( X X. { B } ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2167   {crab 2479   _Vcvv 2763   C_ wss 3157   ~Pcpw 3605   {csn 3622   {cpr 3623   class class class wbr 4033   |-> cmpt 4094   X. cxp 4661   dom cdm 4663   |` cres 4665   o. ccom 4667   Rel wrel 4668   Fun wfun 5252   Fn wfn 5253   -->wf 5254   ` cfv 5258  (class class class)co 5922   ^pm cpm 6708   CCcc 7877   RRcr 7878   - cmin 8197   # cap 8608   / cdiv 8699   abscabs 11162   ↾_t crest 12910   MetOpencmopn 14097   Topctop 14233   intcnt 14329   -cn->ccncf 14806   lim CC climc 14890   _D cdv 14891

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998  ax-caucvg 7999

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 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-isom 5267  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-map 6709  df-pm 6710  df-sup 7050  df-inf 7051  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-n0 9250  df-z 9327  df-uz 9602  df-q 9694  df-rp 9729  df-xneg 9847  df-xadd 9848  df-seqfrec 10540  df-exp 10631  df-cj 11007  df-re 11008  df-im 11009  df-rsqrt 11163  df-abs 11164  df-rest 12912  df-topgen 12931  df-psmet 14099  df-xmet 14100  df-met 14101  df-bl 14102  df-mopn 14103  df-top 14234  df-topon 14247  df-bases 14279  df-ntr 14332  df-cn 14424  df-cnp 14425  df-cncf 14807  df-limced 14892  df-dvap 14893

This theorem is referenced by:  dvconstss  14934